Faculteit WetenschappenVakgroep Wiskunde

Topological Properties of the Spaces of

Ultradifferentiable Functions

Igor Voulis

Promotor: Prof. dr. J. Vindas Dıaz

Masterproef ingediend tot het behalen van de academische graad van master in dewiskunde, afstudeerrichting zuivere wiskunde.

Academiejaar 2013–2014

Preface

Dear reader,

This thesis will take you on a journey through the world of ultradifferentiale functions.Before we start this journey, I would like to thank my promoter Prof. Dr. Jasson VindasDıaz. He has provided me with this fascinating subject. His advise has been very helpfulat every stage, starting from the sources which he advised as a starting point and endingwith advice on how to finalize the thesis. His input on challenges that I faced while writingthis thesis was very valuable. For this I sincerely thank him. I would also like to thankmy friends and family for their support and encouragement.

Igor Voulis

De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellenen delen van de masterproef te kopieren voor persoonlijk gebruik. Elk ander gebruikvalt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking totde verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uitdeze masterproef.

Datum: 30 Mei 2014 Handtekening:

1

Contents

Introduction 4

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Topology and ultradifferentiable functions 6

1.1 General definitions from topology . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Projective limit topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Inductive limit topology and final topology . . . . . . . . . . . . . . . . . . 16

1.4 Ultradifferentiable functions . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Properties controlled by the defining sequence 32

2.1 The behavior of sequences of positive reals . . . . . . . . . . . . . . . . . . 32

2.2 The hierarchy of ultradifferentiable functions . . . . . . . . . . . . . . . . . 37

2.3 Products of ultradifferentiable functions . . . . . . . . . . . . . . . . . . . 40

2.4 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5 Quasi-analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 On the assumption (M.1) 52

3.1 Inequalities of Landau-Kolmogorov type and Chebychev polynomials . . . 52

3.2 Two counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Positive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Paley-Wiener Theorem 66

4.1 Paley-Wiener Theorem for ultradifferentiable functions . . . . . . . . . . . 66

4.2 A theorem of Paley-Wiener type for the torus . . . . . . . . . . . . . . . . 69

5 The first structure theorem 77

5.1 The first structure theorem for the torus . . . . . . . . . . . . . . . . . . . 77

5.2 The first structure theorem for Ω ⊂ Rd . . . . . . . . . . . . . . . . . . . . 79

2

Conclusion 82

A Abstract in Dutch 83

References 85

3

Introduction

In the first chapter we introduce topological concepts that will be useful throughout thefollowing chapters. In the last part of this chapter we introduce the spaces of ultradiffer-entiable functions. This chapter is based on the work of H. Komatsu [4, 5].

In the second chapter we study the properties of a sequence (Mp)p∈N which defines a spaceof ultradifferentiable functions. The properties we prove will be useful in the followingchapters and will motivate some assumptions one can make on the regularity of (Mp)p∈N.

In the third chapter we use different versions of the Landau-Kolmogorov inequality tostudy the conditions under which we can replace an arbitrary sequence (Mp)p∈N by itsgreatest logarithmically convex minorant without changing the corresponding spaces ofultradifferentiable functions. For periodic functions and functions with compact sup-port, we obtain results that were already know by S. Mandelbrojt [7, Chapter 6] in theone dimensional case and their topological consequences. For other spaces of functionsS. Mandelbrojt used other regularizations of the sequence (Mp)p∈N. We, however, willinvestigate what happens when we replace (Mp)p∈N by its greatest log-convex minorant.

In the fourth chapter we will show the Paley-Wiener Theorem for ultradifferentiable func-tions with compact support. This theorem was already well know (see [5, Section 9]). Wewill also show an analogue of the Paley-Wiener Theorem for periodic ultradifferentiablefunctions. The topological implications of this theorem will prove very useful in the lastchapter.

In the fifth chapter we give a proof of the first structure theorem for ultradistributions onfunctions with compact support and periodic functions. The theorem for functions withcompact support was already know but required the Mittag-Leffler Lemma to prove it.We will, however, not use the strength of this theorem (or any results which encompassthe strength of this theorem) to prove the first structure theorem. We develop a newmethod here.

Notations

We will use the following notations and conventions in this thesis.

• The natural numbers, including 0, are denoted by N. We also write N+ = N \ 0.

• For α, β ∈ Zd, we write |α| = |α1|+ · · ·+ |αd| and αβ = αβ11 · · ·αβdd .

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• By convention, we define the empty sum to be equal to zero and the empty productto be equal to one:

∑s∈∅ s = 0 and

∏t∈∅ t = 1. We will define 00 = 1.

• If A is a subset of Rd or a more general topological space, we denote the interior ofA by int (A) and the closure of A by A.

• If A ⊂ Rd, we write B b A if B ⊂ int (A). We write K ⊂comp

A if K is compact

(bounded and closed) and has no isolated points. This convention is useful to definespaces of differentiable functions on compact sets.

• For differentiable function ϕ : Rd → C, we denote differentiation with respect tothe k-th variable (k ∈ 1, . . . , d) by Dk. For (α1, . . . , αd) = α ∈ Nd we writeDα = Dα1

1 · · ·Dαdd .

• For an open set Ω ⊂ Rd we denote E(Ω) = ϕ : Ω → C| ϕ is smooth and D(Ω) =ϕ ∈ E(Ω)| suppϕ is compact.

• For K ⊂comp

Rd we denote E(K) = ϕ ∈ E(int (K))| ∀α ∈ Nd : Dαϕ ∈ C(K) and

D(K) = ϕ ∈ E(Rd)| suppϕ ⊂ K.

• For r > 0, we denote the torus with radius r by Tdr = [0, 2πr]d/ ∼ where x ∼ y ifand only if x− y ∈ −2πr, 0, 2πrd. We also write Td = Td1.

• The space of smooth functions on a torus is

E(Tdr) = ϕ ∈ E([0, 2πr]d)| ∀α ∈ Nd, ∀x ∼ y ∈ [0, 2πr]d : Dαϕ(x) = Dαϕ(y),

it can be identified with ϕ ∈ E(Rd)| ∀β ∈ Zd,∀x ∈ Rd : ϕ(x) = ϕ(x+ 2πrβ).

• The Fourier-Laplace transform of a function ϕ : Rd → C is will be denoted by

F(ϕ) : Cd → C : ζ 7→∫Rdϕ(x) exp(−ix · ζ)dx.

Its inverse is, for ϕ : Cd → C,

F−1(ϕ) : Rd → C : x 7→∫Rdϕ(x) exp(iy · x)dy.

• The Lebesgue measure of a set A ⊂ Rd is denoted by |A|.

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Chapter 1

Topology and ultradifferentiablefunctions

1.1 General definitions from topology

In this chapter we will recall some definitions from topology. We will also prove someproperties that will be relevant in what follows. We assume that the reader is familiar withbasic concepts like convex sets, neighborhoods, (relatively) compact sets, completeness,... See e.g. [13, 6].

We will work with topological vector spaces, i.e. spaces X such that X is a vector space(over C) and X has a topology which is invariant under linear transformations. A topologyon a space X can be either given by defining the set of open subsets of X explicitly, orby defining the neighborhoods of zero in X. Note that a neighborhood of a point doesnot need to be open, it rather contains an open set that contains this point. We recallthe following definitions.

Definition 1.1.1. Let X be a topological vector space and let T be the set open subsetsof X. We call B ⊂ T a basis of X if⋃

U∈b

U : b ⊂ B

= T .

We call S ⊂ T a sub-basis of X if⋂U∈s

U : s ⊂ S, s is finite

is a basis of X.

Definition 1.1.2. Let X be a topological space and let F be the set of neighborhoods ofzero in X. We call B ⊂ F a basis of neighborhoods of zero in X if

V ⊂ X| ∃U ∈ b : U ⊂ V = F .

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We call S ⊂ F a sub-basis of neighborhoods of zero in X if⋂U∈s

U : s ⊂ S, s is finite

is a basis of neighborhoods of zero in X.A space X is called locally convex if it has a (sub-)basis of neighborhoods of zero thatcontains only convex sets.

Remark 1.1.3. Let f : X → Y be any map between topological spaces. We have forany set S of subsets of X

f

(⋃s∈S

s

)=⋃s∈S

f(s) and f

(⋂s∈S

s

)=⋂s∈S

f(s)

and for any set T of subsets of Y we have

f−1

(⋃t∈T

t

)=⋃t∈T

f−1(t) and f−1

(⋂t∈T

t

)=⋂t∈T

f−1(t).

So, in particular, if we want to check whether a map is open (or continuous), it sufficesto work with a sub-basis or a sub-basis of neighborhoods of zero.

Definition 1.1.4. A linear map f : X → Y between locally convex topological vectorspaces X and Y is called compact if for every bounded set B ⊂ X the set f(B) ⊂ Y isrelatively compact (i.e. f(B) ⊂ Y is compact).

Lemma 1.1.5. For any X, Y, U and V , locally convex Hausdorff topological vector spacesand f : X → U , g : Y → V , compact maps, we have that

f × g : X × Y → U × V

is compact.

Proof. Any bounded set A ⊂ X × Y is contained in a set of the form B × C where Bis a bounded subset of X and C is a bounded subset of Y (we can take B and C to bethe projections of A). It follows that (f × g)(B × C) = f(B) × g(C) is the product oftwo relatively compact sets and thus relatively compact. It follows that f(A) is a closedsubset of the compact set f(B)× g(C). This implies that f(A) is compact.

Lemma 1.1.6. If Y and X are compact, Hausdorff topological spaces and f : X → Y isa continuous bijection, then f is a homeomorphism between X and Y .

Proof. To prove the statement, we need to prove that f−1 is continuous. This is equivalentto proving that for every closed U ⊂ X the set f(U) ⊂ Y is closed. Take any closedU ⊂ X, since X is compact and Hausdorff we know that U is compact. By the continuityof f , we conclude that f(U) is compact. Since Y is Hausdorff, we conclude that thecompact set f(U) is closed. The set U ⊂ X was an arbitrary closed set, so we find thatf is a homeomorphism.

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Definition 1.1.7. A topological vector space X is a called a Frechet space if it is complete,metrizable and locally convex.

Remark 1.1.8. Alternatively one can define a Frechet space as a complete space forwhich there is a countable sequence of semi-norms p0, p1, . . . (which one can assume tobe increasing) and the topology is generated by the metric

d : (x, y) 7→∞∑n=0

2−npn(x− y)

1 + pn(x− y).

See [6, p. 205].

Definition 1.1.9. If X is a topological vector space, then we call B ⊂ X a barrel if

(a) B is closed,

(b) B is balanced (i.e. B = zB = zx : x ∈ B for all z ∈ C, |z| ≤ 1),

(c) B is absorbing (i.e. (∀x ∈ X)(∃c ∈ R+)(x ∈ cB)),

(d) B is convex (i.e. (∀λ ∈ [0, 1])(∀x, y ∈ B)(λx+ (1− λ)y ∈ B)).

Definition 1.1.10. If X is a Hausdorff topological vector space, we call X barreled ifevery barrel B ⊂ X is a neighborhood of 0.

Remark 1.1.11. It follows from Baire’s Theorem that every Banach space and everyFrechet space is barreled. See [13, p. 347].

Definition 1.1.12. A locally convex topological vector space X is called a semi-Montelspace if every bounded, closed set K ⊂ X is compact. If on top of that X barreled, we callX a Montel space.

Notation 1.1.13. For a topological vector space X we denote the dual of X by X ′ andwe denote the dual of X endowed with the strong topology by X ′b. For x′ ∈ X ′ we writefor all x ∈ X

〈x′, x〉 = x′(x).

Definition 1.1.14. We define for a linear map between topological vector spaces f : X →Y the transpose of f as

f t : Y ′ → X ′ : y′ 7→ y′ f.

Definition 1.1.15. We call a topological vector space semi-reflexive if the canonical in-jection X → (X ′b)

′

x 7→ (X ′ 3 x′ 7→ 〈x′, x〉)

is surjective. If on top of that this is an isomorphism between X and (X ′b)′b, then we call

X reflexive.

Remark 1.1.16. A Montel space is always reflexive and a semi-Montel space is alwayssemi-reflexive. See [13, p. 373-376].

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1.2 Projective limit topology

Definition 1.2.1 (Projective limit). Let (P,<) be a (strict) partial order and let (Xα)α∈Pbe a collection of locally convex topological vector spaces. Let for each α, β ∈ P with α > βuαβ : Xα → Xβ be a continuous linear map such that

(∀α, β, γ ∈ P )(α > β > γ ⇒ uβγ uαβ = uαγ ).

We call

lim←−α∈P

Xα =

(xα)α∈P ∈

∏α∈P

Xα : (∀α > β ∈ P )(uαβ(xα) = xβ)

the projective limit of (Xα)α∈P defined by the maps uαβ . The projective limit topology isthe weakest locally convex topology such that all projections

uα : lim←−β∈P

Xβ → Xα : (xβ)β∈P 7→ xα

are continuous.

Notation 1.2.2. In this section we will use the notation from the previous definitionimplicitly (unless we say otherwise). We will assume P , (Xα)α∈P , uαβ and uα to be as inthe previous definition.

Remark 1.2.3. Explicitly we have the following sub-basis of lim←−α∈P Xα:

u−1α (U)| α ∈ P,U ⊂ Xα is open.

Indeed, this defines the weakest topology for which uα is continuous for all α ∈ P . Weneed to verify that it is locally convex. For any neighborhood of zero U in Xα (for someα ∈ P ) we have a convex neighborhood of zero U ′ ⊂ U . By the linearity of uα we getthat u−1

α (U ′) ⊂ u−1α (U) is a convex neighborhood of 0 in X = lim←−α∈P Xα.

In particular, we find that X is Hausdorff when Xα is Hausdorff for all α ∈ P .

Example 1.2.4. Let P be a trivial partial order, i.e. ∀α, β ∈ P : α 6< β. We have

lim←−α∈P

Xα∼=∏α∈P

Xα.

This follows immediately from the definitions of both spaces: both spaces are defined tocarry the weakest topology such that the projections

uβ :∏α∈P

Xα → Xβ

for β ∈ P are continuous. Explicitly we have the following sub-basis in this topologyU × ∏α∈P\β

Xα| β ∈ P,U ⊂ Xβ is open

.

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Before we study various properties of projective limit topologies, we will make someremarks about partial orders.

Remark 1.2.5. Sometimes it is convenient to replace (P,<) by (P,≺) where α > β ifand only if α ≺ β. If P ⊂ P ′ and supP = A ∈ P ′ we will sometimes write lim←−α→A insteadof lim←−α∈P .

Definition 1.2.6. We call a (strict) partial order P a (strictly) directed set if for everyα, β ∈ P there is a γ ∈ P such that α ≤ γ and β ≤ γ.

Definition 1.2.7. In a partial order P we call a subset D ⊂ P dense1 in P , if

(∀α ∈ P )(∃β ∈ D)(β ≤ α)

and reverse-dense if(∀α ∈ P )(∃β ∈ D)(β ≥ α).

Example 1.2.8. Assume that P is a directed set. Assume for all α > β ∈ P thatXα ⊂ Xβ and that uαβ is the trivial injection. In this case we have a canonical bijectionbetween X = lim←−α∈P Xα and

⋂α∈P Xα. We have, by definition

X =

(xα)α∈P ∈

∏α∈P

Xα : (∀α > β ∈ P )(xα = xβ)

.

Using the fact that for ever α, β ∈ P there is a γ ∈ P such that α ≤ γ and β ≤ γ we findthat

X =

(xα)α∈P ∈

∏α∈P

Xα : (∀α, β ∈ P )(xα = xβ)

=

(x)α∈P ∈

∏α∈P

Xα : (∀α ∈ P )(x ∈ Xα)

.

We conclude that (x)α∈P 7→ x is a canonical bijection between X and⋂α∈P Xα. We

identify these two set; this is often a natural and very useful identification. By Remark1.2.3, we conclude that

U ∩⋂α∈P

Xα| ∃α ∈ P : U ⊂ Xα is open

is a sub-basis of X.

Theorem 1.2.9 (Universal property of the projective limit topology). Let X = lim←−α∈P Xα

and let Y be topological space. A map f : Y → X is continuous if and only if uα f :Y → Xα is continuous for all α ∈ P .

1One can define a topology on P in which these subsets are the dense subsets. See [2, p.202].

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Proof. If f : Y → X is continuous, then uα f is continuous as the composition of twocontinuous maps. Conversely, assume that uα f is continuous for all α ∈ P and U ⊂ Xis open. It suffices to prove that f−1(U) is open in Y for all U in some sub-basis of X (seeRemark 1.1.3). We take U arbitrary in the sub-basis from Remark 1.2.3. We have thatU = u−1

β (V ) for some β ∈ P and V ⊂ Xβ, open. It follows by the continuity of uβ f that

(uβ f)−1(V ) = f−1(u−1β (V )) = f−1(U) is open, which proves the continuity of f .

Corollary 1.2.10. Let X = lim←−α∈P Xα, Y = lim←−α∈P Yα be projective limit topologies onX and Y with respect to the maps uαβ : Xα → Xβ, vαβ : Yα → Yβ for α > β ∈ P . Let

fα : Xα → Yα be continuous maps such that ∀α < β ∈ P : vβα fβ = fα uβα. It followsthat ∏

α∈P

fα :∏α∈P

Xα →∏α∈P

Yα : (xα)α∈P 7→ (fα(xα))α∈P

induces a continuous map f : X → Y .

Proof. By the property which we imposed on fα, for α ∈ P , we know that for anyα > β ∈ P , xβ ∈ Xβ, xα ∈ Xα we have

fβ(xβ − uαβ(xα)) = fβ(xβ)− vαβ (fα(xα)).

So x = (xα)α ∈ X implies that the left hand side of the previous equation is zero, makingthe right hand side zero, which implies f(x) ∈ Y . So

∏fα induces a map f : X → Y . It

remains to prove that f is continuous. Observe that for any α ∈ P

vα f = fα uα : X → Yα.

This implies that vα f is continuous for any α ∈ P ; we conclude by the previous theoremthat f is continuous.

Proposition 1.2.11. If for all α ∈ P we have that Yα ⊂ Xα is a closed subspace, thenthe topology of Y = lim←−α∈P Yα coincides with the topology on Y ⊂ X = lim←−α∈P Xα inducedby X and Y ⊂ X is closed.

Proof. By the universal property we know that the injection ι : Y → X is continuous:indeed vα := uα ι = uα|Y is continuous by the definition of the projective limit topologyon Y . We now prove that ι is an open map. It suffices to prove for all V in a fixedsub-basis of Y that ι(V ) is open in the relative topology on Y induced by X (see Remark1.1.3). Let V ⊂ Y be any element of the sub-basis of Y which we described in Remark1.2.3. It follows that for some α ∈ P and U ⊂ Yα, open, V = v−1

α (U). Since Yα ⊂ Xα

is closed, it follows that Yα \ U is closed in the topology of Xα. So it follows that thatY \ V = v−1

α (Yα \ U) ⊂ X is closed. We have proven that the canonical injection Y → Xis bicontinuous. If we repeat this argument with V = ∅, we find that Y ⊂ X is closed.

Proposition 1.2.12. Let X = lim←−α∈P Xα. If D is reverse-dense in P then we have acanonical identification

lim←−α∈P

Xα∼= lim←−

α∈DXα.

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Proof. For any α ∈ P we have a α ≤ βα ∈ D such that

lim←−γ∈D

Xγ

uβα−→ Xβαuβαα−→ Xα

are continuous maps. By taking the projective limit over all α ∈ P and applying theuniversal property, we obtain that the injection∏

α∈P

uβαα uβα : lim←−γ∈D

Xγ −→ lim←−α∈P

Xα

is continuous and it is independent of the choice of βα. Conversely, we know that for anyγ ∈ D the map

lim←−α∈P

Xαuγ−→ Xγ

is continuous. By taking the projective limit over all γ ∈ D and applying the universalproperty, we obtain that the canonical projection

lim←−α∈P

Xα −→ lim←−γ∈D

Xγ

is continuous. It is clear that the two maps are each others inverses, so the proposition isproven.

Remark 1.2.13. Using the previous proposition we will often be able to replace a givenpartial order P by N, but this not possible in general (in general one can use Zorn’sLemma to replace a directed set P by an ordinal). In such a case we can write

X −→ · · · −→ Xn −→ · · · −→ X3

u32−→ X2

u21−→ X1

u10−→ X0.

The other maps umn (m > n ∈ N) are then determined by composing the given maps.

Proposition 1.2.14. Let P be a directed set and let X = lim←−α∈P Xα (defined by the maps

uαβ : Xα → Xβ for α > β ∈ P ) and Y = lim←−α∈P Yα (defined by the maps vαβ : Yα → Yβ for

α > β ∈ P ). Then we haveX × Y ∼= lim←−

α∈PXα × Yα.

Proof. The identification between the sets is trivial. The topology on X×Y is the weakestlocally convex topology such that for all α ∈ P the maps

uα : X → Xα and vα : Y → Yα

are continuous. By the definition of the product topology we know that this is equivalentwith the weakest locally convex topology such that for all α, β ∈ P

uα × vβ : X × Y → Xα × Yβ

is continuous. This implies that X ×Y ∼= lim←−(α,β)∈P×P Xα×Yβ where the order on P ×Pis defined as

(α, β) < (α′, β′) ∈ P ⇐⇒ α < α′ and β < β′.

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Since (α, α)| α ∈ P is reverse-dense in P × P , we conclude by Proposition 1.2.12 that

X × Y ∼= lim←−α∈P

Xα × Yα.

Proposition 1.2.15. If Xα is complete for all α ∈ P , then X = lim←−α∈P Xα is complete.

Proof. Let C be a Cauchy filter on X. It follows that for any α ∈ P the collection uα(c) :c ∈ C is a basis of a Cauchy filter uα(C) on Xα. Since Xα is complete, we concludethat, for α ∈ P arbitrary, uα(C)→ xα for some xα ∈ Xα. By the continuity of uαβ for allα > β ∈ P , we know that (xα)α ∈ X. We get from Remark 1.2.3 that any neighborhoodof (xα)α contains a neighborhood of the form (uα1)

−1(Uα1) ∩ · · · ∩ (uαk)−1(Uαk) for some

k ∈ N, α1, . . . , αk ∈ P and Uα1 , . . . , Uαk neighborhoods of xα1 , . . . , xαk , respectively. Itfollows that C → (xα)α in topology of X because the intersection of a finite number ofelements of C is again in C.

Proposition 1.2.16. Let Q and P be partial orders, and let P × Q be the partial orderdefined by

(α, β) < (α′, β′) ∈ P ×Q ⇐⇒ α < α′ or (α = α′ and β < β′).

Let X = lim←−(α,β)∈P×QXα,β, defined by the maps

uα,βα′,β′ : Xα,β → Xα′,β′

for (α, β) > (α′, β′) ∈ P ×Q. We have that X ∼= lim←−α∈P lim←−β∈QXα,β, defined by the maps

uα,βα,β′ : Xα,β → Xα,β′ and uαα′ =∏β∈Q

uα,βα′,β : lim←−β∈P

Xα,β → lim←−β∈P

Xα′,β

for α > α′ ∈ P, β > β′ ∈ Q.

Proof. It follows from the definition that they coincide as sets. We use Remark 1.2.3 toverify that the topologies coincide. For all α ∈ P we define Xα = lim←−β∈QXα,β. We have

the following sub-basis of lim←−α∈P Xα:

u−1α (U)| α ∈ P,U ⊂ Xα is open

where uα : X → lim←−β∈QXα,β is the canonical projection (for any α ∈ P ). For any α ∈ Pwe have the following sub-basis of Xα = lim←−β∈QXα,β:

u−1α,β(V )| β ∈ Q, V ⊂ Xα,β is open

where uα,β : lim←−β∈QXα,β → Xα,β is the canonical projection (for any α, β). Combining

these two we find the following sub-basis of lim←−α∈P Xα:

u−1α (u−1

α,β(V ))| α ∈ P, β ∈ Q : V ⊂ Xα,β is open.

13

Defininguα,β = uα,β uα : X → Xα,β

to be the canonical projections. We get the following sub-basis of lim←−α∈P Xα:

u−1α,β(V )| α ∈ P, β ∈ Q, V ⊂ Xα,β is open.

This is, by Remark 1.2.3, a sub-basis of X = lim←−(α,β)∈P×QXα,β.

Corollary 1.2.17. Let P be a directed set. We have

lim←−α∈P

lim←−β∈P

Xα,β∼= lim←−

α∈PXα,α.

(The defining maps are as in the previous proposition.)

Proof. Let P ×P be as in the previous proposition. We have, by the previous proposition

lim←−α∈P

lim←−β∈P

Xα,β∼= lim←−

(α,β)∈P×PXα,α.

Since P is a directed set, we have that (α, α)|α ∈ P is reverse-dense in P ×P , thus thecorollary follows from Proposition 1.2.12.

Proposition 1.2.18. A space X is a Frechet space if and only if there are Banach spaces(Xn)n∈N such that

X ∼= lim←−n∈N

Xn.

Proof. Assume that Xn are Banach spaces for n ∈ N, with norms x 7→ ‖x‖n respectively.Let X = lim←−n∈NXn. The canonical projections un : X → Xn are continuous for all n ∈ N,

so pn : x 7→ ‖un(x)‖n (for n ∈ N) define countably many continuous semi-norms on X.Moreover, by Remark 1.2.3 we have the following sub-basis of neighborhoods of zero inX:

U ⊂ X| ∃n ∈ N,∃rn > 0 : x : rn ≥ pn(x) ⊂ U.

This expresses that the topology on X is defined by countable family of semi-normsp0, p1, . . . . Combining this with Proposition 1.2.15 we find that X is a Frechet space.Conversely, let X be a Frechet space defined by the semi-norms pn for n ∈ N. Withoutloss of generality we assume that pn ≤ pn+1 for all n ∈ N (by replacing pn by

∑nm=0 pm).

Define for all n ∈ N the normed spaces

Xn = X/x ∈ X : pn(x) = 0

with norm [x]n 7→ pn(x), where [x]n denotes the equivalence class of x ∈ X. Sincewe assume the semi-norms p0, p1, p2, . . . to be increasing, we conclude that the injectionXn+1 → Xn is continuous for any n ∈ N. We have by construction that X = lim←−n∈NXn.

Let Xn be the completion of Xn. By construction, we know that the canonical maps

Xn → Xn

14

are continuous for all n ∈ N. So by Corollary 1.2.10 we know that lim←−n∈NXn → lim←−n∈N Xn

is continuous. Moreover, using that Xn ⊂ Xn is dense, we can conclude that

X ∼= lim←−n∈N

Xn → lim←−n∈N

Xn

has a dense image. Since both spaces are complete, we find that X ∼= lim←−n∈N Xn.

Definition 1.2.19. A Frechet space X = lim←−n→∞Xn, where (Xn)n∈N are Banach spaces,

is called a Frechet-Schwartz space or an (FS)-space if all defining maps un+1n for n ∈ N

are compact.

Theorem 1.2.20. Let X = lim←−α∈P Xα be a projective limit defined by the maps uαβ : Xα →Xβ, for all α > β ∈ P . If for each β ∈ P we have an α > β such that uαβ : Xα → Xβ

maps bounded sets to relatively compact sets, then X is semi-Montel.

Proof. Let B ⊂ X be bounded, then ∀β ∈ P the set uβ(B) ⊂ Xβ is bounded. For anyβ ∈ P we have a β < α such that uβ(B) = uαβ(uα(B)) is relatively compact. So byTychonoff’s Theorem (see [6, p. 18]) we find that B =

∏β∈P uβ(B) is relatively compact

in∏

α∈P Xα. It follows from Example 1.2.4 and Remark 1.2.3 that the canonical injectionlim←−α∈P Xα →

∏α∈P Xα is an isomorphism into its image, so we conclude that B =∏

uβ(B) is relatively compact in lim←−α∈P Xα.

Corollary 1.2.21. The projective limit of semi-Montel spaces is semi-Montel.

Proof. Indeed, a sequence of semi-Montel spaces trivially satisfies the condition fromthe previous theorem, because every bounded set in a semi-Montel space is relativelycompact.

Corollary 1.2.22. An (FS)-space X = lim←−n→∞Xn is Montel.

Proof. Indeed, by the previous theorem X is semi-Montel. Since X is a Frechet space, itis barreled. So X is Montel.

Lemma 1.2.23. Let X =∏

α∈I Xα for some index set I and some (locally convex)topological vector spaces Xα. For any f ∈ X ′, we have a finite set J ⊂ I and fα ∈ X ′α forα ∈ J such that

f =∑α∈J

fα uα

where uα denote the usual projections. The converse is trivial.

Proof. Take f ∈ X ′ as in the statement. We have that U = f−1(z : |z| < 1) is aneighborhood of zero in X. From Example 1.2.4 it is clear that∏

α∈J

Uα ×∏α∈I\J

Xα ⊂ U

for some finite J ⊂ I and some neighborhoods of zero Uα ⊂ Xα for all α ∈ J . Thefunctional f is bounded on U , so it follows that for x ∈

∏α∈J0 ×

∏α∈I\J Xα we have

15

f(x) = 0. We define for α ∈ J the subspace Xα = Xα ×∏

β 6=α0 ⊂ X and fα = f |Xαand we obtain that f =

∑α∈J fα. For any α ∈ J we have

f−1α (z : |z| < 1) = f−1(z : |z| < 1) ∩ Xα = U ∩ Xα,

this proves that fα ∈ X ′α. The statement we wanted to prove follows by observing thatuα : Xα → Xα is an isomorphism.

Proposition 1.2.24. Assume that X = lim←−α∈P Xα is complete and that Xα is Hausdorff

for all α ∈ P . For any f ∈ X ′, there is a finite Q ⊂ P and fα ∈ X ′α for α ∈ Q such that

f =∑α∈Q

fα uα.

Moreover, if P is a directed set, then we can take α ∈ P and fα ∈ X ′α such that f = fαuα.The converse is trivial.

Proof. Let X = lim←−α∈P Xα and f ∈ X ′ be as in the statement and define Y =∏

α∈P Xα.By Remark 1.2.3 and Example 1.2.4 it is clear that the canonical injection X → Y is anisomorphism into its image. The space X is complete, so X ⊂ Y is a closed subset. Bythe Hahn-Banach Theorem we can extend f to an element f of Y ′. Using the previouslemma, we can take a finite Q ⊂ P and fα ∈ X ′α for α ∈ Q such that

f =∑α∈Q

fα uα.

We obtain the first part of the proposition by restricting this equality to X.Assuming that P is a directed set and using the fact that Q is finite, we can take β ∈ Psuch that ∀α ∈ Q : α ≤ β. We see that

f =∑α∈Q

fα uα =

(∑α∈Q

fα uβα

) uβ.

So the second part of the proposition now follows by observing that∑

α∈Q fαuβα ∈ X ′β.

1.3 Inductive limit topology and final topology

Definition 1.3.1 (Inductive limit). Let (P,<) be a (strict) partial order and let (Xα)α∈Pbe a collection of locally convex topological vector spaces. Let for each α, β ∈ P with α < βuαβ : Xα → Xβ be a continuous linear map such that

(∀α, β, γ ∈ P )(α < β < γ ⇒ uβγ uαβ = uαγ ).

Let ⊕α∈P

Xα ⊃ H0 = 〈xβ − uαβ(xα)| α < β ∈ P, xβ ∈ Xβ, xα ∈ Xα〉.

16

We call

lim−→α∈P

Xα =

(⊕α∈P

Xα

)/H0

the inductive limit of (Xα)α∈P defined by the maps uαβ : Xα → Xβ. The inductive limittopology is the strongest locally convex topology such that for all α ∈ P the maps

uα : Xα → lim−→α∈P

Xα : xα 7→ [xα]

are continuous, where [xα] denotes the equivalence class of xα in lim−→α∈P Xα.

Notation 1.3.2. In this section we will use the notation from the previous definitionimplicitly (unless we say otherwise). We will assume P , (Xα)α∈P , uαβ and uα to be as inthe previous definition.

Remark 1.3.3. Explicitly, we can give the following basis of neighborhoods of 0 inlim−→α∈P Xα

U ⊂ X : U is convex, ∀α ∈ P : (uα)−1(U) is a neighborhood of 0 in Xα

.

Indeed, the first condition expresses that the topology is locally convex and the secondone expresses that it is the strongest topology such that for all α ∈ P the map uα iscontinuous, so the given set is a sub-basis of neighborhoods of 0. It follows from Remark1.1.3 that this is a basis.

Definition 1.3.4. Let

X =

(⊕α∈P

Xα

)/H0

be as in the previous definition (as a set). We define the final topology on X as thestrongest topology for which the maps uα (for all α ∈ P ) are continuous, without therequirement that this topology is locally convex. In general this topology does not coincidewith the topology from the previous definition. We denote X endowed with this topologyby lim−→α∈P Xα.

Remark 1.3.5. Explicitly the open sets in lim−→α∈P Xα are

U ⊂ X| ∀α ∈ P : (uα)−1(U) ⊂ Xα is open.

It is indeed clear that these sets are a sub-basis in lim−→α∈P Xα and from Remark 1.1.3 wecan conclude that this set is the entire set of open sets.

Example 1.3.6. Assume that P is a directed set. Assume for all α < β ∈ P thatXα ⊂ Xβ and that uαβ is the trivial injection. In this case we have a canonical bijection

between X = lim−→α∈P Xα (or lim−→α∈P Xα) and⋃α∈P Xα. Indeed, we get that

X =

(⊕α∈P

Xα

)/H0

= [xα1 + · · ·+ xαn ] : n ∈ N,∀j : αj ∈ P, xαj ∈ Xαj.

17

For any n ∈ N, α1, . . . , αn ∈ P we have some α ∈ P such that ∀j : α ≥ αj. We get that

X = [xα] : α ∈ P, xα ∈ Xα,

thus the canonical bijection is [xα] 7→ xα. We identify these two set; it is often very usefulto make this identification. Using the previous remark, we conclude that

U ⊂⋃α∈P

Xα| ∀α ∈ P : U ∩Xα ⊂ Xα is open

is the set of open subsets of lim−→α∈P Xα.Under the same identification we have, by Remark 1.3.3, the following basis of neighbor-hoods of 0 in lim−→α∈P Xα

U ⊂ X : U is convex, ∀α ∈ P : U ∩Xα ⊂ Xα is a neighborhood of 0 .

Theorem 1.3.7 (Universal property of the inductive limit topology). Let X = lim−→α∈P Xα

and let Y be a locally convex topological space. A linear map f : X → Y is continuous ifand only if f uα : Xα → Y is continuous for all α ∈ P .

Proof. If f : X → Y is continuous, then f uα is continuous as the composition oftwo continuous maps. Conversely, if f uα is linear and continuous for all α ∈ P andU ⊂ Y is a convex neighborhood of 0, then (f uα)−1(U) = (uα)−1(f−1(U)) is a convexneighborhood of 0 for all α ∈ P . Since U is convex, f−1(U) is also convex. Combiningthese two observations with Remark 1.3.3 we conclude that f−1(U) ⊂ X is a neighborhoodof zero.

Corollary 1.3.8. Let X = lim−→α∈P Xα, Y = lim−→α∈P Yα be inductive limit topologies onX and Y with respect to the maps uαβ : Xα → Xβ, vαβ : Yα → Yβ for α < β ∈ P . Let

fα : Xα → Yα be continuous linear maps, such that ∀β < α ∈ P : vβα fβ = fα uβα. Itfollows that ⊕

α∈P

fα :⊕α∈P

Xα →⊕α∈P

Yα :∑α∈P

xα 7→∑α∈P

fα(xα)

induces a continuous map f : X → Y .

Proof. By the property which we imposed on fα, for α ∈ P , we know that for anyα < β ∈ P , xβ ∈ Xβ, xα ∈ Xα we have

fβ(xβ − uαβ(xα)) = fβ(xβ)− vαβ (fα(xα)).

So⊕

α fα induces a map f : X → Y . It remains to prove that f is continuous. Observethat for any α ∈ P

f uα = vα fα : Xα → Y.

This implies that f uα is continuous for any α ∈ P ; we conclude by the previous theoremthat f is continuous.

18

Proposition 1.3.9. Let X = lim−→α∈P Xα. If D is reverse-dense in P then we have acanonical identification

lim−→α∈P

Xα∼= lim−→

α∈DXα.

Proof. For any α ∈ P we have a α ≤ βα ∈ D such that

Xα

uαβα−→ Xβαuβα−→ lim−→

γ∈DXγ

are continuous maps. By taking the inductive limit over all α ∈ P and applying theuniversal property, we obtain that the injection⊕

α∈P

uβα uαβα : lim−→α∈P

Xα −→ lim−→γ∈D

Xγ

is continuous and it is independent of the choice of βα. Conversely, we know that for anyγ ∈ D the map

Xγuγ−→ lim−→

α∈PXα

is continuous. By taking the inductive limit over all γ ∈ D and applying the universalproperty, we obtain that the canonical injection

lim−→γ∈D

Xγ −→ lim←−α∈P

Xα

is continuous. It is clear that the two maps are each others inverses, so the proposition isproven.

Remark 1.3.10. Using the previous proposition we will often be able to replace a givenpartial order P by N, but this not possible in general. In such a case we can write

X0

u01−→ X1

u12−→ X2

u23−→ X3 −→ · · · −→ Xn −→ · · · −→ X.

The other maps umn (m < n ∈ N) are then determined by composing the given maps.

Proposition 1.3.11. Assume that P is a directed set. Let X = lim−→α∈P Xα (defined by

the maps uαβ : Xα → Xβ for α < β ∈ P ) and Y = lim−→α∈P Yα (defined by the maps

vαβ : Yα → Yβ for α < β ∈ P ). We have

X × Y ∼= lim−→α∈P

Xα × Yα.

Proof. The identification between the sets is trivial. The topology on X × Y is thestrongest locally convex topology such that for all α ∈ P the maps

uα : Xα → X and vα : Yα → Y

19

are continuous. By the definition of the product topology we know that this is equivalentwith the strongest locally convex topology such that for all α, β ∈ P

uα × vβ : Xα × Yβ → X × Y

is continuous. This implies that X ×Y ∼= lim−→(α,β)∈P×P Xα×Yβ where the order on P ×Pis defined as

(α, β) < (α′, β′) ∈ P ⇐⇒ α < α′ and β < β′.

Since (α, α)| α ∈ P is reverse-dense in P × P , we conclude by Proposition 1.3.9 that

X × Y ∼= lim−→α∈P

Xα × Yα.

Proposition 1.3.12. Let Q and P be partial orders, and let P × Q be the partial orderdefined by

(α, β) < (α′, β′) ∈ P ×Q ⇐⇒ α < α′ or (α = α′ and β < β′).

Let X = lim−→(α,β)∈P×QXα,β, defined by the maps

uα,βα′,β′ : Xα,β → Xα′,β′

for (α, β) < (α′, β′) ∈ P ×Q. We have that X ∼= lim−→α∈P lim−→β∈QXα,β, defined by the maps

uα,βα,β′ : Xα,β → Xα,β′ and uαα′ =⊕β∈Q

uα,βα′,β : lim−→β

Xα,β → lim−→β

Xα′,β

for α < α′ ∈ P, β < β′ ∈ Q.

Proof. It follows from the definition that they coincide as sets. We use Remark 1.3.3 toverify that the topologies coincide. We define Xα = lim−→β∈QXα,β. We have the following

basis of neighborhoods of 0 in lim−→α∈P Xα:U ⊂ X : U is convex, ∀α ∈ P : (uα)−1(U) is a neighborhood of 0

where uα : lim−→β∈QXα,β → X is the canonical injection. We have, for any α ∈ P , the

following basis of neighborhoods of 0 in lim−→β∈QXα,β:V ⊂ lim−→

β∈QXα,β : V is convex, ∀β ∈ Q : (uα,β)−1(V ) is a neighborhood of 0

where uα,β : Xα,β → lim−→β∈QXα,β is the canonical injection. Combining these two we find

the following basis of neighborhoods of 0 in lim−→α∈P Xα:U ⊂ X :

U is convex, ∀α ∈ P :((uα)−1(U) is convex and

∀β ∈ Q : (uα uα,β)−1(U) is a neighborhood of 0) .

20

Note that for any convex U ⊂ X the set (uα)−1(U) is convex (for all α ∈ P ) because uα

is linear. We defineuα,β = uα uα,β : Xα,β → X

to be the canonical injections. We find the following basis of neighborhoods of 0 inlim−→α∈P Xα:

U ⊂ X : U is convex, ∀α ∈ P, ∀β ∈ Q : (uα,β)−1(U) is a neighborhood of 0.

This is, by Remark 1.3.3, a basis of neighborhoods of 0 in X = lim−→(α,β)∈P×QXα,β.

Corollary 1.3.13. Let P be a directed set. We have

lim−→α∈P

lim−→β∈P

Xα,β∼= lim−→

α∈PXα,α.

(The defining maps are as in the previous proposition.)

Proof. Let P ×P be as in the previous proposition. We have by the previous proposition

lim−→α∈P

lim−→β∈P

Xα,β∼= lim−→

(α,β)∈P×PXα,β.

Since P is a directed set, we have that (α, α)|α ∈ P is reverse-dense in P ×P , thus thecorollary follows from Proposition 1.3.9.

Lemma 1.3.14. If Xα are barreled spaces for all α ∈ P and X = lim−→α∈P Xα is Hausdorff,then X is barreled.

Proof. Let B ⊂ X be a barrel. For any α ∈ P we have by the linearity and continuity ofuα that (uα)−1(B) is a barrel in Xα. Indeed, continuity implies that (uα)−1(B) is closedand the other three properties follow from linearity. We find that (uα)−1(B) ⊂ Xα is aneighborhood of 0 in the barreled space Xα, for any α ∈ P . Combining this with the factthat every barrel is convex, we find, by Remark 1.3.3, that B is a neighborhood of 0 inX.

Theorem 1.3.15. Let X = lim−→n∈NXn where the defining maps unn+1 are compact linearinjections and Xn are Banach spaces for all n ∈ N. Firstly, for every bounded set B ⊂ Xthere is an n ∈ N and a relatively compact Bn ⊂ Xn such that un : Bn → B is ahomeomorphism. Secondly, X is a Montel space. Thirdly, the topology on X coincideswith the final topology, i.e.

lim−→n∈N

Xn∼= lim−→

n∈NXn.

Proof. Let X = lim−→n∈NXn be as in the statement of the theorem. For any n ∈ N we will

denote the open ball around x ∈ Xn with radius r ∈ R+ by Bn,r(x) and the norm of x by‖x‖n.Let B ⊂ X be bounded. We will prove by contradiction that there is an n ∈ N anda bounded Bn ⊂ Xn such that B = un(Bn). So we assume that for all n ∈ N and allbounded Vn ⊂ Xn there is an xn ∈ B such that xn /∈ un(Vn). In other words B \ un(Vn)

21

is not empty for all n ∈ N, and all bounded Vn ⊂ Xn. We define U0 = B0,1(0). We define,by recursion, for n ∈ N+

rn = min‖x‖n/n| un(x) = xk for some k < nUn = un−1

n (Un−1) +Bn,rn(0)

xn ∈ B \ nun(Un).

We now takeV =

⋃n∈N

un(Un).

We have, by definition, that V is a convex neighborhood of zero in X. On the other hand,we have xn /∈ nV for all n ∈ N+. It follows that B 6⊂ nV for all n ∈ N+, this contradictsthe fact that B is bounded. So we have established that for every bounded B ⊂ X, wehave an n ∈ N and a bounded Bn ⊂ Xn such that un(Bn) = B. Since unn+1 is compact,we find that unn+1(Bn) is relatively compact. If we apply Lemma 1.1.6 to the closure ofunn+1(Bn) and B, we find that

un+1 : unn+1(Bn)→ B

is a homeomorphism. This proves the first part. In particular one has the following. Takeany sequence (xn)n∈N in X which converges to some x. Since the sequence is convergent,we have that xn : n ∈ N ∪ x is bounded in X. Using what we have just proven, wefind for some m ∈ N and a sequence (yn)n∈N in Xm and y ∈ Xm such that yn → y andum(yn) = xn for all n ∈ N.We now check that X is Hausdorff. Assume that 0 ⊂ X is different from 0, then 0is bounded and thus isomorphic to some subset of Xn (for some n ∈ N), this contradictsthe fact that Xn is Hausdorff.It follows from the previous part that every bounded set in X is compact. Combiningthis with Lemma 1.3.14, we conclude that X is Montel.For the last part we need to prove that the identity map lim−→n∈NXn → lim−→n∈NXn isan isomorphism. It is clear that it is continuous. It suffices to prove that for everyneighborhood of zero U in lim−→n∈NXn = X, there is a neighborhood of zero V in lim−→n∈NXn

such that V ⊂ U . We will construct such a V for any given U . Take a sequence of positivereals (rn)n∈N such that ⋃

n∈N

un(Bn,rn(0)) ⊂ U.

For all m ∈ N we define rn,m = rn1+m2+m

and we take the following increasing sequence of

neighborhoods of zero in X

Um =⋃n∈N

un(Bn,rn,m(0)) ⊂ U.

By the first part of the proof and the fact that X → X is a continuous map, we knowthat for all m ∈ N the set Um is sequentially closed in X and contained in the sequentialinterior of Um+1. In particular we have for any m ∈ N+ that (um)−1(Um−1) is closed andcontained in the interior of (um)−1(Um). We will now define Vm ⊂ (um)−1(Um) recursively,such that Vm is bounded and convex. Take V0 = B0,r0,0

(0). For m ∈ N+, we know that

22

um−1m (Vm−1) ⊂ (um)−1(Um−1) is contained in the interior of (um)−1(Um) so the following

map takes only positive values

gm : um−1m (Vm−1)→ (0, 1] : x 7→ maxr ≤ 1|Bm,r(x) ⊂ (um)−1(Um).

Since gm is continuous and um−1m (Vm−1) is compact in Xm, we can take ρm > 0 to be its

minimum and we define Vm = um−1m (Vm−1) +Bm,ρm(0). By our choice of ρm we have that

Vm ⊂ (um)−1(Um) and that Vm is convex (because Vm−1 and Bm,ρm(0) are convex). Wenow define

V =⋃m∈N

um(Vm) ⊂⋃m∈N

Um ⊂ U.

We have Bm,ρm ⊂ (um)−1(V ) for all m ∈ N, we also have u0(V0) ⊂ u1(V1) ⊂ u2(V2) ⊂ · · · ,which implies that V is convex. So we conclude that V is a neighborhood of zero in Xand V ⊂ U .

Definition 1.3.16. A space which satisfies the conditions from the previous theorem iscalled a (DFS)-space. Explicitly: X is a (DFS)-space if we have Banach spaces Xn andcompact linear injections unn+1 : Xn → Xn+1 for all n ∈ N such that X ∼= lim−→n∈NXn. This

class of spaces was introduced by J. Sebastiao e Silva [11].

Proposition 1.3.17. The product of two (DFS)-spaces is a (DFS)-space.

Proof. This follows directly from Lemma 1.1.5 and Propostion 1.3.11.

Theorem 1.3.18 (Universal property of the final topology). Let X = lim−→α∈P Xα and letY be topological space. A map f : X → Y is continuous if and only if f uα : Xα → Yis continuous for all α ∈ P .

Proof. If f : X → Y is continuous, then f uα is continuous as the composition of twocontinuous maps. Conversely, if f uα is continuous for all α ∈ P and U ⊂ Y is open,then (f uα)−1(U) = (uα)−1(f−1(U)) is open for all α ∈ P . Combining this with Remark1.3.5 we conclude that f−1(U) ⊂ X is open.

Corollary 1.3.19. Let X = lim−→α∈P Xα, Y = lim−→α∈P Yα be final topologies on X and Ywith respect to the maps uαβ : Xα → Xβ, vαβ : Yα → Yβ for α < β ∈ P . Let fα : Xα → Yαbe continuous maps, such that ∀β < α ∈ P : vβα fβ = fα uβα. It follows that⊕

α∈P

fα :⊕α∈P

Xα →⊕α∈P

Yα :∑α∈P

xα 7→∑α∈P

fα(xα)

induces a continuous map f : X → Y .

Proof. By the property which we imposed on fα, for α ∈ P , we know that for anyα < β ∈ P , xβ ∈ Xβ, xα ∈ Xα we have

fβ(xβ − uαβ(xα)) = fβ(xβ)− vαβ (fα(xα)).

So⊕

fα induces a map f : X → Y . It remains to prove that f is continuous. Observethat for any α ∈ P

f uα = vα fα : Xα → Y.

This implies that f uα is continuous for any α ∈ P ; we conclude by the previous theoremthat f is continuous.

23

Proposition 1.3.20. If for all α ∈ P we have that Yα ⊂ Xα is a closed subspace, thenthe topology of Y = lim−→α∈P Yα coincides with the topology on Y ⊂ X = lim−→α∈P Xα inducedby X and Y ⊂ X is closed.

Proof. By the universal property we know that the injection Y → X is continuous.Conversely, let V ⊂ Y be closed in the final topology on Y . It follows that for α ∈ Parbitrary (uα)−1(V ) is closed in the topology of Yα for all α ∈ P . Since Yα ⊂ Xα is closed,it follows that (uα)−1(V ) is closed in the topology Xα. Since this holds for all α ∈ P , weconclude by Remark 1.3.5 (by going to complements) that V ⊂ X is closed. Thus we haveproven that the canonical injection Y → X is bicontinuous. If we repeat this argumentwith V = Y , we find that Y ⊂ X is closed.

Remark 1.3.21. Since the inductive limit topology and final topology coincide on a(DFS)-space, we can apply Corollary 1.3.19 and Proposition 1.3.20 to (DFS)-spaces.

Definition 1.3.22. If, for all n ∈ N the space Xn is an (FS)-space and unn+1 : Xn → Xn+1

is an isomorphism between Xn and unn+1(Xn), then we call X = lim−→n∈NXn an (LFS)-space.

Remark 1.3.23. If we identify in the previous definition Xn with unn+1(Xn) for all n ∈ N,we can identify X with

⋃n∈NXn as in Example 1.3.6.

Lemma 1.3.24. Let Y be a closed linear subspace of a locally convex topological vectorspace X. If V ⊂ Y is a convex neighborhood of zero and x ∈ X \ Y , then we have aconvex neighborhood of zero U ⊂ X such that V = U ∩ Y and x /∈ V .

Proof. Let x ∈ X and V ⊂ Y be as in the statement of the lemma. Since X is locallyconvex and Y is closed, we have that X/Y is locally convex and Hausdorff. So we cantake a convex W ⊂ X such that x /∈ W + Y . Since V is a neighborhood of zero in Y inthe relative topology, there is a convex neighborhood of zero A in X such that Y ∩A ⊂ V .We take B to be the convex hull of A∪ V . Since V and A are convex, any element b ∈ Bis of the form b = λv + (1 − λ)a where v ∈ V , a ∈ A \ Y and λ ∈ [0, 1]. So we haveV = Y ∩B. Taking U = B ∩ (W + Y ) concludes our proof.

Theorem 1.3.25. Let X = lim−→n∈NXn be an (LFS)-space defined by the (FS)-spacesXn and the maps unn+1 : Xn → Xn+1 for n ∈ N. Firstly, for every n ∈ N the mapun : Xn → un(Xn) ⊂ X is an isomorphism and its image is closed. Secondly, for everybounded set B ⊂ X is we have n ∈ N, Bn ⊂ Xn bounded, such that B = un(Bn). Thirdly,X is Montel.

Proof. Let X = lim−→n∈NXn be as in the statement. We first prove the first part of the

statement. Take any m ∈ N. We want to prove that um : Xm → um(Xm) ⊂ X isan isomorphism and that its image is closed in X. Since unn+1 is an injection for alln ∈ N, we can conclude that um is a continuous bijection. So it suffices to prove that forevery convex neighborhood of zero U in Xm and for every x ∈ X \ um(Xm) there is aneighborhood of zero V in X such that V ∩ um(Xm) = U and x /∈ V . For all n ∈ N wedefine ξn = (un)−1(x) (this is a singleton or an empty set). We take Vm = V and defineVn recursively for m < n ∈ N. Note that for any m < n ∈ N the space Xn−1 is completeand thus un−1

n (Xn−1) is a closed subspace of Xn. So for any m < n ∈ N we can use the

24

previous lemma to take Vn ⊂ Xn such that Vn∩un−1n (Xn−1) = un−1

n Vn−1 and Vn is disjointfrom ξn. We define V =

⋃n≥m u

n(Vn). By construction V is convex and a neighborhoodof zero in X, and x /∈ V . So we have proven that um : Xm → X is an isomorphism ontoits image and its image is closed, for any m ∈ N.We now prove the second part. Let B ⊂ X be bounded and assume that B is notcontained in un(Xn) for any n ∈ N. It follows that we have k0 < k1 < . . . such thatukn(Xkn) \ ukn−1(Xkn−1) contains an element xn ∈ B for all n ∈ N+. We use the previouslemma to take convex neighborhoods of zero Vn ⊂ Xkn (for all n ∈ N) such that Vn ⊂(uknkn+1

)−1(Vn+1) and xn/n /∈ ukn(Vn). It follows that V =⋃n∈N u

kn(Vkn) is a neighborhoodof 0 in X but V does not contain xn/n for all n ∈ N+. It follows that B 6⊂ nV for alln ∈ N, this is in contradiction with the assumption that B is bounded. So we haveestablished that for every bounded B ⊂ X, there is an n ∈ N such that B ⊂ un(Xn).From the first part we conclude that B ∼= (un)−1(B) ⊂ Xn.We now check that X is Hausdorff. Assume that 0 ⊂ X is different from 0, then 0is bounded and thus isomorphic to some subset of Xn (for some n ∈ N), this contradictsthe fact that Xn is Hausdorff. From Lemma 1.3.14 we know that X is barreled. And fromthe fact that any bounded set in X is homeomorphic to a bounded set in the (FS)-spaceXn we conclude that X is Montel.

1.4 Ultradifferentiable functions

Definition 1.4.1 (Ultradifferentiable function). Let (Mp)p∈N be a sequence in R+ andΩ ⊂ Rd an open set. We say that ϕ ∈ E(Ω) is an ultradifferentiable function of classMp if

(∀K ⊂comp

Ω)(∃h > 0)(∃C > 0)(∀α ∈ Nd)(supx∈K|Dαϕ(x)| ≤ Ch|α|M|α|). (1.1)

We denote the set of these functions by EMp(Ω). We call this the ultradifferentiablefunctions of Roumieu type. We say that ϕ ∈ E(Ω) is an ultradifferentiable function ofclass (Mp) if

(∀K ⊂comp

Ω)(∀h > 0)(∃C > 0)(∀α ∈ Nd)(supx∈K|Dαϕ(x)| ≤ Ch|α|M|α|). (1.2)

We denote the set of these functions by E (Mp)(Ω). We call this the ultradifferentiablefunctions of Beurling type.

Definition 1.4.2. For a sequence of positive reals (Mp)p∈N and r > 0. We will denotethe set of periodic ultradifferentiable functions by

E (Mp)(Tdr) = E (Mp)(Rd) ∩ E(Tdr) and EMp(Tdr) = EMp(Rd) ∩ E(Tdr).

Definition 1.4.3. For a sequence of positive reals (Mp)p∈N and an open set Ω ⊂ Rn. Wewill denote the set of ultradifferentiable functions with compact support by

D(Mp)(Ω) = E (Mp)(Ω) ∩ D(Ω) and DMp(Ω) = EMp(Ω) ∩ D(Ω).

25

Notation 1.4.4. When we do not want to specify explicitly which sequence (Mp)p∈Nor which class of ultradifferentiable functions we look at, we write ∗ instead of Mp or(Mp). Within one expression ∗ will always stand for a fixed sequence and a fixed class.For example, we could write the previous definition as D∗(Ω) = E∗(Ω) ∩ D(Ω).

The next thing we want to do is to topologize E∗(Ω) and D∗(Ω). We will do this by usingthe theory which we introduced in the previous sections. From (1.1) and (1.2) we see thatit is necessary to introduce the following norms.

Definition 1.4.5. Let (Mp)p∈N be a sequence of positive reals, r, h > 0 and K ⊂comp

Rd. A

function ϕ ∈ E(K) satisfies

(∃C > 0)(∀α ∈ Nd)(supx∈K|Dαϕ(x)| ≤ Ch|α|M|α|) (1.3)

if and only if the following norm is finite

‖ϕ‖EMp,h(K) = supx∈Kα∈Nd

|Dαϕ(x)|h|α|M|α|

= supα∈Nd

‖Dαϕ‖C(K)

h|α|M|α|.

This defines the normed spaces of functions that satisfy (1.3)

EMp,h(K) = ϕ ∈ E(K) : ‖ϕ‖EMp,h(K) <∞,DMp,h(K) = ϕ ∈ D(K) : ‖ϕ‖EMp,h(K) <∞ and

EMp,h(Tdr) = ϕ ∈ E(Tdr) : ‖ϕ‖EMp,h([0,2πr]d) <∞.

Notation 1.4.6. We will denote the norm ‖ · ‖EMp,h([0,2πr]d) on the space EMp,h(Tdr) by‖ · ‖EMp,h(Tdr).

Lemma 1.4.7. For K ⊂comp

Rd, h ∈ R+, a ∈ Rd and a sequence of positive reals (Mp)p∈N

we haveEMp,h(K) =

x 7→ ϕ(hx+ a)| ϕ ∈ EMp,1

(h−1(K − a)

)and

‖ϕ‖EMp,h(K) = ‖ϕ‖EMp,1(h−1(K−a)).

Equivalentlyϕ 7→ (x 7→ ϕ(hx+ a))

is an isomorphism between the normed vector spaces EMp,h(K) and EMp,1(h−1(K−a)).

Proof. This follows from the fact that for all α ∈ Nd

Dα(x 7→ ϕ(hx+ a)) = x 7→ h|α|Dα(ϕ)(hx+ a).

26

Corollary 1.4.8. For K ⊂comp

Rd, r, h ∈ R+, a ∈ Rd and a sequence of positive reals

(Mp)p∈N we have thatϕ 7→ (x 7→ ϕ(hx+ a))

is an isomorphism between the normed vector spaces DMp,h(K) and DMp,1(h−1(K−a)),it is also an isomorphism between EMp,h(Tdr) and EMp,1(Tdr/h).

Proof. The first part follows by restricting the isomorphism that we found in the previouslemma to the space of functions with compact support. The second part follows byrestricting the isomorphism ϕ 7→ (x 7→ ϕ(hx)) to EMp,h(Tdr/h) and observing that ϕ 7→(x 7→ ϕ(x+ a)) is automorphism on the normed vector space EMp,h(Tdr/h).

Remark 1.4.9. The isomorphism obtained in the previous corollary will allow us workwith Td instead of Tdr (for general r > 0). Because of this Tdr will only appear in argumentswhere scaling makes our argumentation easier.

Lemma 1.4.10. Let (Mp)p∈N be a sequence of positive reals, K ⊂comp

Rd and h > 0. The

natural inclusionEMp,h(K)→ E(K)

is a continuous map. This makes its restrictions

EMp,h(Tdr)→ E(Tdr) and DMp,h(K)→ D(K)

continuous.

Proof. For ϕ ∈ EMp,h(K) arbitrary we have

‖ϕ‖Cm(K) = sup|α|≤m

‖Dαϕ‖C(K) ≤ C‖ϕ‖EMp,h(K)

where C = maxh|α|M|α| : |α| ≤ m does not depend on ϕ. So we have that for any m ∈ Nthe natural inclusion EMp,h(K) → Cm(K) is continuous. The desired result follows bythe universal property of the projective limit topology lim←−m→∞C

m(K) = E(K).

Proposition 1.4.11. Let (Mp)p∈N be a sequence of positive reals. Let K ⊂comp

Rd and

r, h ∈ R+. The space EMp,h(K) is a Banach space. The space DMp,h(K) is a closedsubspace of EMp,h(K) and EMp,h(Tdr) is a closed subspace of EMp,h([0, 2πr]d).

Proof. If (ϕn)n∈N is a Cauchy sequence in EMp,h(K), then it is, by the previous lemma,a Cauchy sequence in E(K) and thus ϕn → ϕ for some ϕ ∈ E(K). Take any ε > 0. TakeN ∈ N such that

∀n ≥ N : ‖ϕn − ϕN‖EMp,h(K) < ε/2. (1.4)

For any α ∈ Nd we have ϕn → ϕ in C |α|(K) and thus we have n > N such that

‖Dα(ϕn − ϕ)‖C(K)

h|α|M|α|< ε/2.

27

Combining this with (1.4) we find that for any α ∈ Nd

‖Dα(ϕN − ϕ)‖C(K)

h|α|M|α|< ε.

From this we conclude that ‖ϕm − ϕ‖EMp,h(K) → 0, so EMp,h(K) is a Banach space.The second part of the proposition follows by using the previous lemma and observingthat D(K) and E(Tdr) are closed subspaces of E(K) and E([0, 2πr]d), respectively.

Proposition 1.4.12. Let (Mp)p∈N be sequence of positive reals, K ⊂comp

Rd, r ∈ R+ and

k > h > 0. The natural inclusions

EMp,h(K) → EMp,k(K),

DMp,h(K) → DMp,k(K) and

EMp,h(Tdr) → EMp,k(Tdr)

are compact maps.

Proof. Let B = ϕ ∈ EMp,h(K) : ‖ϕ‖EMp,h(K) < 1 be the unit ball in EMp,h(K). We

wish to prove that B ⊂ EMp,k(K) is relatively compact. So take any ε > 0 and takem ∈ N such that (h/k)m < ε/2. We will now apply the Arzela-Ascoli Theorem to thefinite set V = Dαϕ : ϕ ∈ B, |α| < m ⊂ C(K). Let M = maxh|α|M|α| : |α| ≤ m,then we get for ϕ ∈ V that ‖ϕ‖C(K) ≤ M and by the mean value theorem we get forx, y ∈ K that |ϕ(x)−ϕ(y)| ≤M |x−y|. So we conclude that V is uniformly bounded andequicontinuous, so we can apply the Arzela-Ascoli Theorem and conclude that V ⊂ C(K)is relatively compact. Take µ = mink|α|M|α| : |α| < m and ϕ1, . . . , ϕn ∈ V such that forany ϕ ∈ V there is a 1 ≤ j ≤ n such that ‖ϕ − ϕj‖C(K) < µε/2. From this we concludethat there are ϕ1, . . . , ϕn′ ∈ B (explicitly, we take for every n-tuple (α1, . . . , αn) ∈ (Nd)n

with |αj| < m for all j one function in ϕ ∈ B : ∀j ‖Dαjϕ − ϕj‖C(K) < µε/2 if this setis not empty) such that

(∀ϕ ∈ B)(∃j ∈ 1, . . . , n′)(∀|α| < m)(‖Dα(ϕ− ϕj)‖C(K) < µε ≤ k|α|M|α|ε).

By our choice of m we have that for all j and for m ≤ |α|

‖Dα(ϕ− ϕj)‖C(K) ≤ 2h|α|M|α| < εk|α|M|α|.

Combining these two, we get

(∀ϕ ∈ B)(∃j ∈ 1, . . . , n′)(∀α ∈ Nd)(‖Dα(ϕ− ϕj)‖C(K) < εk|α|M|α|).

So we can conclude that

(∀ϕ ∈ B)(∃j ∈ 1, . . . , n′)(∀α ∈ Nd)(‖ϕ− ϕj‖EMp,k(K) < ε).

We get that

EMp,h(K) → EMp,k(K) (1.5)

28

is compact. The second and third part follow from the fact that we can see

DMp,h(K) → DMp,k(K) and

EMp,h(Tdr) → EMp,k(Tdr)

as restrictions of (1.5) to functions with support in K or to periodic functions for K =[0, 2πr]d.

Corollary 1.4.13. Let (Mp)p∈N be a sequence of positive reals, K ⊂comp

K ′ ⊂comp

Rd and

k > h > 0. The inclusions

EMp,h(K ′) → EMp,k(K) : ϕ 7→ ϕ|KDMp,h(K) → DMp,k(K ′) : ϕ 7→ ϕ

are compact.

Proof. This follows from the previous proposition, with the additional observations thatfor ϕ ∈ EMp,h(K ′)

‖ϕ‖EMp,k(K) ≤ ‖ϕ‖EMp,k(K′)

and for ϕ ∈ DMp,h(K)‖ϕ‖EMp,k(K′) = ‖ϕ‖EMp,k(K).

Definition 1.4.14. Let (Mp)p∈N be sequence of positive reals, K ⊂comp

Rd and r, h ∈ R+.

The previous proposition allows us to define the following (FS)-spaces

E (Mp)(K) = lim←−h→0

EMp,h(K) = lim←−n→∞n∈N

EMp,1/n(K),

D(Mp)(K) = lim←−h→0

DMp,h(K) = lim←−n→∞n∈N

DMp,1/n(K) and

E (Mp)(Tdr) = lim←−h→0

EMp,h(Tdr) = lim←−n→∞n∈N

EMp,1/n(Tdr).

As well as the following (DFS)-spaces

EMp(K) = lim−→h→∞EMp,h(K) = lim−→

n→∞n∈N

EMp,n(K),

DMp(K) = lim−→h→∞DMp,h(K) = lim−→

n→∞n∈N

DMp,n(K) and

EMp(Tdr) = lim−→h→∞EMp,h(Tdr) = lim−→

n→∞n∈N

EMp,n(Tdr).

Note that, as a set, E∗(Tdr) can be identified with the definition we gave in 1.4.2 (seeExample 1.2.8 and 1.3.6).

29

Proposition 1.4.15. For K ⊂comp

Rd we have that the natural injections

D∗(K)→ E∗(K) and E∗(Td)→ E∗([0, 2π]d)

are isomorphisms onto their image and their images are closed. This allows us to identifyD∗(K) (and E∗(Td)) with a closed subspace of E∗(K) (of E∗([0, 2π]d)).

Proof. Depending on which case ∗ represents we have either a map between inductivelimit topologies or between projective limit topologies. In the first case the propositionfollows from Remark 1.3.21 and Proposition 1.3.20. In the second case the propositionfollows from Proposition 1.2.11.

Proposition 1.4.16. Let K ⊂comp

Rd, k ∈ R+ and a ∈ Rd. The scaling and translation of

functionsϕ 7→ (x 7→ ϕ(kx+ a))

defines an isomorphism between E∗(K) and E∗(k−1(K − a)). And in particular betweenD∗(K) and D∗(k−1(K − a)) and between E∗(Tdr) and E∗(Tdr/k).

Proof. This follows directly from Lemma 1.4.7 and Corollary 1.4.8 by taking projectiveor inductive limits of isomorphic spaces.

We are now ready to topologize E∗(Ω) and D∗(Ω). Note that, as sets, we have an identifi-cation of D∗(Ω) with

⋃K ⊂

compΩD∗(K). We can also identify E∗(Ω) with

⋂K ⊂

compΩϕ : ϕ|K ∈

E∗(K). So the following definition topologizes the correct sets.

Definition 1.4.17. On an open set Ω ⊂ Rd the topological space of ultradifferentiablefunctions of class ∗ is defined as

E∗(Ω) = lim←−K ⊂

compΩ

E∗(K) = lim←−n→∞

E∗(Kn)

and the topological space of ultradifferentibiale functions of class ∗ with compact supportis defined as

D∗(Ω) = lim−→K ⊂

compΩ

D∗(K) = lim−→n→∞

D∗(Kn).

In these definitions K1 ⊂comp

K2 ⊂comp

· · · ⊂comp

Ω can be any sequence of compact sets such that⋃∞n=1Kn = Ω.

Theorem 1.4.18. Let (Mp)p∈N be a sequence of positive reals and let Ω ⊂ Rd be open.We give the following characterization of spaces.

• E (Mp)(Ω) is an (FS)-space.

• EMp(Ω) is a complete semi-Montel space.

• D(Mp)(Ω) is an (LFS)-space.

30

• DMp(Ω) is a (DFS)-space.

Proof. Let K1 ⊂comp

K2 ⊂comp

· · · ⊂comp

Ω be a sequence of compact sets such that⋃∞n=1Kn = Ω.

• We have that

E (Mp)(Ω) = lim←−n→∞

E (Mp)(Kn) = lim←−n→∞

lim←−m→∞

EMp,1/m(Kn).

It follows from Corollary 1.2.17 that

E (Mp)(Ω) = lim←−n→∞

EMp,1/n(Kn).

Since Corollary 1.4.13 states that the maps that define this projective limit arecompact, we conclude that E (Mp)(Ω) is an (FS)-space.

• Note for any n ∈ N the space EMp(Kn) is a (DFS)-space and thus it is completeand Montel by Theorem 1.3.15. So EMp(Ω) = lim←−n→∞ E

Mp(Kn) is complete byProposition 1.2.15 and semi-Montel by Corollary 1.2.21.

• The spaces D(Mp)(Kn) are (FS)-spaces and D(Mp)(Ω) = lim←−n→∞D(Mp)(Kn), so we

need to verify that the natural inclusions

D(Mp)(Kn)→ D(Mp)(Kn+1)

for any n ∈ N are isomorphisms into their image. For any h > 0, n ∈ N we have forall ϕ ∈ DMp,h(Kn)

‖ϕ‖EMp,h(Kn) = ‖ϕ‖EMp,h(Kn+1)

and DMp,h(Kn) ⊂ DMp,h(Kn+1) is closed. We conclude by Proposition 1.2.11that

D(Mp)(Kn) = lim←−h→∞DMp,h(Kn)→ lim←−

h→∞DMp,h(Kn+1) = D(Mp)(Kn+1)

is an isomorphism into its image for any n ∈ N. So D(Mp)(Ω) is, by definition, an(LFS)-space.

• We haveDMp(Ω) = lim−→

n→∞DMp(Kn) = lim−→

n→∞lim−→m→∞

DMp,m(Kn).

It follows from Corollary 1.3.13 that

DMp(Ω) = lim−→n→∞

DMp,n(Kn).

Since Corollary 1.4.13 states that the maps that define this inductive limit arecompact, we conclude that DMp(Ω) is a (DFS)-space.

31

Chapter 2

Properties controlled by the definingsequence

In this chapter (Mp)p∈N will denote a sequence of positive reals. We will first study somerelevant properties of such sequences. Subsequently, we will study which implicationsthe behavior of the sequence (Mp)p∈N can have on classes of ultradifferentiable functionsdefined by this sequence.

2.1 The behavior of sequences of positive reals

Definition 2.1.1 (Logarithmic convexity). We call a sequence of positive reals (Mp)p∈Nlog-convex if

(∀p ∈ N+)(M2p ≤Mp−1Mp+1). (M.1)

Remark 2.1.2. (M.1) expresses that ∀p ∈ N : 2 log(Mp) ≤ log(Mp−1) + log(Mp+1) whichmeans that (logMp)p∈N is convex.

Lemma 2.1.3. For a log-convex sequence (Mp)p∈N we have for p, q ∈ N and λ ∈ [0, 1]such that λp+ (1− λ)q ∈ N

Mλp+(1−λ)q ≤MλpM

1−λq .

In particular, if we take q = 0, we have for all p ∈ N and all λ ∈ [0, 1] such that λp ∈ N

Mλp

M0

≤(Mp

M0

)λ.

Proof. For p ∈ N we define µ(p) = log(Mp). Since (Mp)p∈N is log-convex, we get that µ isconvex. For p, q ∈ N and λ ∈ [0, 1] such that λp+ (1− λ)q ∈ N, we get by convexity that

µ(λp+ (1− λ)q) ≤ λµ(p) + (1− λ)µ(q),

which proves the lemma.

32

Definition 2.1.4 (Associated function). Let (Mp)p∈N be a sequence of positive reals, itsassociated function M : [0,∞)→ [0,∞] is defined as

M : ρ 7→M(ρ) = supp∈N

logρpM0

Mp

= log

(supp∈N

ρpM0

Mp

).

It is indeed clear that ∀ρ ∈ R+ : M(ρ) ≥ log ρ0M0

M0= 0. We assume, by convention, that

00 = 1, so M(0) = 0.

Remark 2.1.5. Let (Mp)p∈N be a sequence of positive reals and let M be its associatedfunction. If infp∈N+

p√Mp = 0, then we have for any ρ ∈ R+, ε ∈ (0, 1] a p ∈ N such that

Mp ≤ εpρp ≤ ερp. We conclude that for any ρ ∈ R+ we have a p ∈ N such that

∀ε ∈ (0, 1] :1

ε≤ ρp

Mp

.

It follows that ∀ρ ∈ R+ : M(ρ) = ∞. Because of this we will usually assume thatinfp∈N+

p√Mp > 0.

Lemma 2.1.6. Let M be the associated function of a sequence of positive reals (Mp)p∈N.

Then, M is increasing, limρ→∞M(ρ)log ρ

=∞ and M exp is convex.

Proof. For any ρ′ > ρ ≥ 0 and for any p ∈ N we have that ρ′pM0

Mp≥ ρpM0

Mp, so it follows that

M is increasing.For any p ∈ N we have M(ρ) ≥ p log ρ+ log M0

Mp. So for ρ0 > 0 such that log Mp

M0≤ p

2log ρ0

we get that

∀ρ > ρ0 :M(ρ)

log ρ≥ p

2.

Since this holds for any p ∈ N, it follows that limρ→∞M(ρ)log ρ

=∞.

The convexity of M exp : x 7→ supp∈N (px+ log(M0/Mp)) follows from the fact that forany x, x′ ∈ R we have, by the subadditivity of sup

1

2

(supp∈N

(px+ log

M0

Mp

)+ sup

p∈N

(px′ + log

M0

Mp

))≥ sup

p∈N

(px+ x′

2+ log

M0

Mp

).

Lemma 2.1.7. Let (Mp)p∈N be a sequence of positive reals and let M be its associatedfunction. If infp∈N+

p√Mp > 0, then there is a b > 0 such that M |[0,b] = 0.

Proof. From the assumption on (Mp)p∈N we conclude that there is some 1 > c > 0 such

that M1/pp ≥ c for all p ∈ N+. We take b = minc, c/M0, this implies that Mp/M0 ≥ bp

for all p ∈ N. We find that for ρ ∈ (0, b]

∀p ∈ N : logρpM0

Mp

≤ log(ρpb−p) ≤ log 1 = 0.

So since ∀ρ : M(ρ) ≥ 0, we conclude that ∀ρ ∈ (0, b] : M(ρ) = 0.

33

Definition 2.1.8 (Greatest log-convex minorant). For a sequence of positive real numbers(Mp)p∈N with infp∈N+

p√Mp > 0 we define its greatest log-convex minorant as the sequence

M cp = M0 sup

ρ≥0

ρp

expM(ρ)= sup

ρ≥0infq∈N

(ρp−qMq) for p ∈ N

where M is the associated function of (Mp)p∈N.

Remark 2.1.9. When we assume that the greatest log-convex minorant of a sequence(Mp)p∈N exists, we implicitly assume that infp∈N+ M

1/pp > 0 (Remark 2.1.5 shows why this

is a necessary assumption). The following lemma shows that M cp is positive for all p ∈ N.

Lemma 2.1.10. Let (Mp)p∈N be a sequence of positive reals, let M be its associated func-tion and let (M c

p)p∈N be its greatest log-convex minorant. We have that M cp > 0 for all

p ∈ N.Let (Np)p∈N be a sequence of positive reals and let (N c

p)p∈N be its greatest log-convex mi-norant. Assume that ∀p ∈ N : Np ≤Mp, it follows that ∀p ∈ N : N c

p ≤M cp .

Proof. We first show the second part. We have for all p, q ∈ N and all ρ ∈ R

ρp−qNq ≤ ρp−qMq.

Since taking suprema and infima preserves the inequality, we get

N cp = sup

ρ≥0infq∈N

(ρp−qNq) ≤ supρ≥0

infq∈N

(ρp−qMq) = M cp .

We now prove the first part of the lemma. Let 0 < b = infp∈NM1/pp . We have that for

p ∈ N and ρ ≥ 0

infq∈N

ρp−qbq = ρp infq∈N

ρ−qbq =

ρp if ρ ∈ [0, b]

0 if ρ ∈ (b,∞).

This implies that(bp)cp = sup

ρ≥0infq∈N

ρp−qbq = bp.

From the first part of the proof we obtain 0 < bp = (bp)cp ≤M cp .

Proposition 2.1.11. Let (Mp)p∈N be a sequence of positive reals and let (M cp)p∈N be its

greatest log-convex minorant. We have that (M cp)p∈N is log-convex, that (M c

p)p∈N is smallerthen (Mp)p∈N in the following sense: M0 = M c

0 and M cp ≤ Mp for all p ∈ N and that it

is the maximal sequence with this property, i.e. for any log-convex (Np)p∈N with Np ≤Mp

for all p ∈ N we have Np ≤M cp for all p ∈ N.

Proof. We first prove that (M cp)p∈N is log-convex. This follows from the fact that for any

p ∈ N+ (supρ≥0

ρp

expM(ρ)

)2

= supρ≥0

ρp−1ρp+1

expM(ρ)2≤ sup

ρ≥0

ρp−1

expM(ρ)supρ≥0

ρp+1

expM(ρ).

34

Note that for any p ∈ N and for any ρ ≥ 0

infq∈N

(ρp−qMq) ≤ ρ0Mp = Mp

which implies that ∀p ∈ N : M cp ≤ Mp. We know that infρ≥0M(ρ) = 0 so for p = 0 we

get

M c0 = M0

1

infρ≥0 exp(M(ρ))= M0.

Let (Np)p∈N be a log-convex sequence with Np ≤ Mp for all p ∈ N. By what we haveproved, we know that ∀p : N c

p ≤ Np and N0 = N c0 . We now wish to prove that ∀p : N c

p ≥Np. For this it suffices to find for each p ∈ N a ρ ∈ R+ such that for all q ∈ N

ρp−qNq ≥ Np. (2.1)

If we take ρ = Np+1/Np, then we know by Lemma 2.1.3 that

Np−qp+1Nq ≥ Np−q+1

p for q ≤ p and N q−p−1p Nq ≥ N q−p

p+1 for q > p.

This implies (2.1). We conclude that ∀p ∈ N : Np = N cp . Combining this with Lemma

2.1.10 we get ∀p ∈ N : M cp ≥ N c

p = Np; this concludes the proof.

Definition 2.1.12. We define for the sequence (Mp)p∈N the following sequence of ratios

∀p ∈ N+ : mp =Mp

Mp−1

.

We also define the counting function of this sequence m : R+ → N ∪ ∞

m : λ 7→ #p ∈ N+ : mp ≤ λ =∞∑p=1

χ[mp,∞)(λ).

Proposition 2.1.13. Let M be the associated function and m be the counting functionof the ratios (mp)p∈N+ of some log-convex sequence (Mp)p∈N. We have

∀ρ ∈ [0,∞) : M(ρ) =

∫ ρ

0

m(λ)

λdλ.

In particular: M is finite and continuous on [0, supp∈Nmp).

Proof. Note that (M.1) implies that ∀p ∈ N+ : mp ≤ mp+1. For any ρ ≥ 0, take q ∈ Nminimal such that ρ < mq+1 if ρ < supp∈Nmp and q =∞ otherwise. By using Lebesgue’sMonotone Convergence Theorem, we get∫ ρ

0

m(λ)

λdλ =

∫ ρ

0

∞∑p=1

χ[mp,∞)(λ)1

λdλ =

q∑p=1

∫ ρ

mp

1

λdλ

=

q∑p=1

(log ρ− logmp) =

q∑p=1

logρ

mp

= supr∈N

r∑p=1

logρ

mp

= supr∈N

logρr

m1 · · ·mr

= supr∈N

logρrM0

Mr

= M(ρ).

This proves the first part. The second part follows from the fact that (mp)p∈N+ is increas-ing, which implies that m is finite on [0, supp∈Nmp).

35

Corollary 2.1.14. Let M be the associated function and m be the counting function ofthe ratios (mp)p∈N+ of some log-convex sequence (Mp)p∈N. We have on (0, supp∈N+ mp)

D(M exp) = m.

Proof. By the previous proposition we have

∀σ ∈ (0, supp∈N+

mp) : M(exp(σ)) =

∫ exp(σ)

0

m(λ)

λdλ =

∫ σ

0

m(µ)dµ.

Differentiating both sides with respect to σ gives us the desired result.

Definition 2.1.15 (Stability under differentiation). For a sequence of positive real num-bers (Mp)p∈N the following condition will play an important role when we will study thebehavior of ultradifferentiable functions under differentiation:

(∃A,H > 0)(∀p ∈ N)(Mp+1 ≤ AHpMp). (M.2’)

Proposition 2.4.2 and Theorem 2.4.3 will explain the terminology.

Lemma 2.1.16. Let (Mp)p∈N be a sequence of positive reals that satisfies (M.2’). Wehave A,H > 0 such that

∀p, q ∈ N : Mp+q ≤ AqHq q−12 HpqMp.

Proof. We have for any p, q ∈ N by applying (M.2’) q times

Mp+q ≤ AqHp+q−1 · · ·Hp+1HpMp = AqHpq+q q−12 Mp.

Lemma 2.1.17. Let (Mp)p∈N be a sequence of positive reals and let M be its associatedfunction. If (M.2’) is satisfied, then we get

(∀n ∈ N+)(∃C, k > 0)(∀ρ > 0)

(exp

(M(ρk

)−M (ρ)

)≤ C

ρn

).

Conversely, if (Mp)p∈N is log-convex, then

(∃C, k > 0)(∀ρ > 0)

(exp

(M(ρk

)−M (ρ)

)≤ C

ρ

)implies (M.2’).

Proof. Assume that (M.2’) is witnessed by some A,H > 1. Take any n ∈ N+. We get bythe definition of M for all ρ > 0

exp(M(ρk

)−M (ρ)

)=

supp∈N(ρk

)pM−1

p

supq∈N ρqM−1

q

= supp∈N

infq∈N

(ρp−qk−pM−1

p Mq

)= sup

p∈N

(infq∈N

(ρp−q+nMq)ρ−nk−pM−1

p

).

36

We take q = p+ n in this expression and we define k = Hn. We get:

exp(M(ρk

)−M (ρ)

)≤ sup

p∈N

(Mp+nρ

−nk−pM−1p

)≤ sup

p∈N

(AnHnp+n2

k−pρ−n)≤ AnHn2

ρ−n.

The first part of the statement now follows by taking C = AnHn2.

Conversely, assume that for all ρ ∈ R+

exp(M(ρ/k)−M(ρ)) ≤ Cρ−1 (2.2)

for some fixed C, k > 0. Note that (M.1) implies that for all p ∈ N

Mp = M cp = M0 sup

ρ∈R+

ρp

expM(ρ).

Taking the supremum over ρ ∈ R+ is equivalent to taking the supremum over ρ/k ∈ R+,so we get for all p ∈ N+

Mp−1 = M0 supρ/k∈R+

(ρ/k)p−1

expM(ρ/k)= M0k

−p+1 supρ∈R+

ρp−1

expM(ρ/k).

Using (2.2) and the definition of M , we get for all p ∈ N+

Mp−1 ≥M0k−p+1 sup

ρ∈R+

ρp−1

expM(ρ)Cρ−1≥ C−1k−p+1 sup

ρ∈R+

infq∈N

ρp−qMq = C−1k−p+1Mp.

This proves (M.2’).

2.2 The hierarchy of ultradifferentiable functions

In this section we will discuss some conditions under which one class of ultradifferentiablefunctions is contained in another.

Definition 2.2.1. We define the following partial order on sequences of positive reals.For two sequences (Mp)p∈N and (Np)p∈N of positive reals we write

Mp ⊂ Np

if(∃C > 0)(∃L > 0)(∀p ∈ N)(Mp ≤ CLpNp).

This is equivalent to infp∈N+p√Np/Mp > 0.

Remark 2.2.2. Using this definition we can write for K ⊂comp

Rd

EMp(K) = ϕ ∈ E(K) : sup|α|=p‖Dαf‖C(K) ⊂Mp.

37

Proposition 2.2.3. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such thatMp ⊂ Np (witnessed by some C,L > 0). We have for any K ⊂

comp

Rd, h > 0

EMp,h(K) ⊂ ENp,Lh(K), DMp,h(K) ⊂ DNp,Lh(K), EMp,h(Td) ⊂ ENp,Lh(Td)

and the injections are continuous.

Proof. For any ϕ ∈ EMp,h(K) we have

‖ϕ‖EMp,h(K) = supα∈Nd

‖Dαϕ‖C(K)

h|α|M|α|≥ sup

α∈Nd

‖Dαϕ‖C(K)

C(Lh)|α|N|α|=

1

C‖ϕ‖ENp,Lh(K).

This proves the proposition.

Corollary 2.2.4. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such that Mp ⊂Np. We have for any K ⊂

comp

Rd

EMp(K) ⊂ ENp(K), DMp(K) ⊂ DNp(K), EMp(Td) ⊂ ENp(Td)

and the injections are continuous.Analogously, we have

E (Mp)(K) ⊂ E (Np)(K), D(Mp)(K) ⊂ D(Np)(K), E (Mp)(Td) ⊂ E (Np)(Td)

and the injections are continuous.

Proof. By the previous proposition we have, for some L > 0, that the canonical injections

EMp,h(K) −→ ENp,Lh(K),

DMp,h(K) −→ DNp,Lh(K) and

EMp,h(Td) −→ ENp,Lh(Td)

are continuous for all h > 0. By taking the inductive or projective limit and applyingCorollary 1.3.8 or Corollary 1.2.10 we get the desired result.

Corollary 2.2.5. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such that Mp ⊂Np. We have for any Ω ⊂

open

Rd, h > 0

EMp(Ω) ⊂ ENp(Ω) and DMp(Ω) ⊂ DNp(Ω)

and the injections are continuous.Analogously, we have

E (Mp)(Ω) ⊂ E (Np)(Ω) and D(Mp)(Ω) ⊂ D(Np)(Ω)

and the injections are continuous.

38

Proof. By the previous corollary we have that the canonical injections

EMp(K)→ ENp(K) and DMp(K)→ DNp(K)

are continuous for all K ⊂comp

Ω. By taking the projective or inductive limit and applying

Corollary 1.2.10 or Corollary 1.3.8 we get the desired result.

Definition 2.2.6. We define the following strict partial order on sequences of positivereals, which is stronger then ⊂. For two sequences (Mp)p∈N and (Np)p∈N of positive realswe write

Mp ≺ Np

if(∀L > 0)(∃C > 0)(∀p ∈ N)(Mp ≤ CLpNp).

This is equivalent to limp→∞p√Mp/Np = 0.

Remark 2.2.7. Using this definition we can write for K ⊂comp

Rd

E (Mp)(K) = ϕ ∈ E(K) : sup|α|=p‖Dαf‖C(K) ≺Mp.

Example 2.2.8. For 0 ≤ s < t we have psp ≺ ptp. We also have p!s ⊂ psp ⊂ p!s. Theclasses of ultradifferentiable functions corresponding to these sequences are called theGervey classes.

Proof. For the first part is suffices to see that

limp→∞

(psp

ptp

)1/p

= limp→∞

ps−t = 0.

Indeed this implies that for any L > 0 we have a q ∈ N such that for all q < p ∈ Npsp ≤ Lpptp. Taking C = maxp≤q

psp

Lpptpgives us ∀p ∈ N : psp ≤ CLpptp.

We now prove the second part. We have ∀p ∈ N : p!s ≤ psp. This implies p!s ⊂ psp. Onthe other hand we have, by Stirling’s approximation, (e−ppp)s ≤ p!s for all p ∈ N. Thisproves psp ⊂ p!s.

Proposition 2.2.9. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such thatMp ≺ Np. We have for any K ⊂

comp

Rd, h, k > 0

EMp,h(K) ⊂ ENp,k(K), DMp,h(K) ⊂ DNp,k(K), EMp,h(Td) ⊂ ENp,k(Td)

and the injections are continuous.

Proof. Take C > 0 such that Mp ≤ C(k/h)pNp for all p ∈ N. For any ϕ ∈ EMp,h(K) wehave

‖ϕ‖EMp,h(K) = supα∈Nd

‖Dαϕ‖C(K)

h|α|M|α|≥ sup

α∈Nd

‖Dαϕ‖C(K)

Ck|α|N|α|=

1

C‖ϕ‖ENp,k(K).

This proves the proposition.

39

Corollary 2.2.10. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such that Mp ≺Np. We have for any K ⊂

comp

Rd

EMp(K) ⊂ E (Np)(K), DMp(K) ⊂ D(Np)(K), EMp(Td) ⊂ E (Np)(Td)

and the injections are continuous.

Proof. By the previous proposition we have that the canonical injection

EMp,h(K)→ ENp,k(K)

is continuous for all h, k > 0. By taking the inductive limit on the left hand side andapplying Theorem 1.3.7 we get that the canonical injection

EMp(K)→ ENp,k(K)

is continuous for all k > 0. By taking the projective limit and applying Theorem 1.2.9we get the desired result for EMp(K) → E (Np)(K). The other cases now follow byProposition 1.4.15.

Corollary 2.2.11. Let (Np)p∈N and (Mp)p∈N be sequences of positive reals such that Mp ≺Np. We have for any Ω ⊂

open

Rd, h > 0

EMp(Ω) ⊂ E (Np)(Ω), DMp(Ω) ⊂ D(Np)(Ω)

and the injections are continuous.

Proof. By the previous corollary we have that the canonical injections

EMp(K)→ E (Np)(K) and DMp(K)→ D(Np)(K)

are continuous for all K ⊂comp

Ω. By taking the projective or inductive limit and applying

Corollary 1.2.10 or Corollary 1.3.8 we get the desired result.

2.3 Products of ultradifferentiable functions

Proposition 2.3.1. Let (Mp)p∈N be a sequence of positive reals, let K ⊂comp

Rd and let

h, k ∈ R+. If (Mp)p∈N is a log-convex, then the pointwise multiplication of functionsdefines continuous maps

EMp,h(K)× EMp,k(K) → EMp,h+k(K),

EMp,h(K)×DMp,k(K) → DMp,h+k(K) and

EMp,h(Td)× EMp,k(Td) → EMp,h+k(Td).

40

Proof. Let α ∈ Nd, ϕ ∈ EMp,h(K) and ψ ∈ EMp,k(K) be arbitrary. By the Leibniz rulewe find that

‖Dα(ϕψ)‖C(K) =

∥∥∥∥∥∥∥∥∥∑β∈Ndβj≤αj

(α1

β1

). . .

(αdβd

)(Dβϕ)(Dα−βψ)

∥∥∥∥∥∥∥∥∥C(K)

≤∑β∈Ndβj≤αj

(α1

β1

). . .

(αdβd

)‖Dβϕ‖C(K)‖Dα−βψ‖C(K)

≤∑β∈Ndβj≤αj

(α1

β1

). . .

(αdβd

)h|β|M|β|k

|α−β|M|α−β|‖ϕ‖EMp,h‖ψ‖EMp,k .

By using Lemma 2.1.3 we know that

M|β|M0

M|α−β|M0

≤(M|α|M0

) |β||α|(M|α|M0

) |α−β||α|

=M|α|M0

.

We also have that

∀j ∈ 1, . . . d : (h+ k)αj =∑

0≤βj≤αj

(αjβj

)hβjkαj−βj .

By combining these two we find

‖Dα(ϕψ)‖C(K) = (h+ k)|α|M0M|α|‖ϕ‖EMp,h‖ψ‖EMp,k .

We conclude that‖ϕψ‖EMp,h+k ≤M0‖ϕ‖EMp,h‖ψ‖EMp,k ,

which proves the continuity. The proposition follows by two additional observations: ifsuppψ ⊂ K, then supp (ϕψ) ⊂ K and the product of two periodic functions is periodic.

Theorem 2.3.2. Let K ⊂comp

Rd. If (M.1) is satisfied, then E∗(K) is a topological algebra

with respect to pointwise multiplication and addition of functions. In particular, E∗(Td)and D∗(K) are topological algebras.

Proof. The space E∗(K) is a topological vector space so we only need to prove thatpointwise multiplication of functions is continuous.If ∗ = (Mp), we can use Proposition 1.2.14 to write pointwise multiplication as a mapbetween projective limits of spaces

lim←−n→∞

(EMp,1/n(K)× EMp,1/n(K))→ lim←−n→∞

EMp,2/n(K).

41

By the previous proposition we know that for every n ∈ N pointwise multiplication is acontinuous map from EMp,1/n(K)×EMp,1/n(K) to EMp,2/n(K), so the theorem followsfrom Corollary 1.2.10.If ∗ = Mp, we can use Proposition 1.3.11 to write pointwise multiplication as a mapbetween inductive limits of spaces

lim−→n→∞

(EMp,n(K)× EMp,n(K))→ lim−→n→∞

EMp,2n(K). (2.3)

The previous proposition tells us that pointwise multiplication is a continuous map fromEMp,n(K) × EMp,n(K) to EMp,2n(K) for every n ∈ N. Both inductive limits define,by Proposition 1.3.17 and Definition 1.4.14, (DFS)-spaces. So the theorem follows fromRemark 1.3.21 and Corollary 1.3.19.We know by Proposition 1.4.15 that D∗(K) (and E(Td)) are closed topological subspacesof E∗(K) (for K = [0, 2π]d). So the theorem also holds for these subspaces.

Corollary 2.3.3. Let K ⊂comp

Rd. If (M.1) is satisfied, then D∗(K) is a topological E∗(K)-

module with respect to pointwise multiplication and addition of functions.

Proof. The space D∗(K) is a topological vector space so we only need to prove thatpointwise multiplication with functions from E∗(K) is continuous. By Proposition 1.4.15,the natural topology on D∗(K) coincides with the topology induced by E∗(K) ⊃ D∗(K).So by the previous theorem we know that pointwise multiplication D∗(K) × E∗(K) →E∗(K) is the restriction of a continuous map and thus continuous. Since the product of afunction with a function with support in K is a function with support in K, we see thatthe image of this map is contained in D∗(K).

Theorem 2.3.4. Let (Mp)p∈N be a sequence of positive reals. Let Ω ⊂open

Rd. If (M.1) is

satisfied, then E∗(Ω) and DMp(Ω) are topological algebras.

Proof. We know that these spaces are topological vector spaces, so we need to verifythe continuity of pointwise multiplication of functions in E∗(Ω) and DMp(Ω). Take asequence K1 ⊂

comp

K2 ⊂comp

· · · ⊂comp

Ω such that⋃∞n=1Kn = Ω.

By Theorem 2.3.2 we know that pointwise multiplication E∗(Kn)× E∗(Kn) → E∗(Kn) isa continuous map. It follows from Corollary 1.2.10 that pointwise multiplication

E∗(Ω)× E∗(Ω) = lim←−n→∞

(E∗(Kn)× E∗(Kn))→ lim←−m→∞

E∗(Km) = E∗(Ω)

is continuous.From Theorem 2.4.2 we know that DMp,n(Kn) × DMp,n(Kn) → DMp,2n(Kn) is con-tinuous for any n ∈ N+. We know by Proposition 1.3.17 and Theorem 1.4.18 thatDMp(Ω)×DMp(Ω) and DMp(Ω) are (DFS)-spaces. So by Remark 1.3.21 and Corollary1.3.19 we know that pointwise multiplication

DMp(Ω)×DMp(Ω) = lim−→n→∞

(DMp,n(Kn)×DMp,n(Kn))→ lim−→n→∞

DMp,2n(Kn)

is continuous.

42

2.4 Differential operators

The theory of ultradifferentiable functions is used when studying differential equations.For this reason we show the following results.

Lemma 2.4.1. For any sequence of positive reals (Mp)p∈N, h > 0 and K ⊂comp

Rd differen-

tiation of functions defines (for any α ∈ Nd) continuous linear maps

Dα : EMp,h(K) → EMp+|α|,h(K),

Dα : DMp,h(K) → DMp+|α|,h(K) and

Dα : EMp,h(Td) → DMp+|α|,h(Td).

Proof. For an arbitrary β ∈ Nd we have

‖DβDαϕ‖C(K) ≤ h|α|+|β|M|α|+|β|‖ϕ‖EMp,h(K).

So we get ‖Dβϕ‖EMp+|α|,h(K)≤ h|α|‖Dβϕ‖EMp,h(K), which proves the continuity of the

maps.

Proposition 2.4.2. For any sequence of positive reals (Mp)p∈N that satisfies (M.2’) (withgiven A,H > 0), any h > 0 and K ⊂

comp

Rd differentiation of functions defines (for any

α ∈ Nd) continuous linear maps

Dα : EMp,h(K) → EMp,H|α|h(K),

Dα : DMp,h(K) → DMp,H|α|h(K) and

Dα : EMp,h(Td) → DMp,H|α|h(Td).

Proof. We get by Lemma 2.1.16, that for any ϕ ∈ EMp+|α| and any β ∈ Nd

‖Dβϕ‖C(K) ≤ h|β|M|β|+|α|‖ϕ‖EMp+|α|,h(K)≤ A|α|H |α|

|α|−12 (hH |α|)|β|M|β|‖ϕ‖EMp+|α|,h(K)

,

which implies ‖ϕ‖EMp,H|α|h(K)

≤ A|α|H |α||α|−1

2 ‖ϕ‖EMp+|α|,h(K). So the natural inclusions

EMp+|α|,h(K) → EMp,H|α|h(K),

DMp+|α|,h(K) → DMp,H|α|h(K) and

EMp+|α|,h(Td) → DMp,H|α|h(Td)

are continuous maps. We conclude that the proposition follows from the previous lemma.

Theorem 2.4.3. Let K ⊂comp

Rd. If (M.2’) is satisfied, then differentiation of functions is

an endomorphism on E∗(K), on D∗(K) and on E∗(Td). In other words, for any α ∈ Nd

Dα : E∗(K)→ E∗(K), Dα : D∗(K)→ D∗(K) and Dα : E∗(Td)→ E∗(Td)

are continuous linear maps.

43

Proof. If E∗(K), D∗(K) or E∗(Td) is an inductive limit, then this follows from Proposition2.4.2 and Corollary 1.3.8.If E∗(K), D∗(K) or E∗(Td) is a projective limit, then this follows from Proposition 2.4.2and Corollary 1.2.10.

Theorem 2.4.4. Let Ω ⊂open

Rd. If (M.2’) is satisfied, then differentiation of functions is

an endomorphism on E∗(Ω) and on D∗(Ω). In other words, for any α ∈ Nd

Dα : E∗(Ω)→ E∗(Ω) and Dα : D∗(Ω)→ D∗(Ω)

are continuous linear maps.

Proof. The space E∗(Ω) is a projective limit, so the theorem follows from Theorem 2.4.3and Corollary 1.2.10.The space D∗(Ω) is an inductive limit, so the theorem follows from Theorem 2.4.3 andCorollary 1.3.8.

Definition 2.4.5. Let (Mp)p∈N be a sequence of positive reals. Let aα ∈ C : α ∈ Nd bearbitrary coefficients of the differential operator

P (D) =∑α∈Nd

aαDα.

We call P (D) an ultradifferential operator of class Mp if

(∀K ⊂comp

Ω)(∀L > 0)(∃B > 0)(∀α ∈ Nd)

(|aα| ≤ B

L|α|

M|α|

).

And P (D) is an ultradifferential operator of class (Mp) if

(∀K ⊂comp

Ω)(∃L > 0)(∃B > 0)(∀α ∈ Nd)

(|aα| ≤ B

L|α|

M|α|

).

To study such operators we introduce the following condition on (Mp)p∈N, which strength-ens (M.2’).

Definition 2.4.6 (Stability under ultradifferential operators). We impose the followingcondition on a sequence of positive reals (Mp)p∈N:

(∃A,H > 0)(∀p ∈ N)

(Mp ≤ AHp min

0≤q≤pMqMp−q

). (M.2)

Lemma 2.4.7. Let (Mp)p∈N be a sequence of positive reals. Assume that (M.2) is satisfiedand that this is witnessed by some A,H > 0. Let P (D) =

∑α aαD

α be an operator suchthat, for some L,B > 0,

∀α ∈ Nd : |aα| ≤ BL|α|

M|α|.

44

Let K ⊂comp

Rd and let h ∈ (0, (LH)−1). We have that for any ϕ ∈ EMp,h(K)

P (D)ϕ =∑α∈Nd

aαDαϕ

converges absolutely in the EMp,Hh(K)-norm. Therefore

P (D) : EMp,h(K)→ EMp,Hh(K)

is a continuous linear map.

Proof. Take any ϕ ∈ EMp,h(K) and any β, α ∈ Nd. We get

‖Dβ(aαDαϕ)‖C(K) = ‖aαDα+βϕ‖C(K) ≤ |aα|‖Dα+βϕ‖C(K)

≤ BL|α|

M|α|h|α+β|M|α+β|‖ϕ‖EMp,h(K).

By (M.2), we get for the given A,H > 0

‖Dβ(aαDαϕ)‖C(K) ≤ BL|α|h|α+β|AH |α+β|M|β|‖ϕ‖EMp,h(K).

Since this holds for any β ∈ Nd, we get

‖aαDαϕ‖EMp,Hh(K) ≤ BL|α|h|α|H |α|‖ϕ‖EMp,h(K)

for any α ∈ Nd. Since we have 1 > LhH, we can take the sum over α ∈ Nd. We get that∥∥∥∥∥∑α∈Nd

aαDαϕ

∥∥∥∥∥EMp,Hh(K)

≤ B‖ϕ‖EMp,h(K)

∑α∈Nd

L|α|h|α|H |α|

converges absolutely.

Theorem 2.4.8. Assume that (M.2) is satisfied. Let P (D) be an ultradifferential operatorof class ∗ and let K ⊂

comp

Rd. The operator P (D) : E∗(K) → E∗(K) is a continuous linear

operator. The restrictions of this operator

P (D) : D∗(K)→ D∗(K) and P (D) : E∗(Td)→ E∗(Td)are also continuous operators.

Proof. Take A,H > 0 which witness (M.2). Depending on which case ∗ represents wecan either take L > 0 such that LhH < 1 or we can take h > 0 such that LhH <1. The previous lemma applies in both cases. By taking the projective or inductivelimit and applying Corollary 1.2.10 or 1.3.8, we get that P (D) : E∗(K) → E∗(K) is acontinuous map. It follows from Proposition 1.4.15 that the restrictions of this operatorare continuous.

Theorem 2.4.9. Assume that (M.2) is satisfied. Let P (D) be an ultradifferential operatorof class ∗ and Ω ⊂

open

Rd. The operators

P (D) : E∗(Ω)→ E∗(Ω) and P (D) : D∗(Ω)→ D∗(Ω)

are continuous and linear.

Proof. This follows from the previous theorem by applying Corollary 1.2.10 or Corollary1.3.8.

45

2.5 Quasi-analyticity

We will now answer the following important question. When is D∗(K) trivial? If D∗(K)is trivial, then we call E∗(K) quasi-analytic.

Definition 2.5.1. We call a sequence (Mp)p∈N non-quasi-analytic if

∞∑p=1

1

mp

=∞∑p=0

Mp

Mp+1

<∞. (M.3’)

Theorem 2.5.2. Let (Mp)p∈N be a log-convex sequence of positive reals. The followingare equivalent:

(a)∞∑p=1

1

mp

<∞;

(b)

∫ ∞0

m(λ)

λ2dλ <∞;

(c)

∫ ∞0

M(ρ)

ρ2dρ <∞;

(d)∞∑p=1

1p√Mp

<∞.

Proof. (a) ⇒ (b). We have, using the Lebesgue’s Monotone Convergence Theorem

∞ >∞∑p=1

1

mp

=∞∑p=1

∫ ∞0

χ[mp,∞)(λ)

λ2dλ =

∫ ∞0

m(λ)

λ2dλ.

(b) ⇒ (c). By Proposition 2.1.13 we have∫ ∞0

M(ρ)

ρ2dρ =

∫ ∞0

∫ ρ

0

m(λ)

λρ2dλdρ =

∫ ∞0

∫ ∞λ

m(λ)

λρ2dρdλ =

∫ ∞0

m(λ)

λ2dλ <∞.

(c) ⇒ (d). Take p ∈ N. From Lemma 2.1.3 one gets that p√Mp/M0 ≤ p+1

√Mp+1/M0. For

e p√Mp ≤ ρ we have

M(ρ) ≥ logρpM0

Mp

= p logρ

p√Mp

+ logM0 ≥ p log e+ logM0 = p+ logM0.

46

From this we conclude that for any P ∈ N, P > 1∫ ∞0

M(ρ)dρ

ρ2≥

∫ ∞eM1

M(ρ)− logM0

ρ2dρ+

∫ ∞eM1

logM0

ρ2dρ

≥∫ e P

√MP

eM1

M(ρ)− logM0

ρ2dρ+

∫ ∞e P√MP

M(ρ)− logM0

ρ2dρ+

logM0

eM1

≥P−1∑p=1

∫ e p+1√Mp+1

e p√Mp

pdρ

ρ2+

∫ ∞e P√MP

Pdρ

ρ2+

logM0

eM1

=P−1∑p=1

p

(1

e p√Mp

− 1

e p+1√Mp+1

)+

P

e P√MP

+logM0

eM1

=1

e

P∑p=1

1p√Mp

+logM0

eM1

.

So (c) implies (d).

(d) ⇒ (a). From Lemma 2.1.3 one gets that ∀p ∈ N+ : Mp−1/M0 ≤ (Mp/M0)p−1p . So we

get for any p ∈ N+

1

mp

=Mp−1

Mp

≤M1− p−1

p

0 Mp−1p−1

p = M− 1p

p M1p

0 .

So the series in (a) is dominated by the series in (d) in the following way

∞∑p=1

1

mp

≤ max1,M0∞∑p=1

1p√Mp

.

Lemma 2.5.3. If a sequence of positive reals (Mp)p∈N satisfies (M.3’), then the sequencedefined by N0 = 1 and

∀p ∈ N+ : Np = Np−1mp

√√√√ ∞∑q=p

1

mq

also satisfies (M.3’) and Np ≺Mp.

Proof. Define for all p ∈ N+

np =Np

Np−1

= mp

√√√√ ∞∑q=p

1

mq

to be the be the sequence of ratios for the given sequence (Np)p∈N. Take any p ∈ N+.

47

Observe that

0 ≤

√√√√ ∞∑q=p

1

mq

−

√√√√ ∞∑q=p+1

1

mq

2

=∞∑q=p

1

mq

− 2

√√√√ ∞∑q=p

1

mq

√√√√ ∞∑q=p+1

1

mq

+∞∑

q=p+1

1

mq

= 2∞∑q=p

1

mq

− 2

√√√√ ∞∑q=p

1

mq

√√√√ ∞∑q=p+1

1

mq

− 1

mp

.

From this we see that

1

np=

1

mp

√∑∞q=p

1mq

≤ 2

√√√√ ∞∑q=p

1

mq

−

√√√√ ∞∑q=p+1

1

mq

.

So since∑∞

p=1m−1p converges, we have

∞∑p=1

1

np≤ lim

P→∞

P∑p=1

1

np≤ 2

√√√√ ∞∑q=1

1

mq

− 2 limP→∞

√√√√ ∞∑q=P+1

1

mq

<∞.

This proves that (M.3’) holds for (Np)p. It remains to prove that Np ≺Mp. Let L > 0 bearbitrary. Take P ∈ N+ such that √√√√ ∞∑

q=P

1

mq

< L.

For p ≥ P we have that np ≤ Lmp. We take C = maxp≤PNp

MpLp. We have for p ≤ P

Np ≤ CLpMp and for p ≥ P we find the same inequality by induction:

Np = Np−1np ≤ CLp−1MpLmp = CLpMp.

Theorem 2.5.4 (Denjoy-Carleman). Let K ⊂comp

Rd with non-empty interior. If (Mp)p∈N

satisfies (M.3’), then we have 0 6= D(Mp)(K) ⊂ DMp(K). Moreover we can takeϕ ∈ D(Mp)(K) and ∅ 6= Ω ⊂

open

K such that ϕ|Ω ≡ 1.

Conversely, if (Mp)p∈N satisfies (M.1), then D(Mp)(K) 6= 0 implies that (Mp)p∈N satisfies(M.3’). Moreover we have∫ ∞

0

log(M(ρ))

1 + ρ2dρ <∞ for M(ρ) =

∞∑k=0

M0ρk

Mk

. (2.4)

48

Proof. Using Lemma 2.5.3, take a sequence (Np)p∈N which satisfies (M.3’) such that Np ≺Mp and N0 = 1. Let (np)p∈N+ be its sequence of ratios. Set for k ∈ N

Rk =∞∑

p=k+1

n−1p ∈ [0,∞).

Note that Rk → 0 for k →∞. By translation and scaling (see Proposition 1.4.16) we canassume without loss of generality that [−3R0, 3R0]d ⊂ K.Take ϕ0 ∈ D(Rd) positive and with support in [−2R0, 2R0]d such that ϕ0|[−2R0,2R0]d ≡ 1.Set for p ∈ N+, x ∈ Rd

ϕp(x) =(np

2

)d ∫Ip

ϕp−1(x+ t)dt where Ip = [−n−1p , n−1

p ]d.

It is clear that this sequence is in D(Rd). We want to prove that (ϕp)p∈N converges inD(Rd). Before we prove this, we observe that for p ∈ N, α ∈ Nd and x ∈ Rd

Dαϕp(x) =(np

2

)d ∫Ip

Dαϕp−1(x+ t)dt. (2.5)

By the mean value theorem we get

Dαϕp(x) = Dαϕp−1(x+ ξp,α(x)) for some ξp,α(x) ∈ Ip = [−n−1p , n−1

p ]d. (2.6)

Note that this implies that ‖Dαϕp‖C(Rd) ≤ ‖Dαϕp−1‖C(Rd) and thus, by iterating thisrelation, we get for q ≤ p

‖Dαϕp‖C(Rd) ≤ ‖Dαϕq‖C(Rd). (2.7)

We will now prove that (ϕp)p∈N is a Cauchy sequence in D(Rd).First we see that this is indeed a sequence in D(Rd). Using (2.6) for α = 0 we get thatsupp (ϕp) ⊆ x : |x − n−1

p | ∈ supp (ϕp−1). Since supp (ϕ0) ⊆ [−2R0, 2R0]d we get byinduction

supp (ϕp) ⊆

[−

p∑q=1

n−1q − 2R0, 2R0 +

p∑q=1

n−1q

]d.

We conclude that all ϕp have support in I = [−3R0, 3R0]d.Secondly, we prove that the sequence is Cauchy in D(I), i.e. we prove that it is Cauchyfor all the supremum norms of all derivatives on I. By iterating (2.6) we get for α ∈ Nd,q, p ∈ N and x ∈ I

|Dαϕp+q(x)−Dαϕp(x)| =

∣∣∣∣∣Dαϕp

(x+

q−1∑j=0

ξp+q−j,α(x)

)−Dαϕp(x)

∣∣∣∣∣for some ξp+q−j,α(x) ∈ Ip+q−j for j ∈ 0, ..., q − 1. By the mean value theorem (forderivatives in Rd) and (2.7) we conclude

|Dαϕp+q(x)−Dαϕp(x)| ≤d∑

k=1

‖DαDkϕp‖C(I)

∥∥∥∥∥q−1∑j=0

ξp+q−j,α(x)

∥∥∥∥∥≤

d∑k=1

‖DαDkϕ0‖C(I)d

p+q∑j=p+1

n−1j .

49

It follows that for α ∈ Nd, q ∈ N

‖Dαϕp+q −Dαϕp‖C(R) ≤ d

d∑k=1

‖DαDkϕ0‖C(R)Rp → 0 for p→∞.

We conclude that (ϕp)p∈N is a Cauchy sequence in the complete space D(I). Set ϕ =limp→∞ ϕp. It remains to prove that 0 6= ϕ ∈ DNp(I) ⊂ D(Mp)(I). By (2.5) for |α| = 1(assume αj = 1) we get for p ∈ N+ and x ∈ Rd

|Dαϕp(x)| =np2

∣∣∣∣∣(np2 )d−1∫

[−n−1p ,n−1

p ]d−1

ϕp−1(x+ rpα)− ϕp−1(x− rpα)dxj

∣∣∣∣∣≤ Np

Np−1

‖ϕp−1‖C(I)

where xj stands for (x1, . . . xj−1, xj+1, . . . , xd). By iterating this result and combining itwith (2.7), we get for all α ∈ Nd, q ∈ N, q ≥ |α|

‖Dαϕq‖C(I) ≤ ‖Dαϕ|α|‖C(I) ≤Np

Np−1

· · · N1

N0

‖ϕ0‖C(I) = Np‖ϕ0‖C(I).

We conclude that, for α ∈ N, ‖Dαϕ‖C(I) ≤ Np‖ϕ0‖C(I) and thus ϕ ∈ DNp(I) ⊂ D(Mp)(I).By iterating (2.6), for α = 0, p times (for any p ∈ N) we get for x ∈ [−R0, R0]d

ϕp+1(x) = ϕ0(x+ ξ(x)) for some ξ(x) ∈ [−R0 +Rp, R0 −Rp]d ⊂ [−R0, R0]d.

So, since ϕ0|[−2R0,2R0]d ≡ 1, we conclude that ϕp|[−R0,R0]d ≡ 1 for all p ∈ N. This impliesϕ|[−R0,R0]d ≡ 1.

Conversely, assume we have ϕ ∈ D(Mp)(K) non zero. By Theorem 2.3.2, we know thatϕϕ ∈ D(Mp)(K). By restricting ϕϕ to a suitable line in Rd and interpreting this restrictionas a function on R, we have 0 6= ψ ∈ D(Mp)([−a, a]) for some a > 0. In particular, wehave for some C > 0

∀p ∈ N : ‖Dpψ‖∞ ≤ C2−pMp.

For z ∈ C, set

f(z) =

∫ a

−aψ(t)eiztdt. (2.8)

Note that since ψ is positive and non zero, we have that f(i) > 0. By partial integrationwe get for x ∈ R

f(x) = (−1)p(ix)−p∫ a

−a(Dpg)(t)eixtdt.

So we get

|xpf(x)| =∣∣∣∣∫ a

−a(Dpψ)(t)eixtdt

∣∣∣∣ ≤ ∫ a

−a|ψ(t)|dt ≤ C2−p+1a ·Mp.

We conclude that

M(x)|f(x)| = M0

∞∑p=0

|xpf(x)|Mp

≤M04aC.

50

From this we get that∫ ∞0

(log M(x) + log |f(x)|) dx

1 + x2=

∫ ∞0

log(M(x)|f(x)|) dx

1 + x2≤ log(M04aC)π

2. (2.9)

Note that since f : C→ C is continuous (by using that ‖ψ‖C(R) < M0 and Fatou’s Lemma[10, p. 23]), we can apply Morera’s Theorem [10, p. 208] to (2.8) and, since we can exchangeintegrals, conclude that f is entire. The transformation x = (i− ieiθ)/(1+eiθ) = tan(θ/2)gives us ∫ ∞

−∞log |f(x)| dx

1 + x2=

1

2

∫ π

−πlog

∣∣∣∣f (i− ieiθ1 + eiθ

)∣∣∣∣ dθ =1

2

∫ π

−πlog |g(eiθ)|dθ

where

g(z) = f

(i− iz1 + z

).

Note that f is entire and bounded by 2aC on z ∈ C : Imz ≥ 0. So we have that gis homomorphic on z ∈ C : |z| < 1 and g is bounded on z ∈ C : |z| ≤ 1, z 6= −1.These two properties, combined with g(0) = f(i) 6= 0 allow us to apply Fatou’s Lemmaand Jensen’s formula [10, p. 307]:∫ ∞

−∞log |f(x)| dx

1 + x2≥ 1

2limr→1

∫ π

−πlog |g(reiθ)|dθ

=π

2limr→∞

(log |g(0)|+

Nr∑n=1

logr

|xn,r|

)> −∞.

Where xn,r are the zero’s of g in z : |z| < r, which implies that log(r/|xn,r|) > 0.Combining this with (2.9) we find (2.4). Since log(M(ρ)) ≥ M(ρ) for all ρ ∈ R+, weconclude by Theorem 2.5.2 that (M.3’) holds.

51

Chapter 3

On the assumption (M.1)

In this chapter we will investigate whether or not one can, in general, replace (Mp)p∈Nby (M c

p)p∈N without changing the corresponding space of ultradifferentiable functions andthe topology on it. We start with some preliminaries.

3.1 Inequalities of Landau-Kolmogorov type and

Chebychev polynomials

We will state some inequalities for an interval I of the following type. For fixed k < n ∈ Nthere is a constant Ck,n ∈ R+ such that

(∀f ∈ Cn(I))(‖Dkf‖C(I) ≤ Ck,n‖f‖1−k/nC(I) ‖D

kf‖k/nC(I)).

Different versions of such inequalities have been proven by E. Landau, A. Gorny, H.Cartan and A. N. Kolmogorov. A detailed discussion of the history of these inequalitiescan be found in the first chapter of [8].

A proof of the following theorem by A. N. Kolmogorov can be found in [3, p. 255] or in[7, p. 211].

Theorem 3.1.1 (Kolmogorov, 1939). For any k < n ∈ N, we have Ck,n ∈ (1, π/2) suchthat ∀f ∈ Cn(R)

‖Dkf‖C(R) ≤ Ck,n‖f‖1−k/nC(R) ‖D

kf‖C(R).

Remark 3.1.2. Kolmogorov also gives an exact expression for Ck,n and proves a converseof this theorem.

The following two theorems are proven by A. Gorny in [1]. In [7, p. 219] one can find aproof which gives slightly different coefficients.

Theorem 3.1.3 (Gorny, 1939). For any k < n ∈ N and for any closed, finite interval Iof length 2δ and with midpoint µ we have ∀f ∈ Cn(I)

|Dkf(µ)| ≤ 16(2e)k‖f‖1−k/nC(I) max

‖Dkf‖C(I), ‖f‖C(I)n!δ−n

k/n.

52

Theorem 3.1.4 (Gorny, 1939). For any k < n ∈ N and for any closed, finite interval Iof length 2δ we have ∀f ∈ Cn(I)

‖Dkf‖C(I) ≤ 4e2k(nk

)k‖f‖1−k/n

C(I) max‖Dkf‖C(I), ‖f‖C(I)n!δ−n

k/n.

We will not use the latter theorem explicitly. It has been proven that the coefficient (n/k)k

cannot be improved [12]. From this one can deduce that one can, in general, not replace(Mp)p∈N by (M c

p)p∈N when one looks at ultradifferentiable functions on a compact interval.To see this we will construct explicit counterexamples. Since Chebychev polynomials areused to prove this result, we will also use them to construct the counterexamples. Wewill state some results about Chebychev polynomials. Proofs can be found in [9, p. 199].These results are due to R. J. Duffin and A. C. Schaeffer.

Definition 3.1.5. The Chebyshev polynomials are defined as

Tn : x 7→ cos(n arccos(x)) for n ∈ N

or by recursion: for all x ∈ R we define T0(x) = 1, T1(x) = x and

Tn+2(x) = 2xTn+1(x)− Tn(x).

Proposition 3.1.6. The derivatives of the Chebychev polynomials satisfy the followingproperty for all k, n ∈ N

‖DkTn‖C([−1,1]) = DkTn(1).

Proposition 3.1.7. The derivatives of the Chebychev polynomials satisfy the followingproperty for all 0 ≤ k ≤ n ∈ N

DkTn(1) =n2(n2 − 12) · · · (n2 − (k − 1)2)

1 · 3 · · · (2k − 1)= 2k

k!

(2k)!n2(n2 − 12) · · · (n2 − (k − 1)2).

We will use this property in the following form.

Corollary 3.1.8. The derivatives of the Chebychev polynomials satisfy the following prop-erty for all k ≤ n ∈ N

(e2k)−kn2k ≤ DkTn(1) ≤ 2kk−kn2k.

Proof. Take any n ∈ N. Tn(1) = 1, so the statement is true for k = 0. Let k ∈ N+. It is

clear that 2k k!(2k)!≤ 2k

kkand n2(n2 − 12) · · · (n2 − (k − 1)2) ≤ n2k. This gives us

DkTn(1) ≤ 2kk−kn2k.

For the other inequality we first observe that for all k ∈ N+we have that (2k)! ≤ k!(2k)k

and that, by Stirling’s approximation, e−2k+1k2k−1 ≤ (2k−1)!. The last inequality impliesthat for all k ∈ N+

e−2k ≤ (2k − 1)!

k2k−1=

(k2 − 12

k2

)(k2 − 22

k2

)· · ·(k2 − (k − 1)2

k2

).

53

So we find(1

e2k

)kn2k ≤ 2k

k!

(2k)!n2k

(1− 12

k2

)(1− 22

k2

)· · ·(

1− (k − 1)2

k2

)≤ 2k

k!

(2k)!(n2(n2 − 12) · · · (n2 − (k − 1)2)) = DkTn(1).

3.2 Two counterexamples

We will first see some examples in which EMp(Ω) 6= EMcp(Ω) for some open Ω ⊂ Rd. We

construct an explicit example for Ω = R. It is trivial to generalize this to Rd (by takingthe same function and seeing it as a function on Rd which is constant in d− 1 variables).One can also restrict it to smaller (open) subsets of Rd.

Proposition 3.2.1. For s ∈ (0, 1) we have a sequence of positive reals (Mp)p∈N such that(M c

p)p∈N = (1)p∈N and a function

ϕ ∈ EMp(R) \ Ep!s(R).

Proof. Let 1 < c ∈ N and let

ϕ : R→ R : x 7→∞∑n=1

xcn

(cn)cn−1+1.

Sincelimn→∞

cn√cncn−1+n = lim

n→∞cnc−1 cn√cn = lim

n→∞cnc−1

=∞,

we conclude that the convergence radius of such a function is ∞. So ϕ is an analyticfunction on R. Take any sequence of positive reals (Mp)p such that ϕ ∈ EMp(R).(Explicitly we can take Mp = ‖Dpϕ‖C([−p,p]) or we can apply Pringsheim’s Theorem [4,

p. 34] and take Mp = p!.) Note that by Stirling’s approximation

D(ck)ϕ(0) =(ck)!

(ck)ck−1+1≥ e−(ck)c−k

(ck)(ck)

(ck)ck−1 ≥ (2e)−(ck)(ck)ck−1(c−1)

≥ (2e)−(ck)(

(ck)ck)(c−1)/c

≥ (2e)−(ck)((ck)!)(c−1)/c.

So for 1 < c ∈ N such that 1− 1/c > s we have ϕ 6∈ Ep!s(R) (see Example 2.2.8). Takethe sequence

Mp =

1 if p = 0 or p = cn + 1 for some n ∈ NmaxMp, 1 else

.

It is clear that M cp = 1 for all p ∈ N. To see that ϕ ∈ EMp(R) it remains to prove that

for any interval [−a, a] ⊂ R (a > 0) we have C, h ∈ R+ such that

∀k ∈ N :∥∥∥D(ck+1)ϕ

∥∥∥ ≤ Chck+1. (3.1)

54

For [−a, a] ⊂ R (a > 0) let N ∈ N be such that ac

cN≤ 1/2. For N < k < n we find for all

x ∈ [−a, a]∣∣∣∣D(ck+1) x(cn)

(cn)cn−1+1

∣∣∣∣ ≤ |x|(cn−ck−1) (cn)ck+1

(cn)cn−1+1≤ a(cn−ck−1)(cn)c

k−cn−1

≤ ack(c−1)−1(ac)c

n−1−ck(cn)ck−cn−1

≤ a−1(ac−1)ck(c−1)

(ac

cN

)cn−1−ck

≤ a−1(a(c−1)2)ck(

1

2

)cn−1−ck

.

Summing all the terms, we get for all k > N and for all x ∈ [−a, a]∣∣∣D(ck+1)ϕ(x)∣∣∣ ≤ ∞∑

n=k+1

|D(ck+1)x(cn)|(cn)cn−1+1

≤ a−1(a(c−1)2)ck∞∑m=0

2−m = 2a−1(a(c−1)2)ck

.

So if we take

C = maxk≤N‖D(ck+1)ϕ‖C([−a,a]) + 2a−1 and h = a(c−1)2 + 1,

then we see that (3.1) holds.

This provides us with counterexamples for sequences (Mp)p∈N that increase slowly. Wewill later prove that on Ω ⊂

open

Rd this problem does not occur for p! ⊂Mp. On a compact

interval we will not have such a result. The following proposition shows why.

Proposition 3.2.2. For any sequence of positive reals (Mp)p∈N we have a sequence ofpositive reals (Np)p∈N such that (N c

p)p∈N = (1)p∈N and a function

ϕ ∈ ENp([−1, 1]) \ EMp([−1, 1]).

Proof. Without loss of generality we can make Mp larger for all p ∈ N and thus we canassume that Mp ∈ N for all p ∈ N. Define recursively k0 = 2 and for n ∈ N

µn = kknn maxm≤nMkm

kn+1 = 2kµnn + n+ 3.

Note that (kn)n∈N is an increasing sequence larger then (n)n∈N. We denote Kn = kµnn andwe define

ϕ : [−1, 1]→ R : x 7→∞∑n=0

TKn(x)

Kknn

where Tk denotes the k-th Chebychev polynomial. For q ∈ N we have∥∥∥∥∥∞∑n=0

DqTKnKknn

∥∥∥∥∥C([−1,1])

≤∞∑n=0

DqTKn(1)

Knn

≤ 2qq−q∞∑n=0

K2qn

Knn

≤ 2qq−q

(2q∑n=0

K2q−nn +

∞∑n=2q+1

n2q−n

).

55

We conclude (by the root test) that the series that defines ϕ converges absolutely inE([−1, 1]). We define the sequence (Np)p∈N by

Np =

1 if p = 0 or p = Kn + 1 for some n ∈ Nmax‖Dpϕ‖C[−1,1], 1 else

.

It is clear that (N cp)p = (1)p. To prove that ϕ ∈ ENp([−1, 1]), it suffices to look at the

Kp + 1-th derivative of ϕ for all p ∈ N. Observe that for p < n ∈ N∥∥∥∥DKp+1TKnKknn

∥∥∥∥C([−1,1])

≤∣∣∣∣DKp+1TKn(1)

Kknn

∣∣∣∣ ≤ ( 2

Kp + 1

)Kp+1K

2Kp+2n

Kknn

≤ K2Kp+2−knn ≤ K2Kn−1+2−kn

n = K−nn ≤ 2−n.

For n ≤ p we have Kn < Kp + 1, so we get

∥∥D(Kp+1)ϕ∥∥C([−1,1])

=

∥∥∥∥∥∞∑

n=p+1

DKp+1TKnKknn

∥∥∥∥∥C([−1,1])

≤∞∑

n=p+1

2−n ≤ 1.

It follows that ϕ ∈ ENp([−1, 1]).We will now prove that ϕ /∈ EMp([−1, 1]). Take p ∈ N, we observe that∥∥∥∥∥DkpTKp

Kkpp

∥∥∥∥∥ ≥∣∣∣∣∣DkpTKp(1)

Kkpp

∣∣∣∣∣ ≥(

1

e2kp

)kp K2kpp

Kkpp

=

(1

e2

)kpk(µp−1)kpp

≥(

1

e2

)kp2(µp−1) ≥ 1

2

(1

e2

)kp2k

kpp Mkp .

For p > n, we have Kn < kp and thus DkpTKn = 0. For p < n we get∥∥∥∥DkpTKnKknn

∥∥∥∥C([−1,1])

≤∣∣∣∣DkpTKn(1)

Kknn

∣∣∣∣ ≤ ( 2

kp

)kp K2kpn

Kknn

≤ Kkn−nn

Kknn

≤(

1

2

)n.

Combining these results, we find∥∥Dkpϕ∥∥C([−1,1])

≥

∣∣∣∣∣DkpTKp(1)

Kkpp

∣∣∣∣∣−∣∣∣∣∣∞∑

n=p+1

DkpTKp(1)

Kkpp

∣∣∣∣∣≥ 1

2

(1

e2

)kp2k

kpp Mkp −

∞∑n=p+1

2−n ≥ 1

2

(1

e2

)kp2k

kpp Mkp − 1.

Since Mp ≺ 2ppMp , we conclude that ϕ 6∈ EMp([−1, 1]).

3.3 Positive results

In this section we will study the cases in which the spaces of ultradifferentiable functionsassociated with a sequence (Mp)p∈N coincide with the spaces associated with (M c

p)p∈N.

Recall that in order to define (M cp)p∈N, we require that infp∈N+ M

1pp > 0, equivalently

1 ⊂Mp.

56

Lemma 3.3.1. Let (Mp)p∈N be a sequence of positive reals and let (M cp)p∈N be its greatest

log-convex minorant. Let C = r−q√Mr/M c

q for some q < r ∈ N. Let

Np =

M c

p if p ≤ q

M cqC

p−q if p > q.

We have that (Np)p∈N is log-convex.

Proof. Proving that (Np)p∈N log-convex is equivalent to proving that ∀p ∈ N+ : Np/Np−1 ≤Np+1/Np. For p < q this follows by the convexity of (M c

p)p∈N. For p > q we haveNp/Np−1 = C = Np+1/Np. It remains to prove that Nq/Nq−1 ≤ Nq+1/Nq. We get fromLemma 2.1.3 that

(M cq )r−q+1 ≤M c

r (Mcq−1)r−q.

This implies that

Nq

Nq−1

=M c

q

M cq−1

≤(M c

r

M cq

) 1r−q

≤ C =Nq+1

Nq

.

Proposition 3.3.2. Let (Mp)p∈N be a sequence of positive reals such that 1 ≺ Mp, i.e.limp→∞

p√Mp =∞. There is a sequence (pj)j∈N of natural numbers such that Mpj = M c

pj

for all j ∈ N.

Proof. We prove this by contradiction, assume q ∈ N maximal such that M cq = Mq (there

is at least one such q because M0 = M c0). Let

C = infp>q

p−q

√Mp

M cq

.

Define (Np)p∈N by

Np =

M c

p if p ≤ q

M cqC

p−q if p > q.

If we have a r > q such that

C = r−q

√Mr

M cq

,

then it follows from the previous lemma that (Np)p∈N is log-convex. By our choice of Cwe have that Np ≤ Mp for all p ∈ N, this implies that Np ≤ M c

p for all p ∈ N. Thiscontradicts the fact that Nr = Mr > M c

r . We conclude that we cannot have such a r. Sowe have q < q1 < q2 < . . . such that

C = limj→∞

qj−q

√Mqj

M cq

.

57

It follows that

C = limj→∞

qj−q√Mqj

qj−q

√1

M cq

= limj→∞

Mqj

1qj

qjqj−q = lim

j→∞Mqj

1qj.

This contradicts limp→∞p√Mp =∞.

Definition 3.3.3. We call the increasing sequence (pj)j∈N of all natural numbers suchthat Mpj = M c

pjfor all j ∈ N, the sequence of principal indexes of (Mp)p∈N.

Remark 3.3.4. The sequence of principal indexes can also exist for (Mp)p∈N when 1 6≺Mp. For example when Mp ≥ 1 for all p ∈ N and Mpj = 1 for some infinite sequencep0 < p1 < p2 < . . . .

Lemma 3.3.5. Let (Mp)p∈N be a sequence of positive reals. Assume that (Mp)p∈N has asequence of principal indexes (pj)j∈N. We have for pj ≤ q < p ≤ pj+1

M cp = M c

q

(Mpj+1

M cq

) p−qpj+1−q

.

Proof. Fix a principal index pj and its successor pj+1. By Lemma 2.1.3 we know that

M cq ≤ (M c

pj)pj+1−qpj+1−pj (M c

pj+1)

q−pjpj+1−pj = Mpj

(Mpj+1

Mpj

) q−pjpj+1−pj

for any pj ≤ q ≤ pj+1. We now want to prove the opposite inequality. Take

A = minpi<r≤pi+1

r−pj

√Mr

Mpj

.

Assume that this minimum is reached for pj < r ≤ pj+1. It follows from Lemma 3.3.1that the sequence defined by

Np =

M c

p if p ≤ r

M cpjAp−pj if r < p

is log-convex. By construction (Np)p∈N is smaller then (Mp)p∈N and passes trough Nr =Mr. If r 6= pj+1, then this equality violates the maximality of (M c

p)p. Thus we conclude

that the minimal such A can only be pj+1−pj√Mpj+1

/Mpj . Since (Np)p is still a log-convexminorant of (Mp)p for this A, we find that

M cq ≥ Nq = Mpj

(Mpj+1

Mpj

) q−pjpj+1−pj

for any pj ≤ q ≤ pj+1. So we have found that (M cp)p is of the form

M cq = Mpj

(Mpj+1

Mpj

) q−pjpj+1−pj

for pj ≤ q ≤ pj+1. Using this twice, we get the result we wanted.

58

Remark 3.3.6. The previous lemma expresses that (log(M cp))p is linear on the segments

pj ≤ q ≤ pj+1, for all principal indexes pj.

Theorem 3.3.7. Let K ⊂ Rd, r, h > 0 and let (Mp)p∈N be a sequence of positive realswith a sequence of principal indexes (pj)j∈N. We have

DMp,h(K) = DMcp,h(K)

andEMp,h(Tdr) = EMc

p,h(Tdr).

In both cases the equalities have to be interpreted as equivalences between normed spaces,thus the sets are equal and the norms are equivalent.

Proof. Since M cp ≤Mp for all p ∈ N it is clear that

DMp,h(K) ⊇ DMcp,h(K) and EMp,h(Tdr) ⊇ EM

cp,h(Tdr).

Moreover, the inclusions are continuous (see Proposition 2.2.3). Take ϕ ∈ DMp,1(K) orϕ ∈ EMp,1(Tdr) with norm 1. In both cases we can see ϕ as a function on Rd either bydefining it to be 0 outside K or by seeing it as a periodic function. For α ∈ Nd take theprincipal index pj such that pj ≤ |α| < pj+1. Take γ ∈ Nd such that |α − γ| = pj andβ ∈ Nd such that

∀n ∈ 1, . . . d : pj+1 = βn + |α| −n−1∑k=1

γk.

To simplify notation we define for n ∈ 1, . . . , d

µn =γn

βn + γn.

By applying Theorem 3.1.1 to

x 7→ Dα1−γ11 Dα2

2 · · ·Dαdd ϕ(x, x2, . . . , xd) for all x2, . . . , xd ∈ R,

x 7→ Dα1−γ11 Dα2−γ2

2 Dα33 · · ·D

αdd ϕ(x1, x, x3, . . . , xd) for all x1, x3, . . . , xd ∈ R,

...

x 7→ Dα1−γ11 Dα2−γ2

2 · · ·Dαd−γdd ϕ(x1, . . . , xd−1, x) for all x1, . . . , xd−1 ∈ R

59

we get:

‖Dαϕ‖C(Rd) = ‖Dα11 · · ·D

αdd ϕ‖C(Rd)

≤ π

2‖Dα1−γ1

1 Dα22 · · ·D

αdd ϕ‖

1−µ1C(R)‖D

α1+β11 Dα2

2 · · ·Dαdϕ‖µ1C(Rd)

≤ π

2

(π

2‖Dα1−γ1

1 Dα2−γ22 Dα3

3 · · ·Dαdd ϕ‖

1−µ2C(Rd)

·‖Dα1−γ1Dα2+β22 Dα3

3 · · ·Dαdd ϕ‖

µ2C(Rd)

)1−µ1Mµ1

pj+1

≤(π

2

)2

‖Dα1−γ11 Dα2−γ2

2 Dα33 · · ·D

αdd ϕ‖

(1−µ2)(1−µ1)

C(Rd)Mµ2(1−µ1)

pj+1Mµ1

pj+1

=(π

2

)2

‖Dα1−γ11 Dα2−γ2

2 Dα33 · · ·Dαdϕ‖(1−µ2)(1−µ1)

C(Rd)Mµ2(1−µ1)+µ1

pj+1

≤ . . .

≤(π

2

)d‖Dα−γϕ‖(1−µd)···(1−µ1)

C(Rd)Mµd(1−µd−1)···(1−µ1)+···+µ2(1−µ1)+µ1

pj+1

≤(π

2

)dM (1−µd)···(1−µ1)

pjMµd(1−µd−1)···(1−µ1)+···+µ2(1−µ1)+µ1

pj+1.

If we compute that

(1− µd) · · · (1− µ1) =d∏

k=1

(1− γk

pj+1 − |α|+ γ1 + · · ·+ γk

)

=d∏

k=1

(pj+1 − |α|+ γ1 + · · ·+ γk−1

pj+1 − |α|+ γ1 + · · ·+ γk

)=

pj+1 − |α|pj+1 − |α|+ γ1 + · · ·+ γd

=pj+1 − |α|pj+1 − pj

and

µd(1− µd−1) · · · (1− µ1) + · · ·+ µ2(1− µ1) + µ1 =∑

∅6=S⊂1,...,d

−∏s∈S

(−µs)

= 1−∑

S⊂1,...,d

∏s∈S

(−µs)

= 1− (1− µd) · · · (1− µ1)

=|α| − pjpj+1 − pj

,

we can conclude by Lemma 3.3.5 that

M (1−µd)···(1−µ1)pj

Mµd(1−µd−1)···(1−µ1)+···+µ2(1−µ1)+µ1pj+1

= M c|α|.

So we find that for all α ∈ Nd

‖Dαϕ‖C(Rd) ≤(π

2

)dM c|α|.

60

If we take ϕ ∈ DMp,1(K) arbitrary and apply this to ϕ‖ϕ‖EMp,1(K)

, we get

∀ϕ ∈ DMp,1(K), ∀α ∈ Nd :‖Dαϕ‖C(K)

M c|α|

≤(π

2

)d‖ϕ‖EMp,1(K).

Analogously, we get

∀ϕ ∈ EMp,1(Tdr),∀α ∈ Nd :‖Dαϕ‖C(Tdr)

M c|α|

≤(π

2

)d‖ϕ‖EMp,1(Tdr).

This concludes the proof for h = 1 (and for arbitrary K ⊂comp

Rd, r > 0) by taking the supre-

mum over all α ∈ Nd. For general h > 0 the theorem follows for the spaces DMp,h(K)and EMp,h(Tdr) by applying Lemma 1.4.8.

Corollary 3.3.8. For any sequence of positive reals (Mp)p∈N such that (M cp)p∈N is defined,

we haveEMp(Td) = EMc

p(Td) and E (Mp)(Td) = E (Mcp)(Td).

And for any K ⊂comp

Rd we have

DMp(K) = DMcp(K) and D(Mp)(K) = D(Mc

p)(K).

Proof. We first prove this in the case that (Mp)p∈N has a sequence of principal indexes(pj)j∈N. The previous theorem says that for every h ∈ R+ the space EMp,h(Td) isisomorphic to EMc

p,h(Td) and DMp,h(K) is isomorphic to DMcp,h(K), so we conclude

that the projective and inductive limits of these spaces are also isomorphic.If (Mp)p∈N does not have a sequence of principal indexes, then we know that 1 6≺Mp. Sowe have that lim infp→∞ p

√Mp = A < ∞. This allows us to take all p0 < p1 < p2 < . . .

such that Mpj ≤ (1 + A)pj for all j ∈ N. Recall that (M cp)p∈N is defined if and only if

infp∈N+p√Mp = a > 0. We define

Np =

1 if p = pj for some j ∈ NMp else

.

It follows from this definition that we have for all p ∈ N

(min1, a)pNp ≤Mp ≤ (1 + A)pNp.

It follows that Np ⊂ Mp ⊂ Np and N cp ⊂ M c

p ⊂ N cp . Note that p0, p1, p2, . . . are principal

indexes of (Np)p∈N so we can apply the first part of this proof to (Np)p∈N. The statementfollows for (Mp)p∈N by combining this with Corollary 2.2.4.

Corollary 3.3.9. For any sequence of positive reals (Mp)p∈N and any open Ω ⊂ Rd, wehave

DMp(Ω) = DMcp(Ω) and D(Mp)(Ω) = D(Mc

p)(Ω).

61

Proof. By the previous corollary we have that for any K ⊂comp

Ω

DMp(K) = DMcp(K) and D(Mp)(K) = D(Mc

p)(K).

So the inductive limits of these spaces are also isomorphic.

Lemma 3.3.10. Let (Mp)p∈N be a sequence of positive reals such that ∀p ∈ N+ : pp ≤Mp.Let (pj)j∈N be the principal indexes of (Mp)p∈N, we have for j ∈ N and pj ≤ q ≤ p ≤ pj+1

M cpM0 ≥M c

qpp−qj+1.

Proof. Take any j ∈ N and any pj ≤ q ≤ pj+1. We have

ppj+1

j+1 ≤Mpj+1= M c

q

(Mpj+1

M cq

) −qpj+1−q

(Mpj+1

M cq

) pj+1pj+1−q

. (3.2)

By Lemma 2.1.3 we get that

(M cq )pj+1 ≤M

pj+1−q0 M q

pj+1.

This implies

M cq

(Mpj+1

M cq

) −qpj+1−q

= (M cq )

pj+1pj+1−qM

−qpj+1−qpj+1 ≤M0.

Combining this with (3.2), we get

pj+1 ≤M1

pj+1

0

(Mpj+1

M cq

) 1pj+1−q

.

This implies for pj ≤ q < p ≤ pj+1, by Lemma 3.3.5 (and M0 ≥ 1 = 00)

M cqp

p−qj+1 ≤M c

q

(Mpj+1

M cq

) p−qpj+1−q

Mp−qpj+1

0 = M cpM

p−qpj+1

0 ≤M cpM0.

Theorem 3.3.11. Let p! ⊂ Mp and K ⊂comp

R. We have for every K ′ bcomp

K a k > 0

(depending only on K and K ′) such that for all h ∈ R+ and h′ ∈ [kh,∞) the restrictionmapping

EMp,h(K)→ EMcp,h′(K ′) : ϕ 7→ ϕ|K′

is continuous.

Proof. By Example 2.2.8, we have pp ⊂ p! ⊂ Mp. We take C,L > 0 such that pp ≤CLpMp. And by Proposition 2.2.3 we can replace for all p ∈ N , Mp by CLpMp (by doingthis we replace M c

p by CLpM cp). So we can assume without loss of generality that pp ≤Mp

for all p ∈ N.Fix 1 ≥ δ > 0 such that x + [−δ, δ] ⊂ K for all x ∈ K ′. Throughout this proof x will

62

stand for an arbitrary element of K ′. Let h > 0 be arbitrary.We will prove the following statement:

∀K ′δb

comp

K ⊂comp

Rd,∀ϕ ∈ EMp,h(K) : ‖Dαϕ‖C(K′) ≤ 16dMd0

(2eh)|α|

δ|α|M|α|‖ϕ‖EMp,h(K)

(3.3)

where K ′δb

comp

K stands for K ′ ⊂comp

K such that K ′ + [−δ, δ]d ⊂ K. We distinguish two

cases for α ∈ Nd, pj ≤ |α| < pj+1: either α1 ≥ pj or α1 < pj.We start with the first case and denote k = |α| − pj and n = pj+1 − pj. We have byTheorem 3.1.3

|Dαϕ(x)| ≤ 16(2e)k supy∈[−δ,δ]

|Dα−ke1ϕ(x+ ye1)|1−k/nM

where M = max

sup

y∈[−δ,δ]|Dα+ne1ϕ(x+ ye1)|k/n, n

k

δksup

y∈[−δ,δ]|Dα−ke1ϕ(x+ ye1)|k/n

.

Depending on what value M takes, we get two cases. We have either, by Lemma 3.3.5(and δ ≤ 1 ≤M0),

|Dαϕ(x)| ≤ 16(2e)k‖Dα−ke1ϕ(x+ ye1)‖1−k/nC(K) ‖D

α+ne1ϕ(x+ ye1)‖k/nC(K)

≤ 16(2e)kM

|α|−pjpj+1−pjpj M

pj+1−|α|pj+1−pjpj+1

(h|α|‖ϕ‖EMp,h(K)

)1− kn

+ kn

≤ 16dMd0 (2ehδ−1)|α|M c

|α|‖ϕ‖EMp,h(K).

Or, in the other case, we have (using Lemma 3.3.10)

|Dαϕ(x)| ≤ 16(2e)k‖Dα−ke1ϕ(x+ ye1)‖C(K)nkδ−k

≤ 16(2eδ−1)kMpjp|α|−pjj+1 h|α|‖ϕ‖EMp,h(K)

≤ 16d(2ehδ−1)|α|M0Mc|α|‖ϕ‖EMp,h(K)

≤ 16dMd0 (2ehδ−1)|α|M c

|α|‖ϕ‖EMp,h(K).

This concludes the proof for (3.3) for α1 ≥ pj. In particular we have proven (3.3) ford = 1.If α1 < pj, we prove (3.3) by induction on d. Let α = (0, α2, . . . , αd). We denoten = pj+1 − |α| and k = α1. We get by Theorem 3.1.3

|Dαϕ(x)| ≤ 16(2e)k supy∈[−δ,δ]

|Dαϕ(x+ ye1)|1−k/nM (3.4)

where M = max

sup

y∈[−δ,δ]|Dα+ne1ϕ(x+ ye1)|k/n, n

k

δksup

y∈[−δ,δ]|Dαϕ(x+ ye1)|k/n

.

We define K ′1 = z1 : (z1, . . . , zd) ∈ K ′. For all z1 ∈ K ′1, y ∈ [−δ, δ] we define K ′z1+y =

(z2, . . . , zd) : (z1, ..., zd) ∈ K ′ and for all z1 ∈ K, we define Kz1 = (z2, . . . , zd) :(z1, . . . , zd) ∈ K. Note that for any z1 ∈ K ′1, y ∈ [−δ, δ] we have that

K ′z1+y

δb

comp

Kz1 .

63

So for z1 ∈ K ′1 and y ∈ [−δ, δ], we can see ϕ as a function on Kz1 ⊂ Rd−1:

ϕz1,y : (z2, . . . , zd) 7→ ϕ((z1 + y, z2, . . . , zd)

).

For any z1 ∈ K ′1 and y ∈ [−δ, δ], we apply our induction hypothesis (3.3) to ϕz1,y ∈EMc

p,h(Kz1) and we get

‖Dαϕz1,y‖C(K′z1+y) ≤ 16d−1Md−1

0 (2eh/δ)|α|−kM c|α|−k‖ϕz1,y‖EMp,h(Kz1 )

≤ 16d−1Md−10 (2eh/δ)|α|−kM c

|α|−k‖ϕ‖EMp,h(K).

We note that the right-hand side no longer depends on y and z1. We will now combinethis with (3.4), depending on what valueM takes, we get two cases. In one case we have,by Lemma 3.3.5

|Dαϕ(x)| ≤ 16(2e)k supz∈K′

supy∈[−δ,δ]

|Dαϕ(z + ye1)|‖Dα+ne1ϕ‖k/nC(K)

= 16(2e)k supz1∈K′1y∈[−δ,δ]

‖Dαϕz1+y‖1−k/nC(K′z1+y

)‖Dα+ne1ϕ‖k/nC(K)

≤ 16dMd−10 (2eh/δ)|α|(M c

|α|−k)pj+1−|α|pj+1−|α|M

α1pj+1−|α|pj+1 ‖ϕ‖EMp,h(K)

≤ 16dMd0 (2eh/δ)|α|M c

|α|‖ϕ‖EMp,h(K).

In the second case, we have, by Lemma 3.3.10

|Dαϕ(x)| ≤ 16(2e)k supz1∈K′1y∈[−δ,δ]

‖Dαϕz1+y‖C(K′z1+y)nkδ−k

≤ 16dMd−10 (2e/δ)kM c

|α|−α1pα1j+1h

|α|‖ϕ‖EMp,h(K)

≤ 16dMd0 (2eh/δ)|α|M c

|α|‖ϕ‖EMp,h(K).

So we have proven that in any case, we have for all x ∈ K ′

|Dαϕ(x)| ≤ 16dMd0

(2eh

δ

)|α|M c|α|‖ϕ‖EMp,h(K).

This implies (3.3).So we have found that

‖ϕ‖EMcp,h′ (K)

≤ 16dMd0 ‖ϕ‖EMp,h(K)

for h′ ≥ 2eh/δ. Thus the theorem follows by taking k = 2e/δ.

Theorem 3.3.12. For Ω ⊂ Rd open and (Mp)p∈N a sequence of positive reals such thatp! ⊂Mp one has

EMp(Ω) = EMcp(Ω) and E (Mp)(Ω) = E (Mc

p)(Ω).

64

Proof. In this proof −→ will denote canonical maps (restriction maps) between the spaces.By Corollary 2.2.5 we know that

EMcp(Ω) −→ EMp(Ω) and E (Mc

p)(Ω) −→ E (Mp)(Ω)

are continuous map. We prove the other directions separately. Take any h′ > 0 and anyK ′ ⊂

comp

Ω. Take K ′ bcomp

K ⊂comp

Ω and take by the previous theorem h > 0 such that

EMp,h(K) −→ EMcp,h′(K ′)

is a continuous map. Since E (Mp)(Ω) is a projective limit, we know that

E (Mp)(Ω) −→ EMp,h(K) −→ EMcp,h′(K ′)

are continuous maps. Since h′ > 0 and K ′ ⊂comp

Ω were arbitrary and E (Mcp)(Ω) is a projective

limit of spaces of the form EMcp,h′(K ′), we find by the universal property of the projective

limit thatE (Mp)(Ω) −→ E (Mc

p)(Ω)

is a continuous map. This concludes the first case.Take K ′ ⊂

comp

Ω arbitrary. Take h > 0 arbitrary. Take K ′ bcomp

K ⊂comp

Ω and take h′ > 0 such

thatEMp,h(K) −→ EMc

p,h′(K ′)

is a continuous map. Since EMcp(K ′) is an inductive limit we find that the following map

is continuousEMp,h(K) −→ EMp,h′(K ′) −→ EMc

p(K ′).

Since h > 0 was arbitrary, we find by taking the inductive limit over h > 0 and using theuniversal property of the inductive limit, that

EMp(K) −→ EMcp(K ′)

is a continuous map. Since EMp(Ω) is a projective limit, we know that

EMp(Ω) −→ EMp(K) −→ EMcp(K ′)

are continuous maps. Since K ′ was arbitrary and EMcp(Ω) is projective limit, we find by

the universal property of the projective limit that

EMp(Ω) −→ EMcp(Ω)

is a continuous map. This concludes the proof.

65

Chapter 4

Paley-Wiener Theorem

4.1 Paley-Wiener Theorem for ultradifferentiable

functions

The Paley-Wiener Theorem is a celebrated theorem from functional analysis. In thissection we will prove an analogue of this theorem for ultradifferentiable functions. Weneed the following definition to state the theorem.

Definition 4.1.1 (Support function). For K ⊂comp

Cd we define the support function of this

set as HK : Cd → RHK : ζ 7→ sup

z∈KIm(z · ζ).

Theorem 4.1.2 (Paley-Wiener-Schwartz Theorem). Let K ⊂comp

Rd be convex and balanced.

An entire function ϕ : Cd → C satisfies

(∀n ∈ N+)(∃C ∈ R+)(∀ζ ∈ Cd : |ϕ(ζ)| ≤ C(1 + |ζ|)−n exp(HK(ζ))

if and only if ∃ϕ ∈ D(K) such that ϕ = F(ϕ).

A proof of this theorem can be found in Chapter 29 of [13].

Lemma 4.1.3. Let (Mp)p∈N be a sequence of positive reals and let M be its associatedfunction. Assume that (Mp)p∈N satisfies 1 ⊂ Mp. Take any K ⊂

comp

Rd, h > 0. If ϕ ∈

DMp,h(K), then

|F(ϕ)(ζ)| ≤M0|K|‖ϕ‖EMp,h(K)

exp

(−M

(|ζ|√dh

)+HK(ζ)

)for all ζ ∈ Cd.

Proof. Let ζ ∈ Cd \ 0 be arbitrary. Take j ∈ 1, . . . , d such that |ζj| is maximal. Takeany p ∈ N. We know that (by partial integration)

(−i)pζpjF(ϕ)(ζ) = F(Dpjϕ)(ζ).

66

From this we get

|F(ϕ)(ζ)| = |ζj|−p|F(Dpjϕ)(ζ)| = |ζj|−p

∣∣∣∣∫RdDpjϕ(x) exp(−ix · ζ)dx

∣∣∣∣ .This implies, by the maximality of |ζj|, that

|F(ϕ)(ζ)| ≤(|ζ|√d

)−p ∫K

|Dpjϕ(x)|| exp(−ix · ζ)|dx

≤

(√d

|ζ|

)p

|K|‖Dpjϕ‖C(K) sup

x∈Kexp(Im(x · ζ))

≤ |K|‖ϕ‖EMp,h(K)

√dpMph

p

|ζ|pexpHK(ζ).

Since this holds for all p ∈ N, we get

|F(ϕ)(ζ)| ≤ M0|K|‖ϕ(x)‖EMp,h(K) infp∈N

√dpMph

p

M0|ζ|pexpHK(ζ)

= M0|K|‖ϕ(x)‖EMp,h(K) exp

(−M

(|ζ|√dh

))expHK(ζ).

This concludes our proof for ζ 6= 0. The lemma follows for ζ = 0 by continuity (seeLemma 2.1.7).

Lemma 4.1.4. Let (Mp)p∈N be a sequence of positive reals and let M be its associatedfunction. Assume that (Mp)p∈N satisfies 1 ⊂ Mp. Let h > 0 and let K ⊂

comp

Ω. Let

ϕ ∈ D(K) such that ∥∥∥∥exp

(M

(|y|h

))F(ϕ)(y)

∥∥∥∥L1(Rd)

<∞

(where y is the integration variable in L1(Rd)). It follows that ϕ ∈ DMp,h(K). Moreover,we have

‖ϕ‖EMp,h(K) ≤1

M0(2π)d

∥∥∥∥exp

(M

(|y|h

))F(ϕ)(y)

∥∥∥∥L1(Rd)

.

Proof. We denote ϕ = F(ϕ). Take any α ∈ Nd and any x ∈ Rd. We know that

i|α|Dαϕ(x) = F−1(yαϕ(y))(x).

67

We get for all α ∈ Nd, x ∈ Rd

|Dαϕ(x)| =1

(2π)d

∣∣∣∣∫Rdyαϕ(y) exp(ix · y)dy

∣∣∣∣=

1

(2π)d

∣∣∣∣∫Rd

yα

exp(M(|y|/h))exp(M(|y|/h))ϕ(y) exp(ix · y)dy

∣∣∣∣≤‖ϕ(y) exp(M(|y|/h))‖L1(Rd)

(2π)dsupy∈Rd

∣∣∣∣ yα

exp(M(|y|/h))

∣∣∣∣≤‖ϕ(y) exp(M(|y|/h))‖L1(Rd)

M0(2π)dh|α| sup

ρ/h∈R+

M0(ρ/h)|α|

exp(M(ρ/h))

=‖F(ϕ)(y) exp(M(|y|/h))‖L1(Rd)

M0(2π)dh|α|M c

|α|.

If we take the supremum over all x ∈ Rd and all α ∈ Nd, then we get

supα∈Nd

‖Dαϕ‖C(K)

h|α|M|α|≤ sup

α∈Nd

‖Dαϕ‖C(K)

h|α|M c|α|

≤ 1

M0(2π)d

∥∥∥∥exp

(M

(|y|h

))F(ϕ)(y)

∥∥∥∥L1(Rd)

.

So the lemma is proven.

Theorem 4.1.5 (Paley-Wiener Theorem). Let (Mp)p∈N be a sequence of positive realsand let M be its associated function. Assume (M.2’) and 1 ⊂Mp. Let K ⊂

comp

Rd be convex

and balanced. An entire function ϕ : Cd → C satisfies

(∃h ∈ R+)(∃C ∈ R+)(∀ζ ∈ Cd : |ϕ(ζ)| ≤ C exp(−M(|ζ|/h) +HK(ζ)))

if and only if ∃ϕ ∈ DMp(K) such that ϕ = F(ϕ). Analogously, an entire functionϕ : Cd → C satisfies

(∀h ∈ R+)(∃C ∈ R+)(∀ζ ∈ Cd : |ϕ(ζ)| ≤ C exp(−M(|ζ|/h) +HK(ζ)))

if and only if ∃ϕ ∈ D(Mp)(K) such that ϕ = F(ϕ).

Proof. Take any h > 0. For ϕ ∈ DMp,h(K) we have by Lemma 4.1.3

|F(ϕ)(ζ)| ≤M0|K|‖ϕ‖EMp,h(K) exp

(−M

(|ζ|√dh

)+HK(ζ)

)for all ζ ∈ Cd. So if we have ϕ ∈ DMp(K), then this holds for some h > 0 and if we haveϕ ∈ D(Mp)(K), then this holds for all h > 0. In both cases we have that F(ϕ) is entire byTheorem 4.1.2.Conversely, note that from Lemma 2.1.6 one can conclude that for any fixed n ∈ N+, h > 0

n log(1 + |ζ|) ≤M(|ζ|/h)

for |ζ| sufficiently large (ζ ∈ C). Since log is continuous on R+ and M is continuous onthe domain where it is finite, we can conclude that there is a C ′ > 0 such that for allζ ∈ C

exp(−M(|ζ|/h))

(1 + |ζ|)−n≤ C ′.

68

Thus one gets that for any n ∈ N+, there is a C ′ > 0 such that

|ϕ(ζ)| ≤ C exp(−M(|ζ|/h) +HK(ζ)) ≤ CC ′(1 + |ζ|)−n exp(HK(ζ)).

By applying Theorem 4.1.2 we conclude that there is a function ϕ ∈ D(K) such thatϕ = F(ϕ).Note that for y ∈ Rd we have

|ϕ (y) | ≤ C exp

(−M

(|y|h

))= C exp

(M

(|y|kh

)−M

(|y|h

))exp

(−M

(|y|kh

))where we choose, by Lemma 2.1.17, C ′′, k ∈ R+ such that

exp

(M

(|y|kh

)−M

(|y|h

))≤ C ′′|y|−d−1.

Combining this with the fact that for some b > 0 we have that M([0, b]) = 0 (Lemma2.1.7) we find that ∥∥∥∥exp

(M

(|y|kh

)−M

(|y|h

))∥∥∥∥L1(Rd)

= A

is finite. From Lemma 4.1.4 we see that

‖ϕ‖EMp,kh(K) ≤1

M0(2π)d

∥∥∥∥exp

(M

(|y|kh

))ϕ

∥∥∥∥L1(Rd)

≤ AC

M0(2π)d.

So we conclude that ϕ ∈ DMp,kh(K). If we have this for some h > 0, then it follows thatϕ ∈ DMp(K) and if we have this for all h > 0, then it follows that ϕ ∈ D(Mp)(K)

4.2 A theorem of Paley-Wiener type for the torus

In this section we will deduce an analogue of the Paley-Wiener Theorem for spaces ofperiodic ultradifferentiable functions.

Lemma 4.2.1. The following series converges for any d ∈ N∑0 6=β∈Zd

|β|−d−1.

Proof. From combinatorics we know that for any n ∈ N we have(d+n−1d−1

)different elements

β ∈ Nd with |β| = n. So we have at most(d+n−1d−1

)2d elements β ∈ Zd such that |β| = n.

We find that∑06=β∈Zd

|β|−d−1 ≤ 2d∞∑n=1

(d+ n− 1

d− 1

)n−d−1 ≤ 2d

∞∑n=1

(d+ n− 1)d−1n−d−1.

Since we know that∑n−2 = π2/6 we find by the comparison test that

∞∑n=1

(d+ n− 1)d−1n−d−1

converges.

69

Lemma 4.2.2. Assume that (Mp)p∈N satisfies (M.2’). Take any h > 0. If ϕ ∈ EMp,h(Td)has Fourier coefficients (aβ)β∈Zd, then we have A,H ≥ 1 (depending only on (Mp)p∈N andd) such that |a0| ≤ ‖ϕ‖EMp,h(Td)M0 and

(∀β ∈ Zd \ 0)(∀p ∈ N)(|aβ| ≤ ‖ϕ‖EMp,h(Td)Ah

p+d+1HpMp|β|−p−d−1).

Proof. Take any β ∈ Zd, we have

aβ =1

(2π)d

∫Tdϕ(x) exp(iβ · x)dx.

If β = 0, then we find

|a0| ≤1

(2π)d

∫Td‖ϕ‖C(Td)dx ≤ ‖ϕ‖EMp,h(Td)M0.

If β 6= 0, take j ∈ 1, . . . , d such that |βj| is maximal; this implies |β| ≤ d|βj|. Integratingby parts gives us for arbitrary p ∈ N

aβ =1

(2π)d

∫Td

(iβj)−pDp

jϕ(x) exp(iβ · x)dx.

It follows that

|aβ| ≤1

(2π)d

∫Tdβ−pj ‖ϕ‖EMp,h(Td)h

pMp ≤ |β|−p‖ϕ‖EMp,h(Td)(dh)pMp.

Since (Mp)p∈N satisfies (M.2’), we can use Lemma 2.1.16 to find A′, H ′ ≥ 1 such that forall p ∈ N

Mp+d+1 ≤ A′H ′pMp.

We find that for all p ∈ N

|aβ| ≤ ‖ϕ‖EMp,h(Td)(dh)p+d+1Mp+d+1|β|−p−d−1

≤ ‖ϕ‖EMp,h(Td)dd+1A′hp+d+1(dH ′)pMp|β|−p−d−1.

Taking A = dd+1A′ and H = dH ′ concludes our proof.

Theorem 4.2.3. Let (Mp)p∈N be a sequence of positive reals and let h > 0. If we haveaβ ∈ C for all β ∈ Zd, such that(

aβ exp

(M

(|β|h

)))β∈Zd

∈ `1(Zd).

Then we can define

ϕ(x) =∑β∈Zd

aβ exp(iβ · x),

and we have that ϕ ∈ EMp,h(Td). Moreover, we have a fixed C > 0 (depending only on(Mp)p∈N) such that

‖ϕ‖EMp,h(Td) ≤ C

∥∥∥∥aβ exp

(M

(|β|h

))∥∥∥∥`1(Zd)

.

70

Conversely, if (Mp)p∈N satisfies (M.2’) and if ϕ ∈ EMp,h(Td) with Fourier coefficients(aβ)β∈Zd, then (

aβ exp

(M

(|β|Hh

)))β∈Zd

∈ `1(Zd)

for some H ≥ 1, depending only on (Mp)p∈N. Moreover, we have a fixed C ′ > 0 (dependingonly on (Mp)p∈N, h and d) such that∥∥∥∥aβ exp

(M

(|β|Hh

))∥∥∥∥`1(Zd)

≤ C ′‖ϕ‖EMp,h(Td).

Proof. Let aβ ∈ C (for all β ∈ Zd) such that(aβ exp

(M

(|β|h

)))β∈Zd

∈ `1(Zd). (4.1)

We defineϕ(x) =

∑β∈Zd

aβ exp(iβ · x)

and we will verify that this defines a function which belongs to EMp,h(Td). Indeed, weget for α ∈ Nd, β ∈ Zd

Dα (aβ exp(iβ · x)) = aβ(iβ)α exp(iβ · x).

So we get for all α ∈ Nd, β ∈ Zd,

|aβ|βα

h|α|M|α|≤ 1

M0

|aβ|M0(|β|/h)|α|

M|α|≤ 1

M0

|aβ| exp

(M

(|β|h

)).

Thus we conclude by (4.1) that the series which we used to define ϕ converges absolutelyin EMp,h(Td). We also have that

‖ϕ‖EMp,h(Td) ≤ supα∈Nd

∑β∈Zd|aβ|

|βα|h|α|M|α|

≤ 1

M0

∥∥∥∥aβ exp

(M

(|β|h

))∥∥∥∥`1(Zd)

.

Conversely, let ϕ ∈ EMp,h(Td) with Fourier coefficients (aβ)β∈Zd . By Lemma 4.2.2 weget some A,H ≥ 1 such that for all 0 6= β ∈ Zd, p ∈ N

|aβ| ≤ ‖ϕ‖EMp,h(Td)Ahp+d+1HpMp|β|−p−d−1.

Fixing 0 6= β ∈ Z and taking the infimum over all p on right-hand side gives us

|aβ| ≤ ‖ϕ‖EMp,h(Td)Ahd+1 inf

p∈N

(hH

|β|

)pMp

= ‖ϕ‖EMp,h(Td)Ahd+1M0

(M0 sup

p∈N

(|β|/(hH))p

Mp

)−1

= ‖ϕ‖EMp,h(Td)Ahd+1M0 exp

(−M

(|β|hH

)).

71

Combing this with |a0| ≤ ‖ϕ‖EMp,h(Td)M0 and with Lemma 4.2.1, we find for some fixed

a ∈ R+

∑β∈Zd|aβ| exp

(M

(|β|hH

))≤ ‖ϕ‖EMp,h(Td)M0 + ‖ϕ‖EMp,h(Td)Ah

d+1M0

∑06=β∈Zd

|β|−d−1

≤ ‖ϕ‖EMp,h(Td)M0(Ahd+1a+ 1).

This concludes our proof.

This motivates the introduction of the following spaces.

Definition 4.2.4. Let (Mp)p∈N be a sequence of positive reals, h > 0 and d ∈ N+. Wedefine

`Mp,h,d1 =

(aβ)β∈Zd :

∥∥∥∥aβ exp

(M

(|β|h

))∥∥∥∥`1(Zd)

<∞

.

The norm on `Mp,h,d1 is the weighted `1-norm

(aβ)β∈Zd 7→∥∥∥∥aβ exp

(M

(|β|h

))∥∥∥∥`1(Zd)

= ‖aβ‖`Mp,h,d1

.

We will identify the dual of `Mp,h,d1 with

(bβ)β∈Zd :

∥∥∥∥bβ exp

(−M

(|β|h

))∥∥∥∥`∞(Zd)

<∞

.

Notation 4.2.5. We will denote the map that maps a periodic function to its Fouriercoefficients by S. So we get for a periodic function

ϕ(x) =∑β∈Zd

aβ exp(iβ · x)

S(ϕ) = (aβ)β. For a periodic (ultra)distribution

f(x) =1

(2π)d

∑β∈Zd

bβ exp(−iβ · x)

we get S−t(f) = (St)−1(f) = (bβ)β.

We can now give a topological interpretation to the previous theorem.

Corollary 4.2.6. We have the following isomorphisms between topological vector spaces

EMp(Td) ∼= lim−→h→∞

`Mp,h,d1 and E (Mp)(Td) ∼= lim←−

h→0

`Mp,h,d1 .

These isomorphisms are witnessed by S.

72

Proof. In this proof we will denote canonical maps between spaces and their induc-tive/projective limits by ι. We take H ≥ 1 as in the previous theorem. We have (bythe previous theorem) that for any h > 0

EMp,h(Td) S−→ `Mp,Hh,d1

ι−→ lim−→k→∞

`Mp,k,d1

are continuous maps. Taking the inductive limit over h > 0, we find by Theorem 1.3.7,that

lim−→h→∞EMp,h(Td) S−→ lim−→

k→∞`Mp,k,d1

is a continuous map. We have that for any h > 0

`Mp,h,d1

S−1

−→ EMp,h(Td) ι−→ EMp(Td)

are continuous maps. Taking the inductive limit over h > 0, we conclude by Theorem1.3.7, that

EMp(Td) S−→ lim−→h→∞

`Mp,h,d1

is an isomorphism.We have that for any h > 0

E (Mp)(Td) ι−→ EMp,h/H(Td) S−→ `Mp,h,d1

are continuous maps. Taking the projective limit over h > 0, we find by Theorem 1.2.9,that

E (Mp)(Td) S−→ lim←−h→∞

`Mp,h,d1

is a continuous map. We have that for any h > 0

lim←−k→∞

`Mp,k,d1

ι−→ `Mp,h,d1

S−1

−→ EMp,h(Td)

are continuous maps. Taking the projective limit over h > 0, we conclude by Theorem1.2.9, that

E (Mp)(Td) S−→ lim←−h→∞

`Mp,h,d1

is an isomorphism.

Proposition 4.2.7. Let (Mp)p∈N be a sequence of positive reals that satisfies (M.2’). We

have f ∈ EMp′(Td) if and only if

(∀L > 0)(∃C > 0)(∀β ∈ Zd)(|bβ| ≤ C exp(M(|β|/L))) (4.2)

where (bβ)β = S−t(f).

73

Proof. Take f ∈ EMp′(Td). We have established that EMp(Td) ∼= lim−→L→∞ `Mp,L,d1 . By

the universal property of the inductive limit topology we know that

S−t(f)|`Mp,L,d1

: `Mp,L,d1 → C

is a continuous functional for all L > 0. So we find that S−t(f) = (bβ)β ∈(`Mp,L,d1

)′for

all L > 0. We conclude that

supβ∈Zd|bβ| exp

(−M

(|β|L

))<∞

for all L > 0. This proves (4.2). Conversely, take (bβ)β such that (4.2) is satisfied. This

implies that (bβ)β defines an element of(`Mp,L,d1

)′for all L > 0. So we conclude by

the universal property of the inductive limit topology that (bβ)β defines an element of(lim−→L→∞ `

Mp,L,d1

)′. Using the isomorphism from the previous corollary, we find that

St((bβ)β∈Zd

)=

1

(2π)d

∑β∈Zd

bβ exp(−iβ · x) ∈ EMp′(Td).

Proposition 4.2.8. Let (Mp)p∈N be a sequence of positive reals that satisfies (M.2’). We

have f ∈ E (Mp)′(Td) if and only if

(∃L > 0)(∃C > 0)(∀β ∈ Zd)(|bβ| ≤ C exp(M(|β|/L))) (4.3)

where (bβ)β = S−t(f).

Proof. We have established that E (Mp)(Td) ∼= lim←−L→∞ `Mp,L,d1 . In particular we have

that lim←−L→∞ `Mp,L,d1 is complete, so we know from Proposition 1.2.24 that the elements

of(

lim←−L→∞ `Mp,L,d1

)′are restrictions of some (bβ)β ∈

(`Mp,L,d1

)′for some L > 0. Using

the isomorphism from Corollary 4.2.6 we can restate this as:

St((bβ)β∈Zd

)∈ E (Mp)′(Td)

if and only if there is an L > 0 such that

supβ∈Zd|bβ| exp

(−M

(|β|L

))<∞.

This proves the proposition.

Lemma 4.2.9. Let (Mp)p∈N be a sequence of positive reals. If we have L,C > 0 andbβ ∈ C (for all β ∈ Zd) such that

∀β ∈ Zd : |bβ| ≤ C exp(M(|β|/L)).

74

Then we can find bβ,p ∈ C, for all β ∈ Zd, p ∈ N, such that

∀β ∈ Zd, p ∈ N : |bβ,p| ≤M0C(|β|/L)p

Mp

and ∀β ∈ Zd : bβ =∞∑p=0

bβ,p.

Moreover, we have for all β ∈ Zd a P ∈ N such that bβ,q = 0 for all q > P .

Proof. Let L,C > 0, (bβ)β be as in the statement of the lemma. Take β ∈ Zd arbitrary.If β = 0, then we have that |bβ| ≤ C. so we can take b0,0 = b0 and b0,p = 0 for p > 0.(Recall that, by convention, 00 = 1 so the required condition is indeed satisfied.)Assume that β 6= 0. If

∞∑p=0

M0C(|β|/L)p

Mp

=∞,

take P ∈ N minimal such that∑P

p=0 C(|β|/L)p

Mp≥ |bβ|. If

∞∑p=0

M0C(|β|/L)p

Mp

<∞,

then we use the fact that

|bβ| ≤ C exp

(M

(|β|L

))= sup

p∈NM0C

(|β|/L)p

Mp

<∞∑p=0

M0C(|β|/L)p

Mp

to take P ∈ N minimal such that∑P

p=0C(|β|/L)p

Mp≥ |bβ| (we can do this because the second

inequality is strict). In both cases we take

cβ,p =

C (|β|/L)p

Mpif p < P

|bβ| −∑P−1

p=0 C(|β|/L)p

Mpif p = P

0 else

.

By our choice P we get that 0 ≤ cβ,p ≤ C (|β|/L)p

Mpfor all p ∈ N. We also get |bβ| =

∑∞p=0 cβ,p.

We get the desired result by taking bβ,p =bβ|bβ |cβ,p (or bβ,p = 0 if bβ = 0) for all p ∈ N.

Lemma 4.2.10. Let (Mp)p∈N be a sequence of positive reals. If we have L,C > 0 andbβ,p ∈ C (for all β ∈ Zd, p ∈ N) such that

∀β ∈ Zd, p ∈ N : |bβ,p| ≤M0C(|β|/L)p

Mp

,

then we have

∀β ∈ Zd :

∣∣∣∣∣∞∑p=0

bβ,p

∣∣∣∣∣ ≤ 2C exp

(M

(2|β|L

)).

75

Proof. Let L,C > 0, (bβ,p)β,p be as in the statement of the lemma. Let β ∈ Zd bearbitrary. We get that∣∣∣∣∣

∞∑p=0

bβ,p

∣∣∣∣∣ ≤∞∑p=0

M0C(|β|/L)p

Mp

≤∞∑p=0

M0C(2|β|/L)p

Mp

1

2p

≤ supq∈N

M0C(2|β|/L)q

Mq

∞∑p=0

1

2p= 2C exp

(M

(2|β|L

)).

This concludes the proof.

Theorem 4.2.11. Assume that (Mp)p∈N satisfies (M.2’), f ∈ E∗′(Td) if and only if wehave bβ,p ∈ C (for all β ∈ Zd, p ∈ N) such that

f(x) =1

(2π)d

∑β∈Zdp∈N

bβ,p exp(−iβ · x) (4.4)

and

(∀L > 0)(∃C > 0)(∀β ∈ Zd)(∀p ∈ N)

(|bβ,p| ≤ C

|β|p

LpMp

)if ∗ = Mp or (4.5)

(∃L > 0)(∃C > 0)(∀β ∈ Zd)(∀p ∈ N)

(|bβ,p| ≤ C

|β|p

LpMp

)if ∗ = (Mp) (4.6)

Moreover, we can take (bβ,p)β,p in such a way that for all β ∈ Zd there is a P ∈ N suchthat bβ,q = 0 for all q > P .

Proof. Take f ∈ EMp′(Td), we use Proposition 4.2.7 to see that S−t(f) satisfies theassumption in Lemma 4.2.9 (for all L > 0) and thus we get (4.5). Conversely, if (4.5) issatisfied, we get by Lemma 4.2.10 (for all L > 0) and Proposition 4.2.7 that (4.4) defines

an element of EMp′(Td).Take f ∈ E (Mp)′(Td), we use Proposition 4.2.8 to see that St(f) satisfies the assumption inLemma 4.2.9 (for some L > 0) and thus we get (4.6). Conversely, if (4.6) is satisfied, weget by applying Lemma 4.2.10 (for some L > 0) and Proposition 4.2.8 that (4.4) defines

an element of E (Mp)′(Td).

76

Chapter 5

The first structure theorem

5.1 The first structure theorem for the torus

The results from the previous chapter allow us to state and prove the following theo-rem immediately. The well know analogue for ultradifferentiable functions with compactsupport will proved in the next part of this chapter.

Theorem 5.1.1 (First structure theorem for the torus). Assume that (Mp)p∈N satisfies(M.2’). We have f ∈ E∗′(Td) if and only if we have periodic functions fα ∈ C(Td) suchthat

(∀L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(Td) ≤

C

L|α|M|α|

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(Td) ≤

C

L|α|M|α|

)if ∗ = (Mp)

(5.1)

andf =

∑α∈Nd

Dαfα. (5.2)

Proof. By Theorem 4.2.11 we have that every f ∈ E∗′(Td) is of the form

f(x) =1

(2π)d

∑β∈Zdp∈N

bβ,p exp(−iβ · x)

and coefficients satisfy

(∀L > 0)(∃C > 0)(∀β ∈ Zd)(∀p ∈ N)

(|bβ,p| ≤ C

|β|p

LpMp

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀β ∈ Zd)(∀p ∈ N)

(|bβ,p| ≤ C

|β|p

LpMp

)if ∗ = (Mp)

(5.3)

Let f ∈ E∗′(Td) be given in this form. We define f0,0 = b0,0. For every β ∈ Zd \ 0 wefix µ(β) such that |βµ(β)| is maximal. We denote Aj = β ∈ Zd \ 0| µ(β) = j (for all

77

j ∈ 1, . . . d). We set for all p ∈ N+, j ∈ 1, . . . d,

fp,j(x) =∑β∈Aj

bβ,p(−iβj)p+d+1

exp(−iβ · x).

Observe that for any p ∈ N+, j ∈ 1, . . . , d,∥∥∥∥∥∥∑β∈Aj

bβ,p(−iβj)p+d+1

exp(−iβ · x)

∥∥∥∥∥∥C(Td)

≤∑β∈Aj

C|β|p

|βj|p+d+1LpMp

≤∑β∈Aj

Cdp+d+1|β|p

|β|p+d+1LpMp

≤ Cdp+d+1

LpMp

∑06=β∈Zd

1

|β|d+1.

So using Lemma 4.2.1 we find that the sequence which we used to define fp,j, defines afunction in C(Td) (for any p, j). Explicitly, we have for all p, j

‖fp,j‖C(Td) ≤dd+1Ca

(L/d)pMp

,

where a =∑|β|−d−1. Using Lemma 2.1.16 we find some H,A > 0 (depending only on

(Mp)p∈N and d) such that

‖fp,j‖C(Td) ≤dd+1CaA

(L/d)pHpMp+d+1

. (5.4)

We have

Dp+d+1j fp,j =

∑β∈Aj

bβ,p(−iβj)p+d+1

Dp+d+1j exp(−iβ · x) =

∑β∈Aj

bβ,µ(β) exp(−iβ · x).

This, together with the fact Zd \ 0 =⋃j∈1,...,dAj is a disjoint union (and the fact that

b0,p = 0 for all p ∈ N+), implies that

f0,0 +∑p∈N+

1≤j≤d

Dp+d+1j fp,j = b0,0 +

∑0 6=β∈Zdp∈N+

bβ,p exp(−iβ · x) = (2π)df.

So (5.2) holds if we take

fα =1

(2π)d

fp,j if α = (p+ d+ j)ej for some p ∈ N+, j ∈ 1, . . . , df0,0 if α = 0

0 else

.

We get (5.1) from (5.4) and (5.3).Conversely, let

f =∑α∈Nd

Dαfα,

78

such that (5.1) is satisfied. We find for ϕ ∈ E∗(Td) (where ∗ denotes the appropriateclass)

〈f, ϕ〉 =∑α∈Nd〈fα, (−D)αφ〉.

We need to verify that the sum on the right-hand side converges absolutely. We have (forsome h > 0 if ∗ = Mp and for all h > 0 if ∗ = (Mp))∑

α∈Nd|〈f, (−D)αφ〉| ≤ h|α|M|α|‖ϕ‖EMp,h(Td)‖fα‖C(Td)

≤ C‖ϕ‖EMp,h(Td)

∑α∈Nd

(h/L)|α|.

This series converges absolutely because we can choose (depending on which case ∗ repre-sents) L > 0 such that h/L < 1 or h > 0 such that h/L < 1. We conclude that f definesa functional on E∗(Td).

5.2 The first structure theorem for Ω ⊂ Rd

Proposition 5.2.1. Let K ⊂comp

Rd and take r > 0, a ∈ Rd such that rK + a ⊂ [0, 2π]d.

We have that D∗(rK + a) is a closed subspace of E∗(Td) and the topology of D∗(rK + a)agrees with the topology of D∗(rK + a), induced by E∗(Td).

Proof. We can identify D∗(rK + a) with a subspace of E∗(Td) because we have that allthe derivatives of any ϕ ∈ D∗(rK + a) vanish on the boundary of [0, 2π]d. Making thisidentification we get that for any h > 0 the injection

DMp,h(rK + a)→ EMp,h(Td)

is an isomorphism onto its image and the image is closed because the norms on theseBanach spaces coincide.If ∗ = (Mp), then we get by Proposition 1.2.11 that D∗(rK + a) → E∗(Td) defines anisomorphism onto its image, and the image is closed.If ∗ = Mp, then we know that both spaces are (DFS)-spaces (Definition 1.4.14), so weget by Remark 1.3.21 and Propostion 1.3.20 that the injection D∗(rK + a) → E∗(Td)defines an isomorphism onto its image, and the image is closed.

This proposition, in combination with the Hahn-Banach Theorem will allow us to formu-late the structural theorems which we had for E∗′(Td), for D∗′(K).

Proposition 5.2.2. Let K ⊂comp

Rd and take r > 0, a ∈ Rd such that rK + a ⊂ Td. We

have f ∈ D∗′(K) if and only if there is an f ∈ E∗′(Td) such that

f |rK+a = f(r−1( · − a)).

In particular, if we assume (M.2’), then we have the characterizations from Proposi-tion 4.2.8 or 4.2.7 (depending on ∗), Theorem 4.2.11 and Theorem 5.1.1 for elements ofD∗′(K). Explicitly we have f ∈ D∗′(K) if and only if f(r−1( · −a)) satisfies the conditionsin these Propositions/Theorems.

79

Proof. We have, by Proposition 1.4.16, D∗(K) ∼= D∗(rK + a) and this isomorphism iswitnessed by

I : D∗(rK + a)→ D∗(K) : ϕ 7→ (x 7→ ϕ(rx+ a)).

So I t : D∗′(K)→ D∗′(rK + a) is also an isomorphism.Take any f ∈ D∗′(K). We have by the previous proposition that D∗(rK + a) is a closedsubspace of E∗(Td), so we can use the Hahn-Banach Theorem to extend I t(f), to anelement of E∗′(Td) which we call f . It follows that f |rK+a = I t(f) = f(r−1( · − a)).Conversely, any f ∈ E∗′(Td) can be restricted to D∗(rK + a) and mapped to an elementf ∈ D∗′(K) by I−t. We conclude that f ∈ D∗′(K) if and only if there is a f ∈ E∗′(Td)such that f |rK+a = I t(f) = f(r−1( · − a)). It remains to observe that Proposition 4.2.8or 4.2.7, Theorem 4.2.11, and Theorem 5.1.1 apply to f .

Corollary 5.2.3. Assume that (Mp)p∈N is a sequence of positive reals that satisfies (M.2’).Let K ⊂

comp

Rd. We have f ∈ D∗′(K) if and only if there are fα ∈ C(K) (for all α ∈ Nd)

such that

(∀L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(K) ≤

C

L|α|M|α|

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(K) ≤

C

L|α|M|α|

)if ∗ = (Mp)

(5.5)

andf =

∑α∈Nd

Dαfα. (5.6)

Proof. Take r > 0, a ∈ Rd such that rK+a ⊂ Td. Take any f ∈ D∗′(K). By the previousproposition and Theorem 5.1.1 we have some functions fα ∈ C(Td) such that

(∀L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(Td) ≤

C

L|α|M|α|

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(Td) ≤

C

L|α|M|α|

)if ∗ = (Mp)

(5.7)

andf(r−1( · − a)) = f |rK+a =

∑α∈Nd

Dαfα|rK+a.

It follows from (5.7) that

(∀L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(rK+a) ≤

C

L|α|M|α|

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(rK+a) ≤

C

L|α|M|α|

)if ∗ = (Mp).

Note that I : C(rK + a)→ C(K) : f 7→ (x 7→ f(rx+ a)) preserves the supremum-norm.So if we take fα = I(fα) for all α ∈ Nd, then we get (5.5) and (5.6).Conversely, let

f =∑α∈Nd

Dαfα,

80

such that (5.5) is satisfied, we find for ϕ ∈ D∗(K) (where ∗ denotes the appropriate class)

〈f, ϕ〉 =∑α

〈fα, (−D)αφ〉.

We need to verify that the sum on the right-hand side converges absolutely. We have (forsome h > 0 if ∗ = Mp and for all h > 0 if ∗ = (Mp))∑

α∈Nd|〈f, (−D)αφ〉| ≤ h|α|M|α|‖ϕ‖EMp,h(K)‖fα‖C(K)

≤ C‖ϕ‖EMp,h(K)

∑α∈Nd

(h/L)|α|.

This series converges absolutely because we can choose (depending on which case ∗ repre-sents) L > 0 such that h/L < 1 or h > 0 such that h/L < 1. We conclude that f definesa functional on D∗(K).

We can also give the following formulation, which is know as the first structure theoremfor D∗(Ω).

Theorem 5.2.4 (First structure theorem). Let Ω ⊂open

Rd. Assume that (Mp)p∈N is a

sequence of positive reals that satisfies (M.2’). We have f ∈ D∗′(Ω) if and only if we havefor every K ⊂

comp

Ω functions fα ∈ C(K) (for α ∈ Nd) such that

(∀L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(K) ≤

C

L|α|M|α|

)if ∗ = Mp or

(∃L > 0)(∃C > 0)(∀α ∈ Nd)

(‖fα‖C(K) ≤

C

L|α|M|α|

)if ∗ = (Mp)

(5.8)

andf |D∗(K) =

∑α∈Nd

Dαfα. (5.9)

Proof. By the universal property of the inductive limit we know that f ∈ D∗′(Ω) ifand only if f |D∗(K) ∈ D∗′(K) for all K ⊂

comp

Ω so this theorem follows from the previous

corollary.

81

Conclusion

In the first chapter we have seen that the spaces of ultradifferentiable functions carryinteresting topologies. We have studied the properties of these topologies in a generalsetting and these properties have been valuable in the other chapters.

In the second chapter we have discussed some general properties of sequences of positivereals. We have seen that, under some restrictions on the behavior of the defining sequence,spaces ultradifferentiable functions posses nice properties. In particular, we have seenwhat restriction are sufficient to make spaces of ultradifferentiable functions closed underthe pointwise multiplication of functions and under differentiation. We have also proventhe Denjoy-Carleman Theorem. This theorem tells us when we have ultradifferentiablefunctions with compact support in terms of the defining sequence.

In the third chapter we have determined under which conditions we can replace a generalsequence of positive reals by a log-convex sequence without changing the correspondingclasses of ultradifferentiable functions.

In the fourth chapter we have seen that the Paley-Wiener Theorem can be formulatedfor spaces of ultradifferentiable functions. This gives a clear connection between the classof ultradifferentiable functions to which a given function with compact support belongsand the decay of the Fourier-Laplace transform of this function. In the second half ofthe fourth chapter we have seen that one can formulate a similar theorem for periodicfunctions.

The topological interpretation of the second part of the fourth chapter allowed us tounderstand the structure of ultradistributions. We have used this insight to develop a newmethod to prove the first structure theorem. First we proved this theorem for periodicultradistributions and afterwards we were able to use this result to prove Komatsu’s firststructure theorem for ultradistributions.

82

Appendix A

Abstract in Dutch

Deze masterproef behandelt de topologische ruimten van ultradifferentieerbare functies.We geven in deze masterproef een toegankelijke inleiding tot de theorie van de ultradiffer-entieerbare functies en we bewijzen een aantal belangrijke eigenschappen van de ruimtenvan deze functies. Hiervoor hebben we het werk van H. Komatsu [4, 5] en ten dele ookop het werk van S. Mandelbrojt [7] als uitgangspunt gebruikt.

In het eerste hoofdstuk introduceren we concepten uit de topologie en een aantal topologis-che ruimten. We bewijzen ook relevante eigenschappen van deze ruimten. In het bijzonderdefinieren we projectieve en inductieve limieten van lokaal convexe topologische vector-ruimten. Vervolgens introduceren we (FS)-ruimten, (DFS)-ruimten en (LFS)-ruimten. Webewijzen een aantal eigenschappen van dergelijke ruimten die een belangrijke rol zullenspelen in de volgende hoofdstukken. Na het opbouwen van de topologische setting waarinwe zullen werken, stappen we over tot de bespreking van ultradifferentieerbare functies.We geven de definities van de ruimten van ultradifferentieerbare functies die gedefinieerdworden door een rij van positieve getallen (Mp)p∈N en we definieren een natuurlijke topolo-gie op deze ruimten. De ruimten van ultradifferentieerbare functies zijn voorbeelden van(FS)-ruimten, (DFS)-ruimten en (LFS)-ruimten, in het bijzonder zijn dit Montelruimten.

In het tweede hoofdstuk bespreken we welke rol (Mp)p∈N speelt. We onderzoeken eerst eenaantal relevante eigenschappen van rijen van positieve getallen. Vervolgens bestuderenwe implicaties van deze eigenschappen op de klassen van ultradifferentieerbare functiesdie (Mp)p∈N definieert. Ten eerste bestuderen we de hierarchie van de ruimten van ultra-differentieerbare functies, i.e. wanneer een ruimte bevat is in een andere. Ten tweedebestuderen we onder welke voorwaarden een ruimte van ultradifferentieerbare functies eentopologische algebra is. Dit blijkt het geval te zijn wanneer (Mp)p∈N logaritmisch convexis. We gaan verder in op deze voorwaarde op (Mp)p∈N in het derde hoofdstuk. Ten derdebestuderen we het gedrag van afleidingsoperatoren op ruimten van ultradifferentieerbarefuncties. Het gedrag van deze operatoren blijkt verbonden te zijn met een voorwaarde op(Mp)p∈N die in de laatste hoofdstukken een belangrijke rol zal spelen. We eindigen hethoofdstuk met de stelling van Denjoy-Carleman. Deze stelling geeft een criterium om aande hand van het gedrag van (Mp)p∈N te bepalen of een klasse van ultradifferentieerbarefuncties functies bevat met een compacte drager.

Het derde hoofdstuk is geweid aan het bestuderen van de voorwaarden waaronder we(Mp)p∈N kunnen vervangen door haar maximale logaritmisch convexe minorante (M c

p)p∈N

83

zonder de klassen van ultradifferentieerbare functies die deze rij definieert te wijzigen.We maken hiervoor gebruik van ongelijkheden van het Landau-Kolmogorov type. Voorruimten van periodieke ultradifferentieerbare functies en ultradifferentieerbare functiesmet compacte drager hebben we een aanpak die analoog is aan wat S. Mandelbrojt [7,hoofdstuk 6] in het 1-dimensionale geval gedaan heeft. Voor deze ruimten komen we toteen positief resultaat. Voor andere ruimten werkt S. Mandelbrojt met andere regular-isaties van (Mp)p∈N, wij bekijken echter wat er gebeurt als we (Mp)p∈N vervangen door(M c

p)p∈N. In dit geval hebben we zowel positieve als negatieve resultaten. Samen scheppendeze een volledig beeld.

In het vierde hoofdstuk bespreken we de stelling van Paley-Wiener voor ultradifferen-tieerbare functies [5, pargraaf 9]. Deze stelling legt een duidelijk verband tussen ultra-differentieerbare functies met compacte drager en hun Fourier-Laplacegetransformeerde.Meer bepaald geeft deze welgekende stelling een uitspraak over de groei van de Fourier-Laplacegetransformeerde van een functie die equivalent is aan het behoren tot een ruimtevan ultradifferentieerbare functies. In het tweede deel bestuderen we het gedrag van deFouriercoefficienten van een periodieke ultradifferentieerbare functie. We komen tot eenstelling die analoog is aan de klassieke versie van de stelling van Paley-Wiener. We for-muleren ook een topologische versie van deze stelling. Dit staat ons toe om de dualeruimte van de periodieke ultradifferentieerbare functies beter te begrijpen. Dit inzicht inhet gedrag van ultradistributies is van groot nut in het laatste hoofdstuk.

Het laatste hoofdstuk gaat over de eerste structuurstelling voor ultradistributies [5, para-graaf 8]. Deze stelling drukt uit welke vorm de functionalen op de ruimte van ultra-differentieerbare functies met compacte drager kunnen aannemen. We ontwikkelen hier,gebruik makend van het vorige hoofdstuk, een nieuwe methode op deze stelling te be-wijzen. Om tot deze stelling te komen, lossen we eerst hetzelfde probleem op voor deduale van de ruimte van periodiek ultradifferentieerbare functies. Het tweede deel van hetvierde hoofdstuk staat ons toe om dit vlot te doen. Vervolgens gebruiken we dit resultaatom het klassieke probleem op te lossen.

84

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