Almost-Periodic Multipliers

Acta Applicandae Mathematicae 65: 137–151, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

137

Almost-Periodic Multipliers

GIORDANO BRUNO, RENATO GRANDE and RITA IANNACCIDipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma‘La Sapienza’ , Via A. Scarpa 16, I–00161 Rome, Italy

(Received: 10 May 2000)

Abstract. In this work we study the necessary and sufficient conditions for a generalized trigonomet-ric series in order for it to be the series of a Stepanoff almost-periodic function f ∈ Sq(R), 1 � q <

∞. We consider analogous conditions for functions belonging to D(�, R). Finally, we characterizethe multipliers of invariance of the (B1(�), B1(�)) type.

Mathematics Subject Classifications (2000): 42A75, 42A45.

Key words: almost-periodic functions, Fourier multipliers.

We denote by Sq(R) and Bq(R), 1 � q < ∞ the space of almost-periodic func-tions in the sense of Stepanoff, respectively of Besicovitch. For the main propertiesof these function spaces, we refer to the monography [1].

If f ∈ Sq(R) or f ∈ Bq(R), there exists finite

a(λ, f ) = limT→∞

1

2T

∫ T

−T

f (x)e−iλx dx

for all λ ∈ R, but it is different from zero, at most, for a countable set. Moreover,for these functions we introduce

σ (f ) = {λ ∈ Rs | a(λ, f ) �= 0},and

f (x) ∼∑

λ∈σ(f )

a(λ, f )eiλx;

a(λ, f ), σ (f ),∑

λ∈σ(f ) a(λ, f )eλ(x) are called, respectively, the Bohr–Fouriertransform, the Spectrum, the Bohr–Fourier series of f .

If � ⊂ R is a countable set, we put

Bq(�) = {f ∈ Bq(R) | σ (f ) ⊂ �

}.

In [2] they study, among other things, necessary and sufficient conditions fora generalized trigonometric series, i.e. a series of the kind

∑λ∈� aλeiλx , where

138 GIORDANO BRUNO ET AL.

� ⊂ R is at most countable, to be the Bohr–Fourier series of a function f belong-ing to S1(R).

In this note, we present a necessary and sufficient condition in order to obtainthe same result for 1 < q < ∞.

For what concerns the spaces Bq(R), 1 � q � ∞, we refer to [4].Let f ∈ L1

loc(R).

DEFINITION 1. We say that f belongs to the class D(�,R) if

lim supT→∞

1

2T

∫ T

−T

|f (x)| dx < +∞,

and there exists finite

limT→∞

1

2T

∫ T

−T

f (x) eiλx dx,

for all λ ∈ R, but it is different from zero only for a subset which is at mostcountable.

If f ∈ D(�,R), one may associate to it a generalized trigonometric series.Also in this case, it gets the necessary and sufficient conditions for a generalizedtrigonometric series to be the series of a function of that class.

This result is then used to obtain a characterization of the multipliers of invari-ence in the space B1(�).

The following result is an extension of Theorem I in [2].

THEOREM 1. Let (fn)n∈N ⊂ Lp

loc(R), 1 < p < ∞, such that

supx∈R

(∫ x+1

x

|fn(t)|p dt

)1/p

� B, ∀n ∈ N.

Then it is possible to find a subsequence (fnk ) of (fn) and a function f ∈ Lp

loc(R)

such that, for every measurable and bounded function ϕ(x) and every boundedinterval (a, b) ⊂ R, one has

limk→∞

∫ b

a

ϕ(x)fnk (x) dx =∫ b

a

ϕ(x)f (x) dx. (1)

Proof. Let us observe that if E is a measurable set with diameter < 1, we haveE ⊂ (x, x + 1), with x ∈ R, suitably chosen, and so∫

E

|fn(t)| dt =∫ x+1

x

χE(t)|fn(t)| dt

�(∫ x+1

x

|fn(t)|p dt

)1/p(∫ x+1

x

χEq(t) dt

)1/q

� supx∈R

(∫ x+1

x

|f (t)|p dt

)1/p

m(E)1/q � Bm(E)1/q,

ALMOST-PERIODIC MULTIPLIERS 139

where 1/p + 1/q = 1 and m is the usual Lebesgue measure on R.Hence∫

E

|fn(t)| dt → 0, if m(E) → 0,

uniformly with respect to n ∈ N.By Theorem I of [2], it is possible to find a subsequence (fnk )k∈N ⊂ (fn)n∈N and

a function f ∈ L1loc(R) such that, for every measurable bounded function ϕ(x) and

for every bounded interval (a, b) ⊂ R, one has (1). Let us prove that f ∈ Lp

loc(R).Let δ ∈ (0, 1). Let us consider

f (δ)(x) = 1

δ

∫ x+δ

x

f (t) dt,

f (δ)n (x) = 1

δ

∫ x+δ

x

fn(t) dt, n ∈ N.

By Lebesgue’s theorem, an almost-everywhere derivability of the integral function,one gets

limδ→0

f (δ)(x) = f (x), x ∈ R a.e.,

limδ→0

f (δ)n (x) = fn(x), x ∈ R a.e.

Furthermore by (1), for each x ∈ R and all δ ∈ (0, 1), one has

limk→∞ f (δ)

nk(x) = f (δ)(x).

Evidently one also has

limk→∞ |f (δ)

nk(x)|p = |f (δ)(x)|p, x ∈ R, δ ∈ (0, 1).

By Fatou’s lemma∫ x+1

x

|f (δ)(t)|p dt � lim infk→∞

∫ x+1

x

|f (δ)nk

(t)|p dt

� lim infk→∞

∫ x+2

x

|f (δ)nk

(t)|p dt � 2Bp,

independently of the choice of x ∈ R. Applying once more Fatou’s lemma,∫ x+1

x

|f (t)|p dt � lim infδ→0

∫ x+1

x

|f (δ)(t)|p dt � 2Bp.

Hence f ∈ Lp

loc(R). ✷


DEFINITION 2. Let F be a subset of the space Sp(R), 1 � p < ∞. We say Fis bounded (with respect to the norm Sp(R)) if there exists a positive constant Csuch that

‖f ‖Sp = supx∈R

(∫ x+1

x

|f (t)|p dt

)1/p

� C, ∀f ∈ F .

We say that F is Sp equi-almost-periodic if, for every ε > 0, there exists a subset(τ )ε ⊂ R, relatively dense, such that

supx∈R

(∫ x+1

x

|f (t + τ) − f (t)|p dt

)1/p

< ε, ∀f ∈ F , ∀τ ∈ (τ )ε.

THEOREM 2. Let (fn)n∈N ⊂ Sp(R), 1 < p < ∞, be bounded in Sp(R) and Sp-equi-almost-periodic. Then it is possible to find a subsequence (fnk ) of (fn) and afunction f ∈ Sp(R) such that, for every measurable and bounded function ϕ(x)

and for every bounded interval (a, b) ⊂ R, one has

limk→∞

∫ b

a

ϕ(x)fnk (x) dx =∫ b

a

ϕ(x)f (x) dx. (2)

Proof. Since (fn)n∈N ⊂ Sp(R) has bounded norm, there exists a constant B > 0such that

‖fn‖Sp = supx∈R

(∫ x+1

x

|fn(t)|p dt

)1/p

� B, ∀n ∈ N,

and this implies, by Theorem 2, that it is possible to find a subsequence (fnk ) of(fn) and a function f ∈ L

p

loc(R) such that, for each measurable bounded functionϕ(x) and each bounded interval (a, b) ⊂ R, (2) holds.

Let us consider δ ∈ (0, 1) and set

f (δ)(x) = 1

δ

∫ x+δ

x

f (t) dt,

f (δ)n (x) = 1

δ

∫ x+δ

x

fn(t) dt, n ∈ N.

We know that

limδ→0

f (δ)(x) = f (x), x ∈ R a.e.,

limδ→0

f (δ)n (x) = fn(x), x ∈ R a.e.

Furthermore by (2), for all x ∈ R and for all δ ∈ (0, 1), one has

limk→∞

f (δ)nk

(x) = f (δ)(x).


Applying Fatou’s lemma one gets, for an arbitrary x ∈ R,∫ x+1

x

|f (δ)(t + τ) − f (δ)(t)|p dt

� lim infk→∞

∫ x+1

x

|f (δ)nk

(t + τ) − f (δ)nk

(t)|p

� lim infk→∞

∫ x+1

x

|fnk(t + τ) − fnk (t)|p � 2εp.

Applying Fatou’s lemma again,∫ x+1

x

|f (t + τ) − f (t)|p dt � lim infδ→0

∫ x+1

x

|f (δ)(t + τ) − f (δ)(t)|p � 2εp.

One gets easily that f ∈ Sp(R). ✷Let us consider a generalized trigonometric series∑

λ∈�aλeiλx, (3)

where � ⊂ R is a countable subset of the real numbers and (aλ)λ∈� ⊂ C. Since �

is countable, we may write

∑λ∈�

aλeiλx =∞∑n=1

aλneiλnx.

Let now (βj )j∈N be a base for �, i.e.

(1) the numbers βj are Z-linearly independent;(2) each λ ∈ � is a linear combination with rational coefficient of a finite number

of elements of the base (βj )j∈N.

DEFINITION 3. We call Bochner sums associated with the series (3), the gener-alized trigonometric polynomials σm(x) defined in the following way:

σm(x) =(m!)2∑

ν1=−(m!)2

· · ·(m!)2∑

νm=−(m!)2

(1 − |ν1|

(m!)2

)· · ·

(1 − |νm|

(m!)2

)aλneiλnx,

where we put

λn = ν1β1

m! + · · · + νmβm

m! .We will occasionally write

σm(x) =∑

d(m)λn

aλneiλnx,


and we have

limm→∞ d

(m)λn

= 1.

THEOREM 3. A necessary and sufficient condition for a generalized trigonomet-ric series∑


to be a Bohr–Fourier series of the function f ∈ Sp(R), 1 < p < ∞, is that theBochner sums (σm(x))m∈N associated with the series (4) give rise to a sequencebounded in Sp(R) and Sp-equi-almost-periodic.

Proof. The condition is necessary. In fact, let f ∈ Sp(R), 1 < p < ∞, and letseries (4) be its Bohr–Fourier series.

Then the Bochner sums (σm(x))m∈N associated with f (or with the series (4)which is the same) are nothing but the classical Bochner–Fejer polynomials andthey may be written in the following way:

σm(x) = limT→∞

1

2T

∫ T

−T

f (x + t)Km(t) dt = Mt{f (x + t)Km(t)},

where Km is the Bochner–Fejer kernel. We may suppose that f assumes real valuesand so∫ τ+1

τ

|σm(x)|p dx

�∫ τ+1

τ

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (x + t)Km(t) dt

∣∣∣∣p

dx

�∫ τ+1

τ

(lim

T→∞1

2T

∫ T

−T

|f (x + t)|Km(t) dt

)p

dx

�∫ τ+1

τ

(lim

T→∞1

2T

∫ T

−T

|f (x + t)|Km(t)1/pKm(t)

1/q dt

)p

dx

�∫ τ+1

τ

(lim

T→∞1

2T

∫ T

−T

|f (x + t)|pKm(t) dt

)×

×(

limT→∞

1

2T

∫ T

−T

Km(t) dt

)p/q

dx

� lim infT→∞

1

2T

∫ τ+1

τ

∫ T

−T

|f (x + t)|pKm(t) dt dx

= lim infT→∞

1

2T

∫ T

−T

Km(t) dt∫ τ+1

τ

|f (x + t)|p dx

� lim infT→∞

∫ T

−T

Km(t) dt · ‖f ‖p

Sp = ‖f ‖p

Sp .


Hence (σm(x))m∈N is a bounded sequence in Sp(R).Analogously, one proves the Sp-equi-almost-periodicity. In fact∫ τ+1

τ

|σm(x + η) − σm(x)|p dx

=∫ τ+1

τ

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (x + η + t)Km(t) dt−

− limT→∞

1

2T

∫ T

−T

f (x + t)Km(t) dt

∣∣∣∣p

dx

=∫ τ+1

τ

∣∣∣∣ limT→∞

1

2T

∫ T

−T

[f (x + η + t) − f (x + t)]Km(t) dt

∣∣∣∣p

dx

� limT→∞

1

2T

∫ T

−T

Km(t) dt · supτ∈R

∫ τ+1

τ

|f (x + η + t) − f (x + t)|p dx

� supτ∈R

∫ τ+1

τ

|f (x + η + t) − f (x + t)|p dx < εp.

The condition is sufficient too, but the proof is more complicated. By Theorem 2,there exists a function σ ∈ Sp(R) and a subsequence (σmk

)k∈N such that, for eachbounded interval (a, b) and for each measurable bounded function ϕ(x), one has

limk→∞

∫ b

a

ϕ(x)σmk(x) dx =

∫ b

a

ϕ(x)σ (x) dx. (5)

Let us consider a number δ ∈ (0, 1) and set

σ (δ)(x) = 1

δ

∫ x+δ

x

σ (t) dt,

σ (δ)m (x) = 1

δ

∫ x+δ

x

σm(t) dt, m ∈ N.

It immediately follows from (5) that

limk→∞ σ (δ)

mk(x) = σ (δ)(x)

pointwise. We want to prove that

limk→∞

σ (δ)mk

(x) = σ (δ)(x) (6)

in the sense of Sp(R). Let us observe that, for every τ ∈ R, one has∫ τ+1

τ

|σ (δ)(t + η) − σ (δ)(t)|p dt �∫ τ+2

τ

|σ (t + η) − σ (t)|p dt � 2εp,

and similarly for every τ ∈ R and for every m ∈ N, one has∫ τ+1

τ

|σ (δ)m (t + η) − σ (δ)

m (t)|p dt � 2εp.


Furthermore, for every τ ∈ R, we have∫ τ+1

τ

|σ (δ)(t) − σ (δ)mk

(t)|p dt

�∫ τ+1

τ

|σ (δ)(t) − σ (δ)(t + η)|p dt +

+∫ τ+1

τ

|σ (δ)(t + η) − σ (δ)mk

(t + η)|p dt +

+∫ τ+1

τ

|σ (δ)mk

(t + η) − σ (δ)mk

(t + η)|p dt

� 2εp +∫ τ+1

τ

|σ (δ)(t + η) − σ (δ)mk

(t + η)|p dt + 2εp.

On the other hand,∫ τ+1

τ

|σ (δ)(t + η) − σ (δ)mk

(t + η)|p dt =∫ τ+η+1

τ

|σ (δ)(t) − σ (δ)mk

(t)|p dt,

and we may choose η ∈ {η}ε such that τ +η ∈ (o, l) for a suitable l > 0. Therefore∫ τ+η+1

τ

|σ (δ)(t) − σ (δ)mk

(t)|p dt �∫ l+1

0|σ (δ)(t) − σ (δ)

mk(t)|p dt.

By the Lebesgue theorem,∫ l+1

0|σ (δ)(t) − σ (δ)

mk(t)|p dt → 0, for k → ∞.

This proves (6).It is well known that the expansion of the polynomial σ (δ)

mk(x) is the following:

σ (δ)mk

(x) =∑

d(m)λn

aλneiλnδ − 1

iλnδeiλnx.

Since σ (δ)mk

(x) → σ (δ)(x) in Sp(R), the Bohr–Fourier series expansion is the formallimit of the expansion of σ (δ)

mk(x) for k → ∞. By (6) we get

σ (δ)(x) ∼∞∑n=1

aλneiλnδ − 1

iλnδeiλnx .

Let us show that limδ→0 σ(δ)(x) = σ (x), in the sense Sp(R). In fact

∫ τ+1

τ

|σ (t) − σ (δ)(t)|p dt


�∫ τ+1

τ

|σ (t) − σ (t + η)|p dt +

+∫ τ+1

τ

|σ (t + η) − σ (δ)(t + η)|p dt +

+∫ τ+1

τ

|σ (δ)(t + η) − σ (δ)(t)|p dt

� εp +∫ τ+1

τ

|σ (t + η) − σ (δ)(t + η)|p dt + 2εp.

If we choose η ∈ (η)ε such that τ + η ∈ (0, l), we get∫ τ+1

τ

|σ (t + η) − σ (δ)(t + η)|p dt �∫ l

0|σ (t) − σ (δ)(t)|p dt,

and there surely exists a number δ0 > 0 such that, for every δ < δ0, the last integralis less than ε. Hence,

supτ∈R

(∫ τ+1

τ

|σ (t) − σ (δ)(t)|p dt

)1/p

� Kε,

where K is a suitable positive constant. Let us observe that

limδ→0

eiλnδ − 1

iλnδ= 1

and, hence, for the function σ ∈ Sp(R), we get the following Bohr–Fourier seriesexpansion:

σ (x) ∼∞∑n=1

aλneiλnx

and this proves the theorem. ✷Let us now consider the class D(�,R).

THEOREM 4. A necessary and sufficient condition for a generalized trigonomet-ric series∑


to be the Bohr–Fourier series of a function f ∈ D(R,�), is that the Bochner sums(σm(x))m∈N associated with the series (7) verify the condition

M{|σm(x)|} = lim supT→∞

1

2T

∫ T

−T

|σm(x)| dx < C, m ∈ N,

where C is a suitable positive constant.


Proof. The condition is necessary. In fact, let f ∈ D(R,�), with the Bohr–Fourier series given by the series (7).

Then, for the Bochner sums (σm(x))m∈N associated with f (or with the series (7)which is equivalent), we have

σm(x) = limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt = Mt{f (t)Km(t − x)},

where Km is the Bochner–Fejer kernel. We may suppose that f assumes real valuesand therefore

lim supS→∞

1

2S

∫ S

−S

|σm(x)| dx

= lim supS→∞

1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx.

For any choice of η > 0, it is possible to find a sufficiently large S such that

lim supS→∞

1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx

� 1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx + η

and by Fatou’s lemma

1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx + η

� lim infT→∞

1

2S

∫ S

−S

∣∣∣∣ 1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx + η

� lim infT→∞

1

2S

∫ S

−S

(1

2T

∫ T

−T

|f (t)|Km(t − x) dt

)dx + η

� lim infT→∞

1

2T

∫ T

−T

|f (t)|(

1

2S

∫ S

−S

Km(t − x) dx

)dt + η.

For sufficiently large values of S, we have

1

2S

∫ S

−S

Km(t − x) dx � lim supS→∞

1

2S

∫ S

−S

Km(t − x) dx + η,

and, hence,

lim supS→∞

1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (t)Km(t − x) dt

∣∣∣∣ dx

� lim infT→∞

1

2T

∫ T

−T

|f (t)| dt

(lim supS→∞

1

2S

∫ S

−S

Km(t − x) dt

)dx +


+ lim infT→∞

1

2T

∫ T

−T

|f (t)| dt · η + η

= lim infT→∞

1

2T

∫ T

−T

|f (t)| dt (1 + η) + η.

Since η > 0 is arbitrary, we conclude that

lim supS→∞

1

2S

∫ S

−S

|σm(x)| dx � lim supT→∞

1

2T

∫ T

−T

|f (t)| dt.

The condition is sufficient. To prove it, it is sufficient to apply the standard argu-ment already used in [3]. ✷

Let (bλ)λ∈� be a sequence of numbers (real or complex).

DEFINITION 4. We say that (bλ)λ∈� is a multiplier of invarience (or a conversionfactor) of type (B1(�), B1(�)) if the Bohr–Fourier series f ∈ B1(�), i.e.

f (x) ∼∑

λ∈σ(f )

a(λ, f )eiλx,

gives the Bohr–Fourier series∑λ∈σ(f )

bλ a(λ, f )eiλx,

of another function of B1(�) by a (formal) multiplication.

THEOREM 5. A necessary and sufficient condition for the sequence (bλ)λ∈� tobe a multiplier of invarience of the (B1(�), B1(�)) type is that the generalizedtrigonometric series∑

λ∈�bλe−iλx, (8)

is the Bohr–Fourier series of a function φ ∈ D(R,�).Proof. The condition is necessary. In fact, let φ ∈ D(R,�), with Bohr–Fourier

series expansion given by the series (8).Then we have for the Bochner sums (τm(x))m∈N associated with f (or with the

series (8), which is the same)

M{τm(x)} = lim supT→∞

1

2T

∫ T

−T

|τm(x)| dx � C, m ∈ N,

where C is a suitable positive constant. Let now f ∈ B1(�) and let

f (x) ∼∑λ∈�

a(λ;f )e−iλx .


Let us consider the trigonometric series∑λ∈�

bλ a(λ;f )e−iλx, (9)

and let us denote by σm(x) the Bochner sums associated with it.Hence

σm(x) =(m!)2∑

ν1=−(m!)2

· · ·(m!)2∑

νm=−(m!)2

(1 − |ν1|

(m!)2

)· · ·

(1 − |νm|

(m!)2

)bλ aλneiλnx.

Let us observe that, for m ∈ N, we have

σm(x) = limT→∞

1

2T

∫ T

−T

f (x + t)τm(t) dt = Mt{f (x + t)τm(t)}.

Then

limS→∞

1

2S

∫ S

−S

|σm(x)| dx

= limS→∞

1

2S

∫ S

−S

∣∣∣∣ limT→∞

1

2T

∫ T

−T

f (x + t)τm(t) dt

∣∣∣∣ dx

� limS→∞

1

2S

∫ S

−S

(lim

T→∞1

2T

∫ T

−T

|f (x + t)||τm(t)| dt

)dx.

Now, independently of the choice of η > 0, it is always possible to find a suffi-ciently large S > 0 such that

limS→∞

1

2S

∫ S

−S

(lim

T→∞1

2T

∫ T

−T

|f (x + t)||τm(t)| dt

)dx

<1

2S

∫ S

−S

(lim

T→∞1

2T

∫ T

−T

|f (x + t)||τm(t)| dt

)dx + η

� lim infT→∞

1

2S

∫ S

−S

1

2T

∫ T

−T

|f (x + t)||τm(t)| dt dx + η,

by Fatou’s lemma, and therefore

lim infT→∞

1

2S

∫ S

−S

1

2T

∫ T

−T

|f (x + t)||τm(t)| dt dx + η

= lim infT→∞

1

2T

∫ T

−T

|τm(t)|(

1

2S

∫ S

−S

|f (x + t)| dx

)dt + η.

But for sufficiently large values of S, we have

1

2S

∫ S

−S

|f (x + t)| dx � limS→∞

1

2S

∫ S

−S

|f (x + t)| dx + η = ‖f ‖1 + η,


and so

lim infT→∞

1

2S

∫ S

−S

1

2T

∫ T

−T

|f (x + t)||τm(t)| dt dx + η

�(

lim infT→∞

1

2T

∫ T

−T

|τm(t)| dt

)· ‖f ‖1+

+(

lim infT→∞

1

2T

∫ T

−T

|τm(t)| dt

)· η + η

�(

lim supT→∞

1

2T

∫ T

−T

|τm(t)| dt

)· ‖f ‖1+

+(

lim supT→∞

1

2T

∫ T

−T

|τm(t)| dt

)· η + η

� C‖f ‖1 + Cη + η,

and since η is arbitrary, we have

lim supS→∞

1

2S

∫ S

−S

|σm(x)| dx < C ′, m ∈ N,

with C ′ = C‖f ‖1. By Theorem B, p. 9 of [4], we easily obtain that the series (9)is the series of a fuction in B1(�).

The condition is sufficient. Let us suppose that the sequence (bλ)λ∈� is a multi-plier of type (B1(�), B1(�)), but the generalized trigonometric series

∑λ∈�

bλe−iλx, (10)

is not the series of a function φ ∈ D(R,�).Then if we consider the Bochner sums τm(x) associated to the series (10), we

have

lim supm→∞

M{τm(x)} = +∞.

By a well-known result ( [2], Lemma, p. 210), it is possible then to find a functionf ∈ C0(R) such that ‖f ‖∞ � 1, for which one has

lim supm→∞

|M{f (x)τm(x)}| = +∞.

Hence, fixed K > 0, we may choose m0 ∈ N such that, for each m > m0 we have

K <

∣∣∣∣lim supT→∞

1

2T

∫ T

−T

f (x)τm(x) dx

∣∣∣∣.


Then for a fixed m, we may choose T0 > 0 such that for T > T0 we have

K

2<

∣∣∣∣ 1

2T

∫ T

−T

f (x)τm(x) dx

∣∣∣∣=

∣∣∣∣ 1

2T

∫ T

−T

f (x)τm(x) dx

∣∣∣∣ |f (ξ)| �∣∣∣∣ 1

2T

∫ T

−T

τm(x) dx

∣∣∣∣,for the mean value theorem.

Hence

K

2<

∣∣∣∣ 1

2T

∫ T

−T

f (x)τm(x) dx

∣∣∣∣ = ∣∣M{τm(x)}∣∣

and so

lim supm→∞

|M{τm(x)}| = +∞.

Since (bλ)λ∈� is a multiplier of type (B1(�), B1(�)), the trigonometric series∑λ∈�

bλ a(λ, g)eiλx, (11)

is the series of a function of B1(�), as long as g ∈ B1(�) and

g(x) ∼∑λ∈�

a(λ, g)eiλx .

But for the Bochner sums associated to the series (11) we have

σm(x) = limT→∞

1

2T

∫ T

−T

g(x + t)τm(t) dt

= limT→∞

1

2T

∫ T

−T

g(t)τm(x + t) dt

and, hence,

limS→∞

1

2S

∫ S

−S

σm(x) dx

= limS→∞

1

2S

∫ S

−S

(lim

T→∞1

2T

∫ T

−T

g(t)τm(x + t) dt

)dx

= limT→∞

1

2Tg(t)

(limS→∞

1

2S

∫ S

−S

τm(x + t) dx

)dt

= limT→∞

1

2T

∫ T

−T

g(t)M{τm(x)} dt

= limT→∞

1

2T

∫ T

−T

g(t) dt · M{τm(x)}.


It follows that

lim supm→∞

|M{σm(x)}| = +∞,

and, hence,

lim supm→∞

M{|σm(x)|} = +∞,

and so the series (11) cannot be the series of a function in B1(�). ✷

References

1. Besicovitch, A. S.: Almost Periodic Functions, Cambridge Univ. Press, Cambridge, 1932.2. Doss, R.: Contribution to the theory of almost-periodic functions, Ann. Math. 46 (1945), 196–

219.3. Doss, R.: Some theorems on almost-periodic functions, Amer. J. Math. 72 (1950), 81–92.4. Følner, E.: On the dual spaces of the Besicovitch almost-periodic spaces, Mat.-fysiske Medd. 29

(1954), 1–27.

Documents

Almost-Periodic Multipliers