ON CONVOLUTION GROUPS OF COMPLETELY MONOTONE

SEQUENCES/FUNCTIONS AND FRACTIONAL CALCULUS

LEI LI AND JIAN-GUO LIU

Abstract. We study convolution groups generated by completely monotone sequences and com-

pletely monotone functions. Using a convolution group, we define a fractional calculus for a certainclass of distributions. When acting on causal functions, this definition agrees with the traditional

Riemann-Liouville definition for t > 0 but includes some singularities at t = 0 so that the group

property holds. Using this group, we are able to extend the definition of Caputo derivatives of or-der in (0, 1) to a certain class of locally integrable functions without using the first derivative. The

group property allows us to de-convolve the fractional differential equations to integral equations

with completely monotone kernels, which then enables us to prove the general Gronwall inequality(or comparison principle) with the most general conditions. This then opens the door of a priori

energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also

established within this frame under very weak conditions. Besides, we also obtain some interestingresults about completely monotone sequences.

1. Introduction

A sequence c = ck∞k=0 is completely monotone if (I − S)jck ≥ 0 for any j ≥ 0, k ≥ 0 whereScj = cj+1. A sequence is completely monotone if and only if it is the moment sequence of aHausdorff measure (a finite nonnegative measure on [0, 1]) ([26]). Completely monotone sequencesare closely related to infinitely divisible probability distributions on N. In [18, 22], a nice descriptionof completely monotone sequences is given:

Lemma 1. A sequence c is completely monotone if and only if the generating function F (z) =∑∞j=0 cjz

j is a Pick function that is analytic and nonnegative on (−∞, 1).

A function f : C+ → C (where C+ denotes the upper half plane, not including the real line) is Pickif it is analytic such that Im(z) > 0 ⇒ Im(f(z)) ≥ 0. Note that if f(z) is Pick and Im(f(z)) = 0for some Im(z) > 0, then Im(f(z)) = 0 for all z. By the theory of continuation, if f(z) is real onsome interval (a, b), then the function can be extended to C+ ∪ (a, b)× 0 ∪ C− by reflection.

Consider the convolution a∗c defined by (a∗c)k =∑n1≥0,n2≥0 δ

n1+n2

k an1cn2

, which is associative

and commutative. If we use Fc(z) to mean the generating function of c, then it is clear that

Fa∗c(z) = Fa(z)Fc(z). (1)

If c is completely monotone, it is shown that there exist c(r), r ∈ R, such that c(r) ∗ c(s) = c(r+s)

and c(1) = c, i.e. there exists a convolution group generated by the completely monotone sequence([18]). If 0 ≤ r ≤ 1, c(r) is completely monotone. Further,

c(0) = δd = (1, 0, 0, . . .),

2010 Mathematics Subject Classification. Primary 47D03, secondary 34A08, 46F10.Key words and phrases. convolution group, fractional calculus, completely monotone sequence, completely monotonefunction, Riemann-Liouville derivative, Caputo derivative, fractional ODE.

1

2 LEI LI AND JIAN-GUO LIU

is the convolution identity. An algorithm has been proposed in [18] to obtain the convolution groupgenerated by c using its canonical sequence. The most interesting sequence is c(−1), the convolutioninverse, which can be used for deconvolution.

Correspondingly, a function g : (0,∞) → R is completely monotone if (−1)nf (n) ≥ 0 for n =0, 1, 2, . . .. The famous Bernstein theorem says that a function is completely monotone if and onlyif it is the Laplace transform of a Radon measure on [0,∞) ([26, 23, 4]). Completely monotonefunctions appear in fractional calculus, which has drawn much attention to model memory effectsin recent years ([12, 19, 25, 1]). To see this, consider the fractional integral

Jγf =1

Γ(γ)

∫ t

0

(t− s)γ−1f(s)ds, 0 < γ ≤ 1.

The kernel 1Γ(γ) t

γ−1 is completely monotone. The fractional integral is just the convolution between

the kernel and f . We may thus expect the fractional derivative to be determined by the convolutioninverse and the fractional calculus may be given by the convolution group generated by these com-pletely monotone functions. However, it is not clear how the convolution group can be generated bya completely monotone function as this must be put in the frame of distributions, while in generalthe convolution between two distributions is not defined. We will aim to define the convolutiongroup generated by the specific kernel 1

Γ(γ) tγ−1.

The Caputo derivatives ([12, 16, 5]) do not have group property, but are suitable for initial valueproblems and share many properties with the ordinary derivative. In the traditional definition,one has to define the γ-th order derivative (0 < γ < 1) using the first order derivative. In [1], adefinition based on integration by parts is proposed and the first order derivative is not needed butthe function has to possess some regularity. We will use the convolution group to generalize theCaputo derivatives so that the first order derivative is not needed either, and they are defined on alarger class of locally integrable functions. In a much weaker sense, we show that all the fundamentalproperties for Caputo derivatives under this new definition still hold.

The rest of the paper is organized as follows. In Section 2 we first investigate the convolutioninverse of a completely monotone sequence and show that the inverse is well-behaved. Based onthis, a preliminary iterative method is proposed for deconvolution. In Section 3, we introduce aspecific class of distributions and generalize the traditional convolution between two distributionswhere one is required to have compact support to this class. A convolution group is then constructedand used to define a fractional calculus for the distributions in this class. When acting on causalfunctions, this definition agrees with the famous Riemann-Liouville fractional calculus for t > 0.At t = 0, some singularities must be included to make the fractional calculus a group. In Section4, we prove a regularity result for the fractional calculus when acting on a special class of Sobolevspaces. In Section 5, using the convolution group, an extension of Caputo derivatives is proposed sothat the ordinary derivative of the function is not needed in the definition. Some properties of thenew Caputo derivatives are proved, which may be used for fractional ODEs (FODE) and fractionalPDEs (FPDE). Especially, the fundamental theorem of the fractional calculus is valid with the mostgeneral conditions by deconvolution using the group property, which allows us to transform thedifferential equations with orders in (0, 1) to integral equations with completely monotone kernels.In Section 6, based on the definitions and properties in Section 5, we prove some fundamental resultsof FODEs with quite general conditions. Especially, we show the existence and uniqueness of theFODEs using the fundamental theorem, and also show the general Gronwall inequalities. Finally,in Section 7, we define a discrete fractional calculus using a discrete convolution group generated bya specific completely monotone sequence and show that it is consistent with the Riemann-Liouvillecalculus.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 3

2. Deconvolution for a completely monotone kernel

Consider the convolution equation

a ∗ c = f, (2)

where c is a completely monotone sequence and c0 > 0. If we find the convolution inverse of c, theequation can be solved. We start with the properties of the convolution inverse.

2.1. The convolution inverse. We now investigate the property of c(−1), whose generating func-tion is 1/Fc(z). To be convenient, we use F (z) to mean Fc(z), the generating function of c.

Theorem 1. Suppose c is completely monotone and c0 > 0. Let c(−1) be its convolution inverse.Then, Fc(−1) is analytic on the open unit disk, and thus the radius of convergence of its power

series around z = 0 is at least 1. c(−1)0 = 1/c0 and the sequence (−c(−1)

1 ,−c(−1)2 , . . .) is completely

monotone. Furthermore,

0 ≤ −∞∑k=1

c(−1)k ≤ 1

c0. (3)

Proof. The first claim follows from that F (z) has no zeros in the unit disk [18].By Lemma 1, F (z) is Pick and it is positive on (−∞, 1). F (−∞) = 0 if the corresponding

Hausdorff measure does not have an atom at 0 (i.e. the sequence c is minimal. See [26, Chap. IV.Sec. 14] for the definition). Since F (−∞) could be zero, we consider

Gε =1

ε− 1

ε+ F (z), ε > 0.

It is easy to verify that Gε a Pick function, analytic and nonnegative on (−∞, 1).SupposeGε is the generating function of d = (dε0, d

ε1, . . .). By Lemma 1, this sequence is completely

monotone. Then,

Hε =1

z[Gε(z)−Gε(0)] =

F (z)− F (0)

z(ε+ F (0))(ε+ F (z)),

is the generating function of the shifted sequence (dε1, . . .), which is completely monotone. Hence,Hε is also a Pick function, nonnegative and analytic on (−∞, 1).

Taking the pointwise limit of Hε as ε→ 0, we find the limit function

H =F (z)− F (0)

zF (0)F (z)(4)

to be nonnegative on (−∞, 1). By the expression of H, it is also analytic since F (z) is neverzero on C \ [1,∞). Finally, since Im(Hε(z)) ≥ 0 for Im(z) > 0, then Im(H(z)), as the limit, isnonnegative. It follows that the sequence corresponding to H is also completely monotone. If c is

in `1, 0 < H(1) = F (1)−F (0)F (0)F (1) < 1

c0. If F (1) = ‖c‖1 = ∞, we have 0 < H(z) ≤ F (z)

zF (0)F (z) = 1z0c0

.

Fix z0 ∈ (0, 1), then for any z ∈ (z0, 1), H(z) ≤ 1z0c0

. H(z) is increasing in z since the sequence

is completely monotone and therefore nonnegative. Letting z → 1−, by the Monotone convergencetheorem, we have H(1) ≤ 1

z0c0. Taking z0 → 1, H(1) ≤ 1

c0.

By the explicit formula of H(z), we see that it is the generating function of −(c(−1)1 , c

(−1)2 , . . .)

since 1/F (z) is the generating function of c(−1) = (c(−1)0 , c

(−1)1 , . . .). The second claim therefore

follows.

We then have the following claim:

4 LEI LI AND JIAN-GUO LIU

Corollary 1. Equation (2) can be solved stably. In particular, ∀f ∈ `p, ∃a ∈ `p such that a ∗ c = fand

‖a‖p ≤2

c0‖f‖p. (5)

The claim follows directly from the fact that ‖c−1‖1 ≤ 2/c0 and Young’s inequality.

2.2. Computing convolution inverse and deconvolution. The deconvolution actually can beperformed directly as the corresponding matrix is lower triangular. Another method is to use thealgorithm in [18] to find c(r). Then, the inverse is computed as a = c(−1) ∗ f . The algorithm for c(r)

reads

• Determine the canonical sequence b that satisfies (n+ 1)cn+1 =∑nk=0 cn−kbk.

• Compute c(r) by (n+ 1)c(r)n+1 = r

∑nk=0 c

(r)n−kbk.

For a completely monotone sequence, bk ≥ 0 ([13]). If c0 = 1, computing the canonical sequence isstraightforward

bn = (n+ 1)cn+1 −n−1∑k=0

cn−kbk. (6)

Note that Fb(z) = F ′c(z)/Fc(z).

If c0 = 1, c(−1)0 = 1 and |c(−1)

n+1 | ≤ 1n+1

∑nk=0 |c

(−1)n−1 |bk. It’s clear by induction that |c(−1)

n+1 | ≤ cn+1.

For general c0, we can apply the above argument to c/c0 and have the bound

|c(−1)k | ≤ 1

c20|ck|. (7)

This is a pointwise bound for the convolution inverse.

0 10 20 30 40 50 60-2

0

2

4

6

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2(a) (b)

Figure 1. A completely monotone sequence and its convolution inverse

Every completely monotone sequence is the moment sequence of a Hausdorff measure. Fix M asa big integer and denote h = 1/M . xi = (i− 1/2)h. Consider the discrete measures

CM =µ : µ = h

M∑i=1

λiδ(x− xi), λi ≥ 0. (8)

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 5

The weak star closure (〈µ, f〉 =∫

[0,1]fdµ where f ∈ C[0, 1]) of ∪M≥1CM is the set of all Hausdorff

measures. Hence, we can generate completely monotone sequences using

dn =

M∑i=1

hλixni , n = 0, 1, 2, . . . , (9)

where λi’s are generated randomly. In Fig. 1, we plot a completely monotone sequence and itsconvolution inverse obtained using this method. In Fig. 2 (a), we have a sequence which is ofsquare shape; in Fig. 2 (b), we plot the convolution between the sequence in (a) and the completelymonotone sequence obtained in Fig. 1. Fig. 2 (c) shows the solution a ∗ c = f by convolving thesequence in Fig. 2(b) with c(−1). The original sequence is recovered accurately.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1(a) (b) (c)

0 10 20 30 40 50 600

0.05

0.1

0.15

0.2

0.25

0.3

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Figure 2. A simple example of deconvolution

2.3. Deconvolution for a general kernel. Consider that the sequence c is no longer completelymonotone. The direct inverting is computationally inexpensive since the matrix is lower triangular.However, if Fc(z) has a zero point near the origin, the generating function of c(−1) has a small radiusof convergence. Then, an iterative method may be desired.

Consider approximating the sequence c by a completely monotone sequence d = dn of the formin Equation (9). Writing d in matrix form, we have

d =1

mAλ = Aη, (10)

where η = 1mλ. A simple iterative method then reads:

ap+1 = f ∗ d(−1) − ap ∗ [(c− d) ∗ d(−1)]. (11)

Clearly, the iteration converges if ‖(c− d) ∗ d(−1)‖1 < 1. A sufficient condition is therefore

‖d(−1)‖1‖c− d‖1 ≤2

‖η‖1‖c−Aη‖1 < 1, (12)

because d is completely monotone and d0 = ‖η‖1. As long as we can find a solution η to thisoptimization problem, the iterative method can be applied to solve the convolution equation (2).

3. Time-continuous groups and a new definition of fractional calculus

The fractional calculus in continuous time has been used widely in physics and engineering formemory effect, viscoelasticity, porous media etc [12, 6, 16, 19, 5, 1, 25]. Given a function f(t), the

6 LEI LI AND JIAN-GUO LIU

fractional integral with order γ > 0 at t > 0 is given by Abel’s formula

Jγf(t) =1

Γ(γ)

∫ t

0

f(s)(t− s)γ−1ds. (13)

For the derivatives, there are two types that are commonly used: the Riemann-Liouville definitionand the Caputo definition (See [16]). Let n−1 < γ < n, the Riemann-Liouville and Caputo fractionalderivatives at t > 0 are given respectively by

Dγrlf(t) =

1

Γ(n− γ)

dn

dtn

∫ t

0

f(s)

(t− s)γ+1−n ds, (14)

Dγc f(t) =

1

Γ(n− γ)

∫ t

0

f (n)(s)

(t− s)γ+1−n ds. (15)

In [20], an idea using distributions to define fractional derivatives for causal functions was mentionedbriefly. Inspired by the idea, we explore a group generated by some completely monotone functionsin detail and define a fractional calculus for a particular class of distributions.

According to (15), the Caputo derivatives can be defined only if f (n) exists in some sense and thisis unnatural since intuitively it can be defined for functions that are ‘γ-th’ order smooth only. In[1], Allen, Caffarelli and Vasseur have introduced an alternative form of Caputo derivative to avoidusing the f (n) derivative. In Section 5, we also provide an alternative definition. Our definition willnot use f (n) either and will cover these definitions if the function has some regularity.

In this section, we first introduce the time-continuous convolution group and then define a frac-tional calculus using this group. This new fractional calculus has the group property. When actingon causal functions, it agrees with the Riemann-Liouville calculus for t > 0. The singularities att = 0 are important for the group property. At last, a group for right derivatives is mentioned briefly.

3.1. A time-continuous convolution group. Consider

C+ =gα : gα =

u(t)tα−1

Γ(α)

. (16)

Note that gα is completely monotone for 0 < α ≤ 1. This set forms a semi-group of convolution forα > 0, where u(t) is the Heaviside step function. This is because∫ t

0

sα−1(t− s)β−1ds = tα+β−1B(α, β) =Γ(α)Γ(β)

Γ(α+ β)tα+β−1.

The Abel’s formula for fractional integral is given by

Jαϕ(t) = gα ∗ (u(t)ϕ(t)) =u(t)

Γ(α)

∫ t

0

ϕ(s)(t− s)α−1ds, ∀α > 0. (17)

This means the Riemann-Liouville integrals can be understood as the convolution between a memberin C+ and a causal function φ = u(t)ϕ (i.e. φ = 0 for t < 0).

As mentioned in the introduction, we aim to find a convolution group generalized by C+. To dothis, we need to generalize the convolution between distributions.

First, let us introduce the following set of distributions

E = v ∈ D ′(R) : ∃Mv ∈ R, supp(v) ⊂ [−Mv,+∞). (18)

D(R) = C∞c (R) is the set of test functions while D ′(R), the dual of D , is the set of distributions.Clearly, E is a linear vector space.

In general, the convolution between two distributions that are not compactly supported is not welldefined. However, we can define the convolution for distributions in E . We first choose a partitionof unit for R, φi (i.e. φi ∈ C∞c ; 0 ≤ φi ≤ 1; On any compact set K, there are only finitely many

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 7

φi’s that are nonzero;∑i φi = 1 for all x ∈ R). Such a partition exists. As an example, consider

ζ ∈ C∞c (−1, 2) that is nonnegative and ζ = 1 on [0, 1]. Let ζi(x) = ζ(x− i). Then,∑i ζi(x) > 0 for

any x ∈ R, where the sum makes sense because for any x, there are only finitely many terms thatare nonzero. Defining φi = ζi/

∑i ζi yields such a partition.

Definition 1. Given f, g ∈ E , we define

〈f ∗ g, ϕ〉 =∑i

〈f ∗ (φig), ϕ〉, ∀ϕ ∈ D = C∞c , (19)

where f ∗ (φg) is given by the usual definition between two distributions when one of them iscompactly supported [9, Chap. 0].

Lemma 2. The definition is independent of φi and agrees with the usual definition of convolutionbetween distributions whenever one of the two distributions is compactly supported. For f, g ∈ E ,f ∗ g ∈ E , and there exists N1, such that

f ∗ g =∑

i≥−N1,j≥−N1

(fφi) ∗ (gφj),

where the sum makes sense because for any compact set K, there are only finitely many pairs (i, j)such that the support of (fφi) ∗ (gφj) has nonempty intersection with K. Moreover,

f ∗ g = g ∗ f, (20)

f ∗ (g ∗ h) = (f ∗ g) ∗ h. (21)

The proof, though tedious, is very straightforward. The key ingredient is that for g ∈ E , thereexists N1 such that when i < −N1, φig = 0 in the distribution sense. We’ll omit the proof here.

Another property is as following and we omit its proof as well:

Lemma 3. We use D to mean the distributional derivative. Then, letting f, g ∈ E , we have

(Df) ∗ g = D(f ∗ g) = f ∗Dg. (22)

With the tools, we are now able extend C+ to a convolution group C , under the convolution inDefinition 1.

Lemma 4. g0 = δ(t) is the convolution identity and for n ∈ N, g−n = Dnδ is the convolutioninverse of gn.

Proof. Note that g0 and g−n are compactly supported. Then, the convolution can be performed

in the traditional way. That δ is the identity is obvious. For g−n, noting gn = u(t)tn−1

(n−1)! , we pick

ϕ ∈ D = C∞c (R) and have

〈Dnδ ∗ (1

(n− 1)!u(t)tn−1), ϕ〉 = (−1)n

1

(n− 1)!〈u(t)tn−1, Dnϕ〉 = ϕ(0).

Hence, g−n = Dnδ is the convolution inverse.

For 0 < γ < 1, inspired by the fact L(gγ) ∼ 1/sγ where L means the Laplace transform, we guessL(g−γ) ∼ sγ . Hence, we guess the convolution inverse is ∼ D(u(t)t−γ), where D is the distributionalderivative. Actually, we have

Lemma 5. Let 0 < γ < 1, the convolution inverse of gγ is given by

g−γ =1

Γ(1− γ)D(u(t)t−γ

). (23)

8 LEI LI AND JIAN-GUO LIU

Proof. We pick ϕ ∈ D and apply Lemma 3:

〈D(u(t)t−γ) ∗ [u(t)tγ−1], ϕ〉 = −〈u(t)t−γ ∗ u(t)tγ−1, Dϕ〉

= −〈B(1− γ, γ)u(t), Dϕ〉 = −B(1− γ, γ)

∫ ∞0

Dϕ(t)dt = B(1− γ, γ)ϕ(0).

This computation verifies that the claim is true.

For n < γ < n+ 1, we define g−γ = Dnδ ∗ gn−γ .Then, we have defined the class

C = gα : α ∈ R. (24)

Theorem 2. C ⊂ E and it is a convolution group under the convolution on E (Definition 1).

Proof. Using the above facts and Lemma 2, Lemma 3, we find that for any γ > 0, g−γ is theconvolution inverse of gγ . The fact that C+ forms a semigroup, the commutativity and associativityin Lemma 2 imply that C− = g−γ forms a convolution semigroup as well.

The group property can then be verified using the semi-group property and the fact that gγ∗g−γ =δ.

By the explicit expressions of the distributions, we have

Lemma 6. For large t, gα ∼ 1Γ(α) |t|

α−1. If α ≤ 0 and is an integer, Γ(α) =∞, the distribution is

compactly supported.

3.2. Time-continuous fractional calculus. In this section, we use the group C to define thefractional calculus and the (modified) Riemann-Liouville fractional calculus.

3.2.1. Fractional calculus for distributions.

Definition 2. For φ ∈ E , we define the operator Iα : E → E by

Iαφ = gα ∗ φ. (25)

The operators Iα give the definition of fractional calculus for distributions in E . By the definition,it is clear that

Lemma 7. The operators Iα form a group, and I−nφ = Dnφ (n = 1, 2, 3, . . .) where D is thedistributional derivative.

It’s clear that for φ ∈ C∞c and α ∈ Z, Iα gives the usual integral (where the integral is from −∞)or derivative. For example,

I1φ = u(t) ∗ φ =

∫ t

−∞φ(s)ds,

I−1φ = (Dδ) ∗ φ = δ ∗Dφ = φ′.

For α = −γ, 0 < γ < 1 and φ ∈ C∞c , we have

I−γφ =1

Γ(1− γ)

d

dt

∫ t

−∞

1

(t− s)γφ(s)ds =

1

Γ(1− γ)

∫ t

−∞

1

(t− s)γφ′(s)ds.

Remark 1. It is possible to act the group on φ /∈ E but some properties mentioned may be invalid.For example, φ = 1 /∈ E . g1 = u, g−1 = Dδ. Both (u ∗Dδ) ∗ 1 and u ∗ (Dδ ∗ 1) are defined whereu(t) is the Heaviside function, but they are not equal. The associativity is not valid.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 9

In many applications, the functions we study may not defined beyond a certain time T > 0. Thismay require us to define the fractional calculus for some distributions in D ′(−∞, T ). Hence, weintroduce the following set

E T = v ∈ D ′(−∞, T ) : ∃Mv ∈ (−∞, T ), supp(v) ⊂ [Mv, T ). (26)

Clearly, E∞ = E . E T is not closed under the convolution (that means if we pick two distributionsform E T , the convolution then is no longer in E T ). Hence, we cannot define the fractional calculusdirectly as we do for D ′(R). Our strategy is to push the distributions into E first and then pull itback.

Let χn ⊂ C∞c (−∞, T ) be a sequence satisfying (i). 0 ≤ χn ≤ 1. (ii). χn = 1 on [−n, T − 1n ].

We introduce the extension operator KTn : E T → E given by

〈KTn v, ϕ〉 = 〈χnv, ϕ〉 = 〈v, χnϕ〉, ∀ϕ ∈ C∞c (R), (27)

where the last pairing is the one between D ′(−∞, T ) and C∞c (−∞, T ). Denote RT : E → E T as thenatural embedding operator. We define the fractional calculus as

Definition 3. For φ ∈ E T , we define the operator ITα : E T → E T by

ITα φ = limn→∞

RT (Iα(KTn φ)) in D ′(−∞, T ). (28)

We check that the definition is well-given.

Lemma 8. Fix φ ∈ E T . For any sequence χn satisfying the conditions given and ε > 0,M > 0,∃N > 0, such that ∀n ≥ N and ϕ ∈ C∞c (−∞, T ) with supp ϕ ⊂ [−M,T − ε],

〈KTn φ, ϕ〉 = 〈φ, ϕ〉.

It follows that the limit in Definition 3 exists.

Proof. The proof for the first claim is standard, which we omit. For the second claim, we pickϕ ∈ C∞c (−∞, T ). Then, ∀n,

〈RT (Iα(KTn φ)), ϕ〉 = 〈gα ∗ (KT

n φ)), ϕ〉 =∑i

〈(φigα) ∗ (KTn φ)), ϕ〉

There are only finitely many terms in the sum. Then, for each term,

〈(φigα) ∗ (KTn φ)), ϕ〉 = 〈KT

n φ, ζi ∗ ϕ〉

where ζi = φigα(−t) is a distribution supported in [−N1, 0] for some N1 > 0. As a result, ζi ∗ ϕ isC∞c (−∞, T ). By the first claim, the limit exists.

Lemma 9. ITα is independent of the choice of extension operators Kn. For any T1, T2 ∈ (0,∞]and T1 < T2,

RT1IT2α φ = IT1

α RT1φ, ∀φ ∈ E T2 . (29)

Further, Iα is a continuous operator under the weak star topology.

Proof. Let ϕ ∈ C∞c (−∞, T1). Then, we need to show

limn→∞

〈gα ∗ (KT2n φ), ϕ〉 = lim

n→∞〈gα ∗ (KT1

n RT1φ), ϕ〉

We use the partition of unit φi for R and the equation is reduced to

limn→∞

∑i

〈(φigα) ∗ (KT2n φ), ϕ〉 = lim

n→∞

∑i

〈(φigα) ∗ (KT1n RT1φ), ϕ〉

10 LEI LI AND JIAN-GUO LIU

Since there are only finite terms that are nonzero for the sum, we can only consider each. Denoteζi(t) = (φigα)(−t) which is supported in (−∞, 0). Then, it suffices to show

limn→∞

〈KT2n φ, ζi ∗ ϕ〉 = lim

n→∞〈KT1

n RT1φ, ζi ∗ ϕ〉

By Lemma 8, this equality is valid.

Lemma 10. ITα forms a group.

Proof. Due the the result in the previous lemma, ITα (ITβ φ) = limn→∞RT (Iα(RT (Iβ(KTn φ))) =

limn→∞RT (Iα+β(KTn φ))) = ITα+βφ.

Due to the discussion here, we adopt the notation Iα for any ITα for convenience without causingmuch confusion. We then introduce the following sloppy notations in the sense of Definition 3:

gα ∗ φ := Iαφ, ∀φ ∈ E T , T ∈ (0,∞], (30)

and the following is true with this notation:

gα ∗ (gβ ∗ φ) = (gα ∗ gβ) ∗ φ = gα+β ∗ φ, ∀φ ∈ E T , T ∈ (0,∞]. (31)

3.2.2. Modified Riemann-Liouville calculus. The above could be regarded as the fractional calculusstarting from t = −∞. However, we are more interested in fractional calculus starting from t = 0.

Consider causal distributions (‘zero’ for t < 0) for T ∈ (0,∞]:

G Tc =

φ ∈ E T : supp φ ⊂ [0, T )

. (32)

The causality is considered because the memory is usually counted from t = 0 in many applications.We now consider the causal correspondence for a general distribution in E . Let un ∈ C∞c (−1/n, T )where n = 1, 2, . . . be a sequence satisfying (1) 0 ≤ un ≤ 1, (2) un(t) = 1 for t ∈ (−1/(2n), T −1/(2n)). Introduce the space

G T =ϕ ∈ E T : ∃φ ∈ G T

c , unϕw∗−−→ φ for any such sequenceun

. (33)

For ϕ ∈ G T , the distribution φ is denoted as u(t)ϕ where u(t) is the Heaviside step function. Clearly,if ϕ(t) ∈ L1

loc(−∞, T ), where the notation L1loc(U) represents the set of all locally integrable function

defined on U , u(t)ϕ can be understood as the usual multiplication.

Lemma 11. G Tc ⊂ G T . ∀ϕ ∈ G T

c , u(t)ϕ = ϕ.

This claim is easy to show and we choose to omit. This then motivates the following definition:

Definition 4. The (modified) Riemann-Liouville operators Jα : G T → G Tc are given by

Jαϕ = Iα(u(t)ϕ(t)) = gα ∗ (u(t)ϕ(t)), (34)

where gα ∗ (u(t)ϕ(t)) should be understood as in Equation (30).

Proposition 1. Fix ϕ ∈ E T . ∀α, β ∈ R, JαJβϕ = Jα+βϕ and J0ϕ = u(t)ϕ. If we make the domainof them to be G T

c (i.e. the set of causal distributions), then they form a group.

Proof. One can verify that supp(Jαϕ) ⊂ [0, T ). Hence, u(t)Jαϕ = Jαϕ. The claims follow from theproperties of Iα. If ϕ ∈ G T

c , then ϕ is identified with u(t)ϕ.

We are more interested in the cases where ϕ is locally integrable. We call them modified Riemann-Liouville because for good enough ϕ they agree with the traditional Riemann-Liouville operators(Equation (14)) at t > 0 while there are some extra singularities at t = 0. Now, let us illustrate thisby checking some special cases.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 11

When α > 0 and ϕ is a continuous function, we have verified that (34) gives the Abel’s formulaof fractional integrals (Equation (17)). It would be interesting to look at the formulas for α < 0 andsmooth ϕ:

• When −1 < α < 0, we have for any t < T

Jαϕ =u(t)

Γ(1− γ)D

∫ t

0

1

(t− s)γϕ(s)ds =

1

Γ(1− γ)(u(t)t−γ) ∗ (u(t)ϕ′ + δ(t)ϕ(0))

=1

Γ(1− γ)

∫ t

0

1

(t− s)γϕ′(s)ds+ ϕ(0)

u(t)

Γ(1− γ)t−γ . (35)

where γ = −α. This is the Riemann-Liouville fractional derivative.• When α = −1, we have

J−1ϕ = D(u(t)ϕ(t)

)= u(t)ϕ′(t) + δ(t)ϕ(0). (36)

We can verify easily that J−1J1ϕ = J1J−1ϕ = ϕ. Traditionally, the Riemann-Liouvillederivatives for integer values are defined as the usual derivatives. J−1ϕ(t) agrees with theusual derivative for t > 0 but it has a singularity due to the jump of u(t)ϕ at t = 0.

• When α = −1− γ. By the group property, we have for t < T

Jαϕ = J−1(J−γϕ) =1

Γ(1− γ)D(u(t)D

∫ t

0

1

(t− s)γϕ(s)ds

)=

1

Γ(2− |α|)D(u(t)D

∫ t

0

1

(t− s)|α|−1ϕ(s)ds

).

This is again the Riemann-Liouville derivative for t > 0.

In this sense, we call Jα the modified Riemann-Liouville operators. Clearly, Jαϕ agrees withthe traditional Riemann-Liouville calculus for t > 0. However, at t = 0, there is some difference. Forexample, J−1 gives an atom ϕ(0)δ(t) at the origin so that J1J−1 = J−1J1 = J0. The singularitiesat t = 0 are expected since the causal function u(t)ϕ(t) usually has a jump at t = 0.

Remark 2. The fractional time derivatives on distributions in E T provides a suitable frame to definefundamental solutions for fractional PDEs.

3.3. Another group for right derivatives. Now consider another group C generated by

gα =u(−t)Γ(α)

(−t)α−1, α > 0. (37)

For 0 < γ < 1, g−γ = − 1Γ(1−γ)D(u(−t)(−t)−γ) (D means the derivative on t and Du(−t) = −δ(t)).

The action of this group is well-defined if we act it on distributions that have supports on (−∞,M ]or on functions that decay faster than rational functions at ∞.

This group can generate fractional derivatives that are noncausal. For example if φ ∈ C∞c (R),

g−γ ∗ φ = − 1

Γ(1− γ)

d

dt

∫ ∞t

(s− t)−γφ(s)ds. (38)

This derivative is called the right Riemann-Liouville derivative in some literature (See e.g.[16]). The derivative at t depends on the values in the future and it is therefore noncausal.

This group is actually the dual of C in the following sense

〈gα ∗ φ, ϕ〉 = 〈φ, gα ∗ ϕ〉, (39)

12 LEI LI AND JIAN-GUO LIU

where both φ and ϕ are in C∞c (R). (If φ and ϕ are not compactly supported or do not decay atinfinity, then at least one group is not well defined for them.) This dual identity actually providesa type of integration by parts.

It is interesting to write explicitly out the case α = −γ.∫ ∞−∞

gα ∗ φ(t)ϕ(t)dt =

∫ ∞−∞

1

Γ(1− γ)

∫ t

−∞(t− s)−γDφ(s)dsϕ(t)dt

= −∫ ∞−∞

1

Γ(1− γ)

D

Ds

∫ ∞s

(t− s)−γϕ(t)dtφ(s)ds =

∫ ∞−∞

gα ∗ ϕ(s)φ(s)ds.

Remark 3. Alternatively, one may define the operator Iα by 〈Iαφ, ϕ〉 = 〈φ, gα ∗ ϕ〉 for φ ∈ D ′(R)and ϕ ∈ D(R) whenever this is well-defined. This definition however is also generally only valid forφ ∈ E = E∞. This is because gα ∗ ϕ is supported on (−∞,M ] for some M . If φ /∈ E , the definitiondoes not make sense.

4. Regularities of the modified Riemann-Liouville operators

By the definition, it is expected that Jα indeed improve or reduce regularities as the ordinaryintegrals or derivatives do. In this section, we check this topic by considering their actions on aspecific class of Sobolev spaces.

Let us fix T ∈ (0,∞) (we are not considering T =∞).Recall that Hs

0(0, T ) is the closure of C∞c (0, T ) under the norm of Hs(0, T ) (Hs(0, T ) itself equalsthe closure of C∞[0, T ]). We would like to avoid the singularities that may appear at t = 0 but

we don’t require much at t = T . We therefore introduce the space Hs(0, T ) which is the closure ofC∞c (0, T ] under the norm of Hs(0, T ) (If ϕ ∈ C∞c (0, T ], supp ϕ ⊂ C(0, T ] and ϕ ∈ C∞[0, T ]. ϕ(T )may be nonzero.).

We now introduce some lemmas for our further discussion:

Lemma 12. Let s ∈ R, s ≥ 0.

• The restriction mapping is bounded from Hs(R) to Hs(0, T ), i.e. ∀v ∈ Hs(R), then v ∈Hs(0, T ) and there exists C = C(s, T ) such that ‖v‖Hs(0,T ) ≤ ‖v‖Hs(R).

• For v ∈ Hs(0, T ), ∃vn ∈ C∞c (R) such that the following conditions hold: (i). supp vn ⊂(0, 2T ). (ii). ‖vn‖Hs(R) ≤ C‖vn‖Hs(0,T ), where C = C(s, T ). (iii). vn → v in Hs(0, T ).

We use D ′(0, T ) to mean the dual of D(0, T ) = C∞c (0, T ). Recall Definition 4.

Lemma 13. If vn → f in Hs(0, T ), (s ≥ 0), then, Jαvn → Jαf in D ′(0, T ), ∀α ∈ R.

Proof. Since vn, f are in Hs, then they are locally integrable functions.Let ϕ ∈ D(0, T ). Let φi a partition of unit for R. By the definition of Jα,

〈gα ∗ (u(t)vn), ϕ〉 =∑i

〈(gαφi) ∗ (u(t)vn), ϕ〉 =∑i

〈u(t)vn, hiα ∗ ϕ〉,

where hiα(t) = (gαφi)(−t). Note that there are only finitely many terms that are nonzero in thesum since ϕ is compactly supported and gα is supported in [0,∞). Since the support of gαφi is in[0,∞), then the support of hiα ∗ ϕ is in (−∞, T ). Further, hiα ∗ ϕ ∈ C∞c (R). Hence, in distribution,

〈u(t)vn, hiα ∗ ϕ〉 =

∫ T

0

vn(t)(hiα ∗ ϕ)(t)dt→∫ T

0

f(t)(hiα ∗ ϕ)(t)dt

= 〈u(t)f, hiα ∗ ϕ〉 = 〈(gαφi) ∗ (u(t)f), ϕ〉.This verifies the claim.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 13

We now consider the action of Jα on Hs(0, T ) and we actually have:

Theorem 3. If mins, s+α ≥ 0, then Jα is bounded from Hs(0, T ) to Hs+α(0, T ). In other words,

if f ∈ Hs(0, T ), then Jαf ∈ Hs+α(0, T ) and there exists a constant C depending on T , s and α suchthat

‖Jαf‖Hs+α(0,T ) ≤ C‖f‖Hs(0,T ), ∀f ∈ Hs(0, T ). (40)

About this topic, some partial results can be found in [16, 15, 11].

Proof. In the proof here, we use C to mean a generic constant, i.e. C may represent differentconstants from line to line, but we just use the same notation.α = 0 is trivial as we have the identity map.Consider α < 0 first. For α = −n (n = 1, . . .), let v ∈ C∞c (0,∞). J−nv ∈ C∞c (0,∞) because

in this case, the action is the usual n-th order derivative. ‖J−nv‖Hs−n(R) ≤ C‖v‖Hs(R) is clear.Taking a sequence vi ∈ C∞c and supp vi ⊂ (0, 2T ) such that ‖vi‖Hs(R) ≤ C‖vi‖Hs(0,T ), and vi → fin Hs(0, T ). It then follows that ‖J−nvi‖Hs−n(R) ≤ C‖vi‖Hs(0,T ). Since the restriction is bounded

from Hs−n(R) to Hs−n(0, T ), J−nvi is a Cauchy sequence in Hs−n(0, T ). The limit in Hs−n(0, T )

must be J−nf by Lemma 13. Hence, J−n sends Hs(0, T ) to Hs−n(0, T ).By the group property, it suffices to consider −1 < α < 0 for fractional derivatives. Let γ = |α|.

We pick first v ∈ C∞c (0, 2T ). We have

J−γv =d

dt

∫ t

0

(t− s)−γv(s)ds =d

dt

∫ t

0

s−γv(t− s)ds =

∫ t

0

(t− s)−γv′(s)ds.

Since J−γv = (u(t)t−γ) ∗ (v′) and v′ ∈ C∞c (0, 2T ), J−γv is C∞ and supp J−γv ⊂ (0,∞). Notethat the last term is the Caputo derivative. The Caputo derivative equals the Riemann-Liouvillederivative for v ∈ C∞c (0, 2T ).

Since |F(u(t)t−γ)| ≤ C|ξ|γ−1, we find |F(J−γv)| ≤ C|ξ|γ |v(ξ)|. Here, F represents the Fouriertransform operator while v is the Fourier transform of v. Hence,∫

(1 + |ξ|2)(s−γ)|F(J−γv)|2dξ ≤∫

(1 + |ξ|2)s|v(ξ)|2dξ,

or ‖J−γv‖Hs−γ(R) ≤ C‖v‖Hs(R). By Lemma 12, the restriction is bounded

‖J−γv‖Hs−γ(0,T ) ≤ C‖v‖Hs(R).

We now take vi ∈ C∞c (0, 2T ) such that vi → f in Hs(0, T ) and ‖vi‖Hs(R) ≤ C‖vi‖Hs(0,T ), then

‖J−γvi‖Hs−γ(0,T ) ≤ C‖vi‖Hs(0,T ) and J−γvi is a Cauchy sequence in Hs(0, T ) ⊂ Hs(0, T ). The

limit in Hs(0, T ) must be J−γf by Lemma 13. Hence, the claim follows for −1 < α < 0.

Consider α > 0 and n ≤ α < n+ 1. Note that Jn sends Hs(0, T ) to Hs+n(0, T ) since this is theusual integral. We therefore only have to prove the claim for 0 < α < 1 by the group property.

For 0 < α < 1, Jαv =∫ t

0sγ−1v(t − s)ds ∈ C∞(0,∞) and supp Jαv ⊂ (0,∞) for v ∈ C∞c (0, 2T ).

We again set γ = |α| = α. The Fourier transform of Jγv is v/(−iξ)γ [16]. There is singularity atξ = 0 because Jγv ∼ tγ−1 as t → ∞. Since we care the behavior on (0, T ), we can pick a cutoff

function ζ = β(x/T ) where β = 1 on [−1, 1] and zero for |x| > 2. ζ is a Schwartz function.

Noting |F(ζJγv)| ≤ |ζ ∗ v|ξ|−γ | ≤ |ζ ∗ |ξ|−γ |‖v‖∞ ≤ C‖v‖∞, we find

‖ζJγv‖2Hs+γ(R) =

∫R(1 + |ξ|2)s+γ |ζ ∗ (v|ξ|−γ)|2dξ =

∫|ξ|<R

+

∫|ξ|≥R

≤ C‖v‖2∞ +

∫|ξ|≥R

.

14 LEI LI AND JIAN-GUO LIU

For |ξ| ≥ R, we split the convolution ζ ∗ (v|ξ|−γ) into two parts and apply the inequality (a+ b)2 ≤2(a2 + b2). It then follows that∫

|ξ|≥R≤ C

∫|ξ|≥R

dξ(1 + |ξ|2)s+γ

((∫|η|≥|ξ|/2

|ζ(ξ − η)||v|(η)|η|−γdη

)2

+

(∫|η|≤|ξ|/2

|ζ(ξ − η)||v|(η)|η|−γdη

)2)= I1 + I2.

For I1 by Holder inequality and Fubini theorem,

I1 ≤ C∫|ξ|≥R

dξ(1 + |ξ|2)s+γ∫|η|≥|ξ|/2

|ζ(ξ − η)||v(η)|2|η|−2γdη

≤ C∫|η|≥R/2

dη|v(η)|2|η|−2γ

∫ξ

|ζ(ξ − η)|(1 + |ξ|2)s+γdξ

≤ C∫|η|≥R/2

dη|v(η)|2|η|−2γ(1 + |η|2)γ+s ≤ C‖v‖2Hs(R).

Here, C depends on R and ζ.

For I2 part, we note that |ζ(ξ− η)| ≤ C|ξ|−N if R is large enough, since ζ is a Schwartz function.∫|η|≤|ξ|/2

|ζ(ξ − η)||v|(η)|η|−γdη ≤ C‖v‖∞|ξ|−N∫|η|≤|ξ|/2

|η|−γdη ≤ C‖v‖∞|ξ|−N+1−γ .

Hence, I2 ≤ C‖v‖2∞.Overall, we have

‖ζJγv‖Hs+γ(R) ≤ C(‖v‖∞ + ‖v‖Hs(R)) ≤ C‖v‖Hs(R).

Note that v is supported in (0, 2T ) and ‖v‖∞ = ‖v‖L1(0,2T ), which is bounded by it’s L2(0, 2T ) normand thus Hs(R) norm. The constant C depends on T and ζ. Using again that the restriction mapis bounded, we find that

‖Jγv‖Hs+γ(0,T ) ≤ C‖ζJγv‖Hs+γ(R) ≤ C‖v‖Hs(R).

The claim is true for C∞c (0, 2T ). Again, using an approximation sequence vi ∈ C∞c (0, 2T ), ‖vi‖Hs(R) ≤C‖vi‖Hs(0,T ) implies that it is true for Hs(0, T ) also.

Enforcing ϕ to be in Hs(0, T ) removes the singularities at t = 0. This then allows us to obtainthe regularity estimates above and the Caputo derivatives will be the same as Riemann-Liouvillederivatives. If v ∈ H0(0, T ) = L2(0, T ), then the value of Jγv at t = 0 is well-defined for γ > 1/2,

which should be zero (See also [15]), because the Holder inequality implies∫ t

0(t − s)γ−1v(s)ds ≤

C(v)tγ−1/2. Actually, Hγ(0, T ) ⊂ C0[0, T ] if γ > 1/2.

5. An extension of Caputo derivatives

By observing the calculation like (35) above, the Caputo derivatives Dγcϕ (γ > 0) (Equation (15))

may be defined using J−γϕ and the terms like ϕ(0) 1Γ(1−γ) t

−γ , and hence may be generalized to a

function ϕ such that only ϕ(m),m ≤ [γ] exist in some sense, where [γ] means the largest integer thatdoes not exceed γ. We then do not need to require that ϕ([γ]+1) exists. In this paper, we only dealwith 0 < γ < 1 cases as they are mostly used in practice. (For general γ > 1, one has to removesingular terms related to ϕ(0), . . . , ϕ[γ](0), the jumps of the derivatives of u(t)ϕ, from J−γϕ.) Weprove some basic properties of the extended Caputo derivatives according to our definition, which

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 15

will be used for the analysis of fractional ODEs (FODEs) in Section 6, and may be possibly usedfor fractional PDEs (FPDEs).

For our discussion, we first introduce a result from real analysis:

Lemma 14. Suppose f, g ∈ L1loc[0, T ) where T ∈ (0,∞], then h(x) =

∫ x0f(x − y)g(y)dy is defined

for almost every x ∈ [0, T ) and h ∈ L1loc[0, T ).

Proof. Fix M ∈ (0, T ). Denote Ω = (x, y) : 0 ≤ y ≤ x ≤ M. F (x, y) = |f(x − y)||g(y)| ismeasurable and nonnegative on Ω. Tonelli’s theorem ([21, 12.4]) indicates that∫∫

D

F (x, y)dA =

∫ M

0

∫ x

0

|f(x− y)||g(y)|dydx =

∫ M

0

|g(y)|∫ M

y

|f(x− y)|dxdy ≤ C(M),

for some C(M) ∈ (0,∞). This means that F (x, y) is integrable on D. Hence, h(x) is defined for

almost every x ∈ [0,M ] and∫M

0|h(x)|dx <∞. Since M is arbitrary, the claim follows.

Now, fix T ∈ (0,∞] (note that we allow T =∞). Suppose f is a distribution supported in [0, T ).We then formally denote (u(t)tγ−1) ∗ f which should again be understood as in Equation (30) by∫ t

0(t− s)γ−1f(s)ds, t ∈ [0, T ). We say a distribution f is locally integrable function if we can find a

locally integrable function f such that 〈f, ϕ〉 =∫fϕdt,∀ϕ ∈ C∞c ((−∞, T )). It is almost trivial that

Lemma 15. If f ∈ L1loc[0, T ),

(u(t)tγ−1) ∗ f =

∫ t

0

(t− s)γ−1f(s)ds, t ∈ [0, T ), (41)

where the integral on the right is understood in Lebesgue sense.

We introduce

XT =

ϕ ∈ L1

loc[0, T ) : ∃C ∈ R, limt→0+

1

t

∫ t

0

|ϕ− C|dt = 0

. (42)

Recall that L1loc[0, T ) is the set of locally integrable functions on [0, T ), i.e. the functions are

integrable on any compact set K ⊂ [0, T ).Clearly, XT is a vector space and C0[0, T ) ⊂ XT ⊂ L1

loc[0, T ). It is easy to see that C is uniquefor every ϕ ∈ XT . We denote

ϕ(0+) := C. (43)

For convenience, we also introduce the following set for 0 < γ < 1:

Y Tγ =

f ∈ L1

loc[0, T ) : limT→0+

1

T

∫ T

0

∣∣∣∣∫ t

0

(t− s)γ−1f(s)ds

∣∣∣∣ dt = 0

, (44)

and also

XTγ = C + JγY

Tγ =

ϕ : ∃C ∈ R, f ∈ Y Tγ , s.t.ϕ = C + Jγ(f)

. (45)

Recall that

Jγ(f) = gγ ∗ (u(t)f) =1

Γ(γ)

∫ t

0

(t− s)γ−1f(s)ds,

where the integral is in Lebesgue sense by Lemma 15.By the definition of Y Tγ , it is almost trivial to conclude that:

Lemma 16. Y Tγ and XTγ are subspaces of L1

loc[0, T ). If f ∈ Y Tγ , then Jγf(0+) = 0 and XTγ ⊂ XT .

16 LEI LI AND JIAN-GUO LIU

Remark 4. If f ≥ 0, a.e., limt→0+1t

∫ t0|∫ τ

0(τ − s)γ−1f(s)ds|dτ = 0 is equivalent to limt→0+

1t

∫ t0(t−

s)γf(s)ds = 0. Hence, t−γ 6∈ Y Tγ and t−γ+δ ∈ Y Tγ ,∀δ > 0. Whether Y Tγ is strictly bigger than the

space determined by limt→0+1t

∫ t0(t− s)γ |f(s)|ds = 0 or not is an interesting real analysis question.

From here on, if T = ∞, we will simply drop the super-index T for convenience. Now, weintroduce our definition of Caputo derivatives:

Definition 5. For 0 < γ < 1, we define the Caputo derivative of order γ, 0 < γ < 1 as Dγc : XT →

E T ,

Dγcϕ = J−γϕ− ϕ(0+)g1−γ = J−γϕ− ϕ(0+)

u(t)

Γ(1− γ)t−γ . (46)

Recall J−γϕ = g−γ ∗(u(t)ϕ(t)) and in the case T <∞, it is understood as in Equations (30). Notethat we have used explicitly the convolution operator J−γ in the definition. The convolution structurehere enables us to establish the fundamental theorem (Theorem 4) below using deconvolution so thatwe can rewrite fractional differential equations using integral equations with completely monotonekernels.

Note that if ϕ does not have regularities, Dγcϕ is generally a distribution in E T . If ϕ ∈ Hγ

0 (0, T1)for some T1 <∞, Dγ

cϕ a function in H00 (0, T1) = L2(0, T1) ⊂ L1(0, T1) as we have seen in Section 4.

Lemma 17. By the definition, we have the following claims:

(1) ∀ϕ ∈ XT , Dγcϕ = J−γ(ϕ− ϕ(0+)). For any constant C, Dγ

cC = 0.(2) Dγ

c : XT → E T is a linear operator.(3) ∀ϕ ∈ XT , 0 < γ1 < 1 and γ2 > γ1 − 1, we have

Jγ2Dγ1c ϕ =

Dγ1−γ2c ϕ, γ2 < γ1,

Jγ2−γ1(ϕ− ϕ(0+)), γ2 ≥ γ1.

(4) Suppose 0 < γ1 < 1. If f ∈ Yγ1 , Dγ2c Jγ1f = Jγ1−γ2f for 0 < γ2 < 1.

(5) If Dγ1c ϕ ∈ XT , then for 0 < γ2 < 1, 0 < γ1 + γ2 < 1,

Dγ2c D

γ1c ϕ = Dγ1+γ2

c ϕ−Dγ1c ϕ(0+)g1−γ2 .

(6) Jγ−1Dγcϕ = J−1ϕ−ϕ(0+)δ(t). If we define this to be D1

c , then for ϕ ∈ C1[0, T ), D1cϕ = ϕ′.

Proof. The first follows from g−γ ∗ (u(t)) = g−γ ∗ g1 = g1−γ . The second is obvious. The thirdclaim follows from Jγ2D

γ1c ϕ = Jγ2(J−γ1ϕ − ϕ(0+)g1−γ1) = Jγ2−γ1ϕ − ϕ(0+)g1−γ1+γ2 , which holds

by the group property. For the fourth, we just note that Jγ1f(0+) = 0 and use the group propertyfor Jα. The fifth statement follows easily from the third statement. The last claim follows fromJγ−1(J−γϕ− ϕ(0+)g1−γ) = J−1ϕ− ϕ(0+)g0 and Equation (35).

Now, we verify that our definition agrees with (15) if ϕ has some regularity:

Proposition 2. For ϕ ∈ XT , if the distributional derivative on (0, T ) D+ϕ is a locally integrablefunction, then

Dγcϕ =

1

Γ(1− γ)

∫ t

0

D+ϕ(s)

(t− s)γds, 0 < γ < 1, (47)

where the convolution integral can be understood in the Lebesgue sense. Further, Dγcϕ ∈ L1

loc[0, T ).

Proof. We first show the claim for T = ∞. Define ϕε = ϕ ∗ ηε where ηε = 1ε η( tε ) and 0 ≤ η ≤ 1

satisfies: (i). η ∈ C∞c (R) with supp(η) ⊂ (−M, 0) for some M > 0. (ii).∫ηdt = 1 . ϕε is clearly

smooth. Then, we have in D ′(R)

D(u(t)ϕε) = δ(t)ϕε(0) + u(t)D(ϕε),

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 17

which can be verified easily. Now, take ε→ 0. Let C = ϕ(0+). Then, |ϕε(0)−C| = |∫M

0η(−x)ϕ(εx)dx−

C| ≤ sup |η|∫M

0|ϕ(εx)− C|dx = sup |η| MεM

∫Mε

0|ϕ(y)− C|dy → 0. Hence, ϕε(0)→ C.

Since u(t)ϕε → u(t)ϕ in L1loc(R), D(u(t)ϕε) → D(u(t)ϕ) in D ′(R). For u(t)D(ϕε), by the

convolution property we have D(ϕε) = (Dϕ)ε in D ′(R). Since D+ϕ is locally integrable, we candefine its values to be zero on (−∞, 0] and then it becomes a distribution in D ′(R). We still denoteit as D+ϕ. If supp(η) ⊂ (−M, 0), then u(t)(Dϕ)ε → u(t)D+ϕ in D ′(R). (Note carefully that if ϕis a causal function, Dϕ as a distribution in D ′(R) generally has an atom Cδ(t), but here we areusing D+ϕ in D ′(0,∞) that does not include the singularity.) This then verifies the distributionalidentity for T =∞.

By the definition of J−γ (Definition 4) and applying Lemma 3,

J−γϕ =1

Γ(1− γ)D(u(t)t−γ

)∗ (u(t)ϕ) =

1

Γ(1− γ)

(u(t)t−γ

)∗D(u(t)ϕ)

=1

Γ(1− γ)

(u(t)t−γ

)∗ (δ(t)ϕ(0+) + u(t)D+ϕ).

The first term gives ϕ(0+) 1Γ(1−γ)u(t)t−γ .

Consider now (u(t)t−γ) ∗ (u(t)D+ϕ). It is clear that

(u(t)t−γ) ∗ (u(t)D+ϕ) = limT→∞

(u(t)t−γχ(t ≤ T )) ∗ (u(t)D+ϕχ(t ≤ T )), in D ′.

With the truncation, each function becomes in L1(R). The convolution hT = (u(t)t−γχ(t ≤ T )) ∗(u(t)D+ϕχ(t ≤ T )) ∈ L1(R) and

hT (t) =

∫ t

0

1

(t− s)γD+ϕ(s)ds, 0 < t < T,

where the integral is in Lebesgue sense. Hence, as T → ∞, we find that (u(t)t−γ) ∗ (u(t)D+ϕ) isa measurable function and for almost every t, Equation (47) holds and the integral is a Lebesgueintegral. By Lemma 14, Dγ

cϕ ∈ L1loc[0,∞).

Then, by Definition 5, we obtain Equation (47). This then finishes the proof for T =∞.For T <∞, consider KT

n ϕ = χnϕ ∈ L1loc[0,∞) and it can be shown easily that

D+(χnϕ) = χ′nϕ+ χnD+ϕ ∈ L1loc[0,∞),

where D+(χnϕ) is understood as the distributional derivative on (0,∞) while D+ϕ is understoodas the distributional derivation on (0, T ). By the result just proved, we have

Dγc (KT

n ϕ) =1

Γ(1− γ)

∫ t

0

χ′nϕ+ χnD+ϕ

(t− s)γds.

We then use the fact KTn ϕ(0+) = ϕ(0+) and the definition of J−γ for ϕ ∈ D ′(−∞, T ), we can see

that the weak star limits of left hand side and right hand side give the desired equality.

Remark 5. Note that if the distributional derivative is locally integrable, then it is also called theweak derivative in PDE theory. Sometimes ϕ is differentiable almost everywhere and the conventionalderivative is denoted as ϕ′. Even if ϕ′ may be defined almost everywhere, one should be careful notto use ϕ′ in the integrand. However, if ϕ is nice enough, Dϕ will be identical to ϕ′. On one hand,if the weak derivative exists and the function is differentiable almost everywhere, then Dϕ = ϕ′ a.e.On the other hand, if ϕ is integrable on [0, T1], T1 > 0 and differentiable everywhere such that ϕ′

is in L1(0, T1), then ϕ is absolutely continuous on [0, T1] and hence the weak derivative exists.

18 LEI LI AND JIAN-GUO LIU

Regarding the Caputo derivatives, we introduce several results that may be applied for fractionalODEs (FODE) and fractional PDEs (FPDE). The first is the fundamental theorem of fractionalcalculus:

Theorem 4. Suppose ϕ ∈ XT and denote f = Dγcϕ (0 < γ < 1) which is supported in [0, T ). Then,

u(t)ϕ(t) = ϕ(0+) + Jγ(f) = ϕ(0+) +1

Γ(γ)

∫ t

0

(t− s)γ−1f(s)ds. (48)

where we only use the integral to mean gγ ∗f as explained in Equation (30). If f is locally integrable,the integral can be understood in Lebesgue sense.

Proof. By the definition (Equation (46)), f(t)+ϕ(0+) u(t)Γ(1−γ) t

−γ = J−γ(ϕ(t)). Then, the convolution

group property yields:

u(t)ϕ(t) = Jγ

(f(t) + ϕ(0+)

u(t)

Γ(1− γ)t−γ)

= Jγf + ϕ(0+)1

Γ(γ)Γ(1− γ)

∫ t

0

(t − s)γ−1s−γds.

Since∫ t

0(t− s)γ−1s−γds = B(γ, 1− γ) = Γ(γ)Γ(1− γ), the second term is just ϕ(0+).

If f ∈ L1loc[0, T ), by Lemma 15, the integral is also a Lebesgue integral.

This theorem is fundamental for fractional differential equations because this allows us to trans-form the fractional differential equations to integral equations with completely monotone kernels(which are nonnegative). Then, we are able to establish the general Gronwall inequalities or thecomparison principle (Theorem 7), which in turn opens the door of a priori estimate of fractionalPDEs.

Using Theorem 4, we conclude that

Corollary 2. Suppose ϕ(t) ∈ XT , ϕ ≥ 0 and ϕ(0+) = 0. If the Caputo derivative Dγcϕ (0 < γ < 1)

is locally integrable, and Dγcϕ ≤ 0, then ϕ(t) = 0. (The local integrability assumption can be dropped

if we understand the inequality in the distribution sense which we will introduce in Section 6.)

Now, we consider functions whose Caputo derivatives are L1loc[0, T ). Actually, we have

Proposition 3. Let f ∈ L1loc[0, T ). Then, Dγ

cϕ = f has solutions ϕ ∈ XT if and only if f ∈ Y Tγ .

If f ∈ Y Tγ , the solutions are in XTγ and they can be written as

ϕ = C +1

Γ(γ)

∫ t

0

(t− s)γ−1f(s)ds, ∀C ∈ R.

Further, ∀ϕ ∈ XTγ , Dγ

cϕ ∈ Y Tγ .

Proof. Suppose that Dγcϕ = f has a solution ϕ ∈ XT . Since f ∈ L1

loc[0, T ), by Theorem 4, we have

ϕ(t) = ϕ(0+) +1

Γ(γ)

∫ t

0

(t− s)γ−1f1(s)ds,

and the integral is in Lebesgue sense. Since ϕ ∈ XT , we have limT→01T

∫ T0|ϕ(s)−ϕ(0+)|ds = 0. It

follows that1

T

∫ T

0

∣∣∣∣∫ t

0

(t− s)γ−1f1(s)ds

∣∣∣∣ dt→ 0, T → 0.

Hence, f1 ∈ Y Tγ . This implies that there are no solutions for Dγcϕ = f if f ∈ L1

loc \ Y Tγ . (For

example, Dγcϕ = t−γ has no solutions in XT .)

For the other direction, now assume f ∈ Y Tγ . We first note Dγcϕ = 0 implies that ϕ is a constant

by Theorem 4. One then can check that Jγf , which is in XTγ by definition, is a solution to the

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 19

equation Dγcϕ = f . Hence any solution can be written as Jγf + C. The other direction and the

second claim are shown.We now show the last claim. If ϕ ∈ XT

γ , by definition (Equation (45)), ∃f ∈ Y Tγ , C ∈ R such that

ϕ = C +1

Γ(γ)

∫ t

0

(t− s)γ−1f(s)ds.

Since f ∈ Y Tγ , Jγf(0+) = 0 by Lemma 16. This means C = ϕ(0+). Now, apply J−γ on both sides.

Note that J−γC = Cg−γ ∗ g1 = Cg1−γ = C u(t)Γ(1−γ) t

−γ . By the group property, we find that

Dγcϕ = J−γJγf = f.

Remark 6. For the purpose of applications in FPDE analysis, we note that a solution to the equationDγcϕ = f where f ∈ L1

loc is understood in the distribution sense or in weak sense: Find ϕ ∈ XTγ ,

such that ∀ζ ∈ C∞c (−∞, T ):

− 1

Γ(1− γ)

⟨∫ t

0

(t− s)−γϕ(s)ds,Dζ⟩− ϕ(0+)

Γ(1− γ)

∫ t

0

(t− s)−γζ(s)ds =

∫ t

0

f(t− s)ζ(s)ds.

Motivated by the discussion in Section 4, we have another claim about the equation Dγcϕ = f

where the solutions are in C0[0, T ):

Proposition 4. Suppose f ∈ L1[0, T ) ∩ Hs(0, T ). If s satisfies (i). s ≥ 0 when γ > 1/2 or (ii).s > 1

2 − γ when γ ≤ 1/2, then ∃ϕ ∈ C0[0, T ) such that Dγcϕ = f . If T = ∞, we can also ask for

f ∈ L1loc[0,∞) ∩ Hs

loc(0,∞).

Let us focus on the mollifying effect on the Caputo derivatives. Let η ∈ C∞c (R), 0 ≤ η ≤ 1 and∫ηdt = 1. We define ηε = 1

ε η( tε ). For a distribution ϕ ∈ D ′(R), it is well known that

ϕε = ϕ ∗ ηε ∈ C∞(R) (49)

and that supp(ϕε) ⊂ supp(ϕ) + supp(ηε) where A+B = x+ y : x ∈ A, y ∈ B.

Proposition 5. Suppose supp(η) ⊂ (−∞, 0). ∀ϕ ∈ X (T = ∞), Dγc (ϕε) → Dγ

cϕ in D ′(R) asε→ 0+. Also,

Dγcϕ

ε =

∫ t

0

Dϕε

(t− s)γds =

ϕε(t)− ϕε(0)

tγ+ γ

∫ t

0

ϕε(t)− ϕε(s)(t− s)γ+1

ds. (50)

If E(·) is a continuous convex function, then

DγcE(ϕε) ≤ E′(ϕε)Dγ

cϕε. (51)

Proof. For any ϕ ∈ X, it is clear that u(t)ϕε → u(t)ϕ in L1loc[0,∞) and hence in distribution. Using

the definition of Jα and the definition of convolution on E , one can readily check J−γϕε → J−γϕ in

D ′(R).For ϕε(0) → ϕ(0+), we need supp(η) ⊂ (−∞, 0). There then exists M > 0 such that η(t) = 0 if

t < −M . Let C = ϕ(0+). Then, |ϕε(0)−C| = |∫M

0η(−x)ϕ(εx)dx−C| ≤ sup |η|

∫M0|ϕ(εx)−C|dx =

sup |η| MεM∫Mε

0|ϕ(y)− C|dy → 0. Hence, ϕε(0)→ C. This then shows that the first claim is true.

For the alternative expressions of Dγcϕ

ε, we have used Proposition 2 and integration by parts.These are valid since ϕε ∈ C∞.

Multiplying E′(ϕε(t)) on ϕε(t)−ϕε(0)tγ + γ

∫ t0ϕε(t)−ϕε(s)

(t−s)γ+1 ds and using the inequality

E′(a)(a− b) ≥ E(a)− E(b)

20 LEI LI AND JIAN-GUO LIU

since E(·) is convex, we find

E′(ϕε)Dγcϕ

ε ≥ E(ϕε(t))− E(ϕε(0))

tγ+ γ

∫ t

0

E(ϕε(t))− E(ϕε(s))

(t− s)γ+1ds.

Since E is convex and ϕε is smooth, the second integral converges for almost every t. Further,

D+E(ϕε) exists and D+E(ϕε) = E′(ϕε)Dϕε. Then, the right hand side must be∫ t

0D+E(ϕε)

(t−s)γ . By

Proposition 2, we end the last claim.

Remark 7. It is interesting to note that we have to choose η such that supp(η) ⊂ (−∞, 0) to mollify.Using other mollifiers, we may not get the correct limit. This reflects that Caputo derivatives onlymodel the dynamics of memory from t = 0+ and the singularities at t = 0 for Riemann-Liouvillederivatives are removed totally. It is exactly this nature that makes Caputo derivatives to havemany similarities with the ordinary derivative and suitable for initial value problems.

Proposition 5 is also useful for ϕ ∈ XT , T <∞. This is because we can extend ϕ to X by definingits values to be zero when t ≥ T , resulting in ϕ. It is not hard to find RTDγ

c ϕ = Dγcϕ. Further, if

supp η ⊂ (−∞, 0), supp ϕε ⊂ (−∞, T ). Hence, we denote

Dγcϕ

ε = RTDγc ϕ

ε ∈ L1loc[0, T ).

As consequences of this simple observation and Proposition 5, we conclude:

Corollary 3. Let supp η ⊂ (−∞, 0). ∀ϕ ∈ XT :

• If there exists a sequence εk, such that Dγcϕ

εk converges in L1loc[0, T ). Then, the limit is

Dγcϕ and Dγ

cϕ ∈ L1loc[0, T ).

• If ϕ is γ + δ (δ > 0) Holder continuous (see [7, Chap. 5]), then Dγcϕ

ε → Dγcϕ uniformly on

[0, T1] for any T1 ∈ (0, T ) and Dγcϕ = ϕ(t)−ϕ(0)

tγ + γ∫ t

0ϕ(t)−ϕ(s)(t−s)γ+1 ds.

The alternative expressions for Dγcϕ

ε are more useful for FPDEs. Unfortunately, for ϕ ∈ XT ,these forms are generally not valid. For example, ϕ must have some regularity for the integral form,which is used in [1], to make sense in the Lebesgue sense. It is possible to show that if ϕ has betterregularity, then Dγ

cϕε converges to Dγ

cϕ in better spaces, but we are not going to do this here.Lastly, we consider the Laplace transform. If ϕ ∈ X (T = ∞), Dγ

cϕ is only a distribution.Recalling that its support is in [0,∞), we then define the Laplace transform of Dγ

cϕ as

L(Dγcϕ) = lim

M→∞〈Dγ

cϕ, ζMe−st〉, (52)

where ζM (t) = ζ0(t/M). ζ0 ∈ C∞c , 0 ≤ ζ0 ≤ 1 satisfies: (i) supp ζ0 ⊂ [−2, 2] (ii) ζ0 = 1 fort ∈ [−1, 1]. This definition clearly agrees with the usual definition of Laplace transform if the usualLaplace transform of function ϕ exists.

To be convenient in discussion below, we will denote the following set

E(L) =ϕ ∈ X : ∃L > 0, s.t. lim

A→∞‖e−Atϕ‖L∞[A,∞) = 0

. (53)

Proposition 6. If ϕ ∈ E(L), then L(Dγcϕ) is defined for Re(s) > L and is given by

L(Dγcϕ) = L(ϕ)sγ − ϕ(0+)sγ−1. (54)

Proof. ζMe−st ∈ C∞c . Then, it follows that

〈Dγcϕ, ζMe

−st〉 =

⟨g−γ ∗ (u(t)ϕ)− ϕ(0+)u(t)t−γ

Γ(1− γ), ζMe

−st⟩

= − 1

Γ(1− γ)

⟨(u(t)t−γ) ∗ (u(t)ϕ), ζ ′Me

−st − sζMe−st⟩− ϕ(0)

Γ(1− γ)

∫ ∞0

t−γζMe−stdt.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 21

Note that supp ζ ′M ∩ [0,∞) ⊂ [M, 2M ]:∣∣∣∣∣∫ 2M

M

∫ t

0

(t− τ)−γϕ(τ)dτζ ′Me−stdt

∣∣∣∣∣ ≤ sup |ζ ′0|M

∫ 2M

M

∫ t

0

(t− τ)−γe−Lτ |ϕ(τ)|dτe−(Re(s)−L)tdt.

By the assumption, there exists T0 such that |e−Lτϕ(τ)| < 1, a.e. if τ > T0. Hence, if M >

2T0, the inner integral is controlled by T−γ0

∫ T0

0|ϕ(τ)|dτ +

∫ tT0

(t − τ)−γdτ ≤ C(1 + t1−γ). Since

limM→∞∫ 2M

M(1 + t1−γ)e−εtdt = 0 for any ε > 0, we find that the term associated with ζ ′M tends to

zero as M →∞.For the second term,

s

Γ(1− γ)

⟨(u(t)t−γ) ∗ (u(t)ϕ), ζMe

−st⟩

=s

Γ(1− γ)

∫ ∞0

∫ t

0

(t− τ)−γϕ(τ)dτζM (t)e−stdt

=s

Γ(1− γ)

∫ ∞0

ϕ(τ)e−sτ∫ ∞τ

(t− τ)−γζM (t)e−(t−τ)sdt.

As M →∞, one finds that ∫ ∞0

t−γζM (t+ τ)e−tsdt→ Γ(1− γ)sγ−1,

for every τ > 0. Since Re(s) > L, the dominate convergence theorem implies that the first termgoes to L(ϕ)sγ .

Similarly, the last term converges to −ϕ(0+)sγ−1.

To conclude, there is no group property for Caputo derivatives. However, the Caputo derivativesremove the singularities at t = 0 compared with the Riemann-Liouville derivatives and have manyproperties that are similar to the ordinary derivative so that they are suitable for initial valueproblems.

6. Fractional ordinary differential equations

Some analysis about fractional ODEs could be found in [6, 5]. In this section, we prove someresults about fractional ODEs using the Caputo derivatives, whose new definition and propertieshave been discussed in Section 5. The assumptions here are sufficiently weak and conclusions aregeneral.

We start with a simple linear FODE:

Proposition 7. Let 0 < γ < 1, λ 6= 0, and suppose b(t) is continuous such that there exists L > 0,lim supt→∞ e−Lt|b(t)| = 0. Then, there is a unique solution of the equation

Dγc v = λv + b(t), v(0) = v0

in E(L) ⊂ X(Recall that this means T =∞) and is given by

v(t) = v0eγ,λ(t) +1

λ

∫ t

0

b(t− s)e′γ,λ(s)ds, (55)

where eγ,λ(t) = Eγ(λtγ) and

Eγ(z) =

∞∑n=0

zn

Γ(γn+ 1)(56)

is the Mittag-Leffler function [14].

22 LEI LI AND JIAN-GUO LIU

Proof. For γ ∈ (0, 1), we have by Proposition 6

L(Dγc v) = sγV (s)− v0s

γ−1,

and V (s) = L(v).By the assumption on b(t), the Laplace transform B(s) = L(b) exists. Hence, any continuous

solution of the initial value problem that is in E(L) must satisfy

V (s) = v0sγ−1

sγ − λ+

B(s)

sγ − λ.

Using the equality ([19, Appendix])∫ ∞0

e−stEγ(sγztγ)dt =1

s(1− z), (57)

and denoting eγ,λ(t) = Eγ(λtγ), we have that

L(eγ,λ) =sγ−1

sγ − λ, L(e′γ,λ) =

λ

sγ − λ.

Taking the inverse Laplace transform of V (·), we get (55). Note that though e′γ,λ blows up at t = 0,it is integrable near t = 0 and the convolution is well-defined. Further, by the asymptotic behaviorof b and the decaying rate of e′γ,λ, the solution is again in E(L). The existence part is proved.

Since the Laplace transform of functions that are in E(L) is unique, the uniqueness part isproved.

Remark 8. For the existence of solutions in X (where T =∞), the condition lim supt→∞ e−Lt|b(t)| =0 can be removed, since for any t > 0, we can redefine b beyond t so that lim supt→∞ e−Lt|b(t)| = 0.The value of v(t) keeps unchanged by the re-definition according to Formula (55). Hence, (55) givesa solution for any continuous function b in X. The uniqueness in X instead of in E(L) will beestablished in Theorem 5 below.

We now consider a general FODE:

Theorem 5. Let 0 < γ < 1 and v0 ∈ R. Consider the initial value problem (IVP)

Dγc v = f(t, v), v(0) = v0, (58)

where v(0) is understood as Equation (43). If there exist T > 0, A > 0 such that f is defined andcontinuous on D = [0, T ]× [v0 −A, v0 +A] such that there exists L > 0,

sup0≤t≤T

|f(t, v1)− f(t, v2)| ≤ L|v1 − v2|,∀v1 v2 ∈ [v0 −A, v0 +A].

Then, the IVP has a unique solution in any X T for T ≤ T1, and T1 is given by

T1 = min

T, sup

t≥0

M

Γ(1 + γ)tγEγ(Ltγ) ≤ A

, (59)

where M = sup(t,v)∈D |f(t, v)| and Eγ(z) =∑∞n=0

zn

Γ(nγ+1) is the Mittag-Leffler function.

Moreover, v(·) ∈ C0[0, T1]. Further, the solution is continuous with respect to the initial value.Indeed, fix t ∈ (0, T1) and ε0 < T1 − t. Then, ∀ε ∈ (0, ε0), ∃δ0 > 0 such that for any |δ| ≤ δ0, thesolution of the FODE with initial value v0 + δ, vδ(·), exists on (0, T1 − ε0) and

|vδ(t)− v(t)| < ε. (60)

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 23

Proof. The proof is just like the proof of existence and uniqueness theorem for ODEs using Picarditeration.

We first show the existence in XT1 and then the restriction to X T will be the solution in thatspace. Consider the sequence constructed by

vn = v0 + gγ ∗ (u(t)f(t, vn−1)), v0 = v0. (61)

where t ∈ [0, T1]. Recall that the convolution in principle is understood as in Equation (30) and it

can be understood as the Lebesgue integral 1Γ(γ)

∫ t0(t− s)γ−1f(s, vn−1(s))ds by Lemma 15.

Consider En = |vn − vn−1|. We then find for t ∈ [0, T1],

E1 = |gγ ∗ (u(t)f(t, v0))| ≤ M

Γ(1 + γ)T γ1 =: MT1

One can verify that MT1 ≤ A by the definition of T1.

Now, we assume that∑n−1m=1E

m ≤ A so that |vn−1−v0| ≤ A. We will then show that this is truefor En as well. With this induction assumption, we can find that for t ∈ [0, T1] and m = 2, 3, . . . , nso that

Em ≤ Lgγ ∗ (u(t)Em−1),

Note that u(t) = g1 and by group property, we find

Em ≤MT1Lm−1g1+(m−1)γ ,m = 1, 2, . . . , n.

It then follows thatn∑

m=1

Em < MT1

∞∑m=1

Lm−1g1+(m−1)γ = MT1

∞∑m=1

Lm−1t(m−1)γ

Γ((m− 1)γ + 1)= MT1

Eγ(Ltγ),

where Eγ is the Mittag-Leffler function. By the definition of T1, we find that∑nm=1E

m ≤ A for allt ∈ [0, T1]. Hence, by induction, we have (t, vn(t)) ∈ D for all t ∈ [0, T1] and n ≥ 0. It then followsthat

∑n |vn − vn−1| converges uniformly on [0, T1]. This shows that vn → v uniformly on [0, T1]. v

is then continuous. Hence,

v(t) = v0 + gγ ∗ (u(t)f(t, v)), t ∈ [0, T1].

where v is continuous. This means that v(·) is a solution.

We now show the uniqueness. Suppose both v1, v2 ∈ X T are solutions. Then, v1(t) and v2(t)must fall into [v0 −A, v0 +A] for t ≤ T ≤ T1 as we consider the solution curve in D.

Let w = v1 − v2. Then, by the linearity of Dγc , we have in distribution sense that

Dγcw = f(t, v1)− f(t, v2).

Since f is Lipschitz continuous and both v1, v2 ∈ L1loc[0, T ), f(t, v1) − f(t, v2) ∈ L1

loc[0, T ). ByTheorem 4, we have in Lebesgue sense that

u(t)w =1

Γ(γ)

∫ t

0

(t− s)γ−1(f(s, v1(s)− f(s, v2(s)))ds.

For all t ≤ T ,

u(t)|w(t)| ≤ L

Γ(γ)

∫ t

0

(t− s)γ−1|w(s)|ds = Lgγ ∗ (u(t)|w|).

Since gα ≥ 0 when α > 0 and u(t)gα = gα, we convolve both sides with gn0γ and have

u(t)w1(t) ≤ LT gγ ∗ (u(t)w1),

where u(t)w1 = gn0γ ∗ (u(t)|w|) ≥ 0. Since gn0γ = C1tn0γ−1, if n0 is large enough, w1 is continuous

on [0, T ].

24 LEI LI AND JIAN-GUO LIU

Then, by iteration and group property, we have

u(t)w1 ≤ Lngnγ ∗ (u(t)w1) ≤ Ln

Γ(nγ + 1)sup

0≤t≤T|w1|

∫ t

0

(t− s)nγds.

Since Γ(nγ + 1) grows exponentially, this tends to zero. Hence, w1 = 0 on [0, T ]. Then, convolveboth sides with g−n0γ on w1 = gn0γ ∗ (u(t)|w|) = 0 and we find |w| = 0. Hence, u(t)v1 = u(t)v2 in

D ′(−∞, T ) and therefore v1 = v2 in X T .For continuity on initial value, we let u = v − v0. Since the Caputo derivative of a constant is

zero, the equation is reduced to

Dγc u = f(t, u+ a), u(0) = 0.

For this question, once again, construct the sequence un+1 like in Equation (61) and show un+1 iscontinuous on a ∈ (v0 − δ0, v0 + δ0) for some chosen δ0. Performing similar argument, un+1 → uuniformly on [0, T1 − ε0]. Then, u is continuous on v0.

Corollary 4. Suppose f is defined and continuous on [0,∞)× R. If ∀T > 0, there exists LT suchthat

sup0≤t≤T

|f(t, v1)− f(t, v2)| ≤ LT |v1 − v2|, ∀v1, v2 ∈ R,

then the solution exists and is unique in X (i.e. T =∞).

Proof. This is true because for any T > 0, A can be chosen arbitrarily large so that T1 = T . Hence,the solution exists and is unique in XT . Since T is arbitrary, the claim follows.

The following corollary, though straightforward, is useful

Corollary 5. Let 0 < γ < 1 and v0 ∈ R. Assume f : [0,∞) × R → R is continuous and Lipschitzin v on any compact subset of [0,∞)× R. If the IVP

Dγc v = f(t, v), v(0) = v0, (62)

has solutions in both Xh1 and Xh2 for 0 < h1 < h2, denoted by vh1 and vh2 , then, the solutionsmust agree on [0, h1] and hence limt→h1

vh1(t) = vh2(h1) which is finite.

Proof. By Theorem 5, both vh1 and vh2 are continuous. By the inverse formula (Theorem), we cansee that vh2 when restricted on [0, h1] is also a solution. By the uniqueness part of Theorem 5,vh1(t) = vh2(t) for all t < h1. The last claim then follows from the continuity of vh2 .

By Corollary 5, we can identify the solutions in different spaces Xh. Hence, we can talk about theinterval of existence of the solution. Now, we establish the following global behavior of the FODE

Theorem 6. Let f : [0,∞)×R→ R is continuous and ∀A > 0,M > 0, there exists LA,M > 0 suchthat

sup0≤t≤A

|f(t, v1)− f(t, v2)| ≤ LA,M |v1 − v2|,∀v1, v2 ∈ [−M,M ]. (63)

Let 0 < γ < 1 and v0 ∈ R. Consider the IVP:

Dγc v = f(t, v), v(0) = v0. (64)

Then, either the solution v(·) ∈ C0[0,∞) or there exists a Tb > 0 such that

lim supt→T−b

|v(t)| =∞.

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 25

Proof. We define

Tb = suph>0The solution of the IVP v(·)exists on[0, h) .

If Tb =∞, the solution then exists on R. We show that if Tb <∞ and lim supt→T−b|v(t)| <∞, then

the solution can be extended to a larger interval and therefore we have a contradiction.We now pick δ > 0. Define another function f(t, v) so that it agrees with f(t, v) on [0, Tb + δ]×

[−M − δ,M + δ] where

M = sup0≤t≤Tb

|v(t)| <∞

by the assumption. Define

f(t, v) =

f(t,M + δ) t ≤ Tb + δ, v ≥M + δ,

f(Tb + δ,M + δ) t ≥ Tb + δ, v ≥M + δ,f(Tb + δ, v) t ≥ Tb + δ, |v| ≤M + δ,f(t,−M − δ) t ≤ Tb + δ, v ≤ −M − δ,

f(Tb + δ,−M − δ) t ≥ Tb + δ, v ≤ −M − δ.

Then, f(t, v) is Lipschitz uniformly. By Corollary 4, the solution to the FODE with f exists on[0,∞) and the solution v is continuous. One can verify that v(·) solves this modified problem as wellon [0, Tb) and hence it agrees with v on this interval. It follows that ∃δ1 ∈ (0, δ) such that so that(t, v(t)) ∈ [0, Tb + δ]× [−M − δ,M + δ] for any t ≤ Tb + δ1. Therefore, on [0, Tb + δ1], v solves theoriginal problem as well, which contradicts with the definition of Tb. Our assumption is thereforenot valid and the claim follows.

We then call Tb the ‘blowup time’. Before further discussion, we introduce the concept of negativedistributions.

Definition 6. We say f ∈ D ′(−∞, T ) is a negative distribution if ∀ϕ ∈ C∞c (−∞, T ) with ϕ ≥ 0,we have

〈f, ϕ〉 ≤ 0. (65)

We say f1 ≤ f2 for f1, f2 ∈ D ′(−∞, T ) if f1 − f2 is negative. We say f1 ≥ f2 if f2 − f1 is negative.

The following lemma is well-known and we omit the proof:

Lemma 18. If f ∈ L1loc[0, T ) ⊂ E T is a negative distribution. Then, f ≤ 0 almost everywhere with

respect to Lebesgue measure.

Lemma 19. Suppose f1, f2 ∈ G Tc for T ∈ (0,∞]. If f1 ≤ f2 (we mean f1 − f2 is a negative

distribution), and that both h1 = Jγf1 and h2 = Jγf2 are functions in L1loc[0, T ), then

h1 ≤ h2, a.e.

Proof. By the definition of G Tc , supp(f1) ⊂ [0, T ) and supp(f2) ⊂ [0, T ).

Suppose the conclusion is not true. Then by Lemma 18, there exists ϕ ∈ C∞c (−∞, T ), ϕ ≥ 0such that ∫ ∞

0

(h1 − h2)ϕdx > 0.

Also, we are able to find ε > 0 such that supp ϕ ⊂ (−∞, T − ε).This means

〈Jγf1, ϕ〉 > 〈Jγf2, ϕ〉.By Lemma 8, we can find an extension operator KT

n so that

〈(u(t)tγ−1) ∗ f1, ϕ〉 > 〈(u(t)tγ−1) ∗ f2, ϕ〉.

26 LEI LI AND JIAN-GUO LIU

where f1 = KTn f1 and f2 = KT

n f2 while

〈fi, ϕ〉 = 〈fi, ϕ〉, i = 1, 2. ∀ϕ ∈ C∞c (−∞, T ), supp ϕ ⊂ (−∞, T − ε].

Let φi be a partition of unit for R. Then, by Definition 1, we have∑i

(〈[φi(u(t)tγ−1)] ∗ f1, ϕ〉 − 〈[φi(u(t)tγ−1)] ∗ f2, ϕ〉

)> 0.

There are only finitely many terms that are nonzero in this sum. Hence, there must exist i0 suchthat ⟨

[φi0(u(t)tγ−1)] ∗ f1, ϕ⟩−⟨

[φi0(u(t)tγ−1)] ∗ f2, ϕ⟩> 0.

Denote ζi0 = φi0(u(t)tγ−1) and ζi0 = ζi0(−t). Then,

〈f1 − f2, ζi0 ∗ ϕ〉 > 0.

ζi0 is a positive integrable function with compact support and ϕ ≥ 0 is compactly supported smooth

function. Then, ζi0 ∗ ϕ ≥ 0 and is C∞c (−∞, T ) with the support in (−∞, T − ε]. It then means

〈f1 − f2, ζi0 ∗ ϕ〉 > 0.

This is a contradiction since we have assumed f1 ≤ f2.

Now, we introduce the general Gronwall inequality (or the comparison principle), which is im-portant for a priori energy estimates of FPDEs:

Theorem 7. Let f(t, v) be a continuous function such that it satisfies the conditions in Theorem 5and ∀t ≥ 0, x ≤ y implies f(t, x) ≤ f(t, y). Let 0 < γ < 1. Suppose v1(t) is continuous satisfying

Dγc v1 ≤ f(t, v1),

where this inequality means Dγc v1 − f(t, v1) is a negative distribution. Suppose also that v2 is the

solution of the equation

Dγc v2 = f(t, v2), v2(0) ≥ v1(0).

v2 is continuous on its largest interval of existence [0, Tb) by Theorem 5 and the corollaries. Then,on [0, Tb), v1(t) ≤ v2(t).

Correspondingly, if

Dγc v1 ≥ f(t, v1),

and v2 solves

Dγc v2 = f(t, v2), v2(0) ≤ v1(0).

Then, v1(t) ≥ v2(t) on the largest interval of existence for v2.

Proof. Fixing T1 ∈ (0, Tb), we show that v1(t) ≤ v2(t) for any t ∈ (0, T1]. Then, since T1 is arbitrary,the claim follows. By Theorem 5, we can find ε0 > 0 such that the solution the equation with initialdata v2(0) + ε, denoted by vε2, exists on (0, T1] whenever ε ≤ ε0.

Define T ε = inft > 0 : vε2(t) ≤ v1(t). Since both v1 and vε2 are continuous and vε2(0) > v1(0),T ε > 0. We claim that T ∗ = T1. For otherwise, we have vε2(T ∗) = v1(T ∗) and v1(t) < v2(t) fort < T ∗. Note that f(t, v1) is a continuous function. By using Theorem 4 and Lemma 19:

v1(T ∗) = v1(0) +1

Γ(γ)

∫ T∗

0

(T ∗ − s)γ−1Dγc v1ds ≤ v1(0) +

1

Γ(γ)

∫ T∗

0

(T ∗ − s)γ−1f(s, v1(s))ds

< v2(0) + ε+1

Γ(γ)

∫ T∗

0

(T ∗ − s)γ−1f(s, v2(s))ds = vε2(T ∗).

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 27

(The first integral is understood as JγDγc v1. However, the obtained distribution is a continuous

function and Lemma 19 guarantees that we can have the first inequality.) This is a contradiction.Hence, v1(t) < vε2(t) for all t ∈ (0, T1]. Taking ε→ 0+ and using the continuity on initial value yieldthe claim. Then, by the arbitrariness of T1, the first claim follows.

Similarly arguments hold for the second claim, except that we perturb v2(0) to v2(0) − ε toconstruct vε2.

We now show another result that may be useful for FPDEs:

Proposition 8. Suppose f is nondecreasing and Lipschitz on any bounded interval, satisfying f(0) ≥0. Then, the solution of Dγ

c v = f(v), v(0) ≥ 0 is continuous and nondecreasing on the interval [0, Tb)where Tb is the blow up time.

Proof. v is continuous by Theorem 5. It is clear that f(v) ≥ 0 whenever v ≥ 0. We first show thatv(t) ≥ v(0) for all t ∈ [0, Tb).

Let vε be the solution with initial data v(0) + ε > v(0). vε is continuous. Fix T1 ∈ (0, Tb). Thereexists ε0 > 0 such that ∀ε ∈ (0, ε0), vε is defined on (0, T1]. Define T ∗ = inft ≤ T1 : vε(t) ≤ v(0).T ∗ > 0 because vε(0) > v(0). We show that T ∗ = T1. If this is not true, vε(T ∗) = v(0) andvε(t) > v(0) ≥ 0 for all t < T ∗. Applying Theorem 4 for t = T ∗ says vε(T ∗) > v(0), a contradiction.Hence, vε ≥ v(t) for all t ∈ (0, T1]. Taking ε → 0 yields v(t) ≥ v(0) for all t ∈ [0, T1], since thesolution is continuous on initial data by Theorem 5. The arbitrariness of T1 concludes the claim.

Now, consider the function sequence

Dγc v

n = f(vn−1), vn(0) = v(0), v0 = v(0) ≥ 0.

All functions are continuous and defined on R. Since v(t) ≥ v0 on [0, Tb), then f(v) ≥ f(v0), and itfollows that v ≥ v1 on [0, Tb) by Theorem 4. Doing this iteratively, we find that v ≥ vn for all n ≥ 0and all t ∈ [0, Tb).

Theorem 4 shows that

v1 = v(0) + f(v(0))1

Γ(1 + γ)tγ ≥ v0.

By Theorem 4 again, it follows that v2 ≥ v1 and hence vn ≥ vn−1 for all n ≥ 1. Therefore, vn

is increasing in n and bounded above by v on [0, Tb). Then, vn → v uniformly on [0, T1] for anyT1 ∈ (0, Tb). v ∈ C0[0, T1]. Since T1 is arbitrary, v ∈ C0[0, Tb) and thus in L1

loc[0, Tb). In distributionsense, we therefore have vn → v and thus Dγ

c vn → Dγ

c v. Hence, v is a solution of Dγc v = f(v) with

v(0) = v(0). Since the solution is unique in XTb , it must be v.This said, now we show that vn is increasing in t. This is clear by induction if we note this fact:

“If h(t) ≥ 0 is a nondecreasing locally integrable function, then gγ ∗ h is nondecreasing in t”, whichcan be verified by direct computation. v, as the the limit of increasing functions, is increasing.

The results in Proposition 5 can be generalized to ϕ ∈ Rm easily and we conclude the following:

Proposition 9. Suppose that E(·) ∈ C1(Rm,R) is convex and that ∇E is locally Lipschitz contin-uous. Let v : [0, Tb)→ Rm solve the FODE:

Dγc v = −∇vE(v) (66)

so that Tb is the blowup time. Assume DγcE(v) is a measurable function and ∃η ∈ C∞c (−∞, 0) and

a sequence εk → 0 such that DγcE(vεk), Dγ

c vεk (See (49)) each converges in L1

loc[0, Tb). Then

E(v(t)) ≤ E(v(0)), t ∈ [0, Tb).

Similarly, with the same conditions except that the equation is of the form:

Dγc v = J∇vE(v), (67)

28 LEI LI AND JIAN-GUO LIU

where J is an anti-Hermitian constant operator, then

E(v(t)) ≤ E(v(0)), t ∈ [0, Tb).

Proof. If E(·) ∈ C1, then ∇vE(v) is continuous. By Theorem 5, v is continuous. Then, Dγc v is

continuous by the equation. vε → v uniformly on [0, T ] for T < Tb, and thus bounded. Then,E(vε) → E(v) uniformly on [0, T ]. Since T is arbitrary, E(vε) → E(v) in L1

loc[0, Tb) and thus indistribution. Hence, Dγ

cE(vε)→ DγcE(v) in distribution.

Passing limit on the subsequences for

DγcE(vε) ≤ ∇vE(vε) ·Dγ

c vε,

by the conditions given, left hand side converges to DγcE(v) in L1

loc[0, Tb) and the right hand sideconverges to ∇vE(v)Dγ

c u in L1loc[0, Tb) because ∇vE(vε)→ ∇vE(v) uniformly on any [0, T ] interval.

Then, it follows thatDγcE(v) ≤ ∇vE(v) ·Dγ

c v = −|∇uE(v)|2.The solution to Dγ

cE(v) = 0 is a constant E(v) = E(v(0)). By Theorem 7 (The function isf(t, E(v)) = 0), we conclude that

E(v(t)) ≤ E(v(0)).

For the second case, we have

DγcE(v) ≤ ∇vE(v) ·Dγ

c u = ∇vE · (J∇vE) = 0.

Theorem 7 again yieldsE(v(t)) ≤ E(v(0)).

As a straightforward corollary, we have

Corollary 6. If lim|v|→∞E(v) = ∞, and the conditions in Proposition 9 hold, then the solutionsexist globally. In other words, Tb =∞.

The physical background of fractional Hamiltonian system Dγc v = J∇vE(v) has been discussed

in [24, 3]. The fractional Hamiltonian system can be rewritten as v(t) = v(0) + Jγ(J∇vE(v)) byTheorem 4, which is of the Volterra type v0 ∈ v(t) + b ∗ (Av). The general Volterra equationswith completely positive kernels and m-accretive A operators have been discussed in [2] and thesolutions have been shown to converge to the equilibrium. In the fractional Hamiltonian system,−J∇vE(v) is not m-accretive and it is not clear whether the solutions converge to one equilibriumsatisfying ∇vE(v) = 0 or not. However, for the following simple example, the energy function Eindeed dissipates and the solutions converge to the equilibrium:

Example: Consider E(p, q) = 12 (p2 + q2) and

Dγc q =

∂E

∂p= p,

Dγc p = −∂E

∂q= −q,

where we assume γ < 1/2, and the initial conditions are p(0) = p0, q(0) = q0.Applying Theorem 5 for u(t) = (p, q) and Corollary 4, we find that both p and q are continuous

functions, and exist globally. By Lemma 17, Dγc (Dγ

c q) = D2γc q − p0g1−γ . Applying Dγ

c on the firstequation yields

D2γc q − p0g1−γ = Dγ

c p = −q.By Proposition 7, we find

q(t) = q0β2γ(t)− p0g1−γ ∗ (β′2γ) = q0β2γ(t)− p0Dγc β2γ ,

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 29

where β2γ = E2γ(−t2γ) is defined as in Proposition 7. Note that g1−γ ∗ (β′2γ) = Dγc β2γ is due to

Proposition 2.Using the second equation, we find

p(t) = p0 + Jγ(−q) = p0β2γ − q0Jγβ2γ .

Actually, from the equation of p, D2γc p = −p− q(0)g1−γ , we find

p(t) = p0β2γ(t) + q0Dγc β2γ .

Since β2γ solves the equation D2γc v = −v, we see Dγ

c β2γ = −Jγβ2γ . Those two expressions for p(t)are identical. Hence, we find that

E(t) = E(0)(β22γ + (Jγβ2γ)2). (68)

(Note that β2γ and Jγβ2γ are the solutions to the following two equations respectively:

D2γc v = −v, v(0) = 1,

D2γc v = −v + g1−γ , v(0) = 0.

Unlike the corresponding ODE system where the two components are both solutions to v′′ = −v,here since D2γ

c = −v only has one solution for an initial value, the two functions are from differentequations.) By the series expression of E2γ = E2γ,1 ([14]):

β2γ = E2γ(−t2γ) =

∞∑n=0

(−1)ng2nγ+1, Jγβ2γ =

∞∑n=0

(−1)ng(2n+1)γ+1 = tγE2γ,γ+1(−t2γ). (69)

According to the asymptotic behavior listed in [10, Eq. (7)],

E2γ,ρ(−t2γ) ∼ −p∑k=1

(−1)kt−2γk

Γ(ρ− 2γk), 0 < γ < 1. (70)

As examples, E1/2,1(−t1/2) = et(

1− 2√π

∫ t1/20

exp(−s2)ds)

and E1/2,1(−t1/2) ∼ 1/t1/2. E1,1(−t) =

e−t decays exponentially while in Equation (70), Γ(1− k) =∞ for all k = 1, 2, 3, . . .. Note that onecannot take the limit γ → 1 in Equation (70) and conclude that E2,1(−t2) and tE2,2(−t2) both decayexponentially. Actually, β2 = E2,1(−t2) = cos(t) and J1β2 = tE2,2(−t2) = sin(t). The singular limitis due to an exponentially small term C1 exp(C2(γ − 1)t) in E2γ,ρ.

If γ < 1/2, Γ(β − 2γ) 6= ∞ and the leading order behavior is t−2γ . Hence, Jγβ2γ ∼ t−γ and Edecays like t−2γ . Following the same method, one can use the last statement in Lemma 17 to solvethe case γ = 1/2 and find that γ = 1/2 case is still right: β1 = E2,1(−t) = e−t decays exponentially

fast while t1/2E2,2(−t) decays like t−1/2. Since E decays to zero, the solution must converge to(0, 0).

Whether this is true for 1/2 < γ < 1 is interesting, which we leave for future study.

Remark 9. According to Theorem 5, the system has a unique solution for any (p(0), q(0)) ∈ R2 and0 < γ < 1. We have only considered 0 < γ < 1/2 here just because we can find the solution easilyfor these cases. If one defines the Caputo derivatives for 1 < α < 2 (α = 2γ) consistently, we guessthat the expressions above are still correct.

7. A discrete convolution group and discrete fractional calculus

We now introduce a special discrete convolution group generated by a completely monotone se-quence to define discrete fractional calculus and show that the discrete fractional calculus is consistentwith the Riemann-Liouville fractional calculus (Definition 4) with appropriate time scaling.

30 LEI LI AND JIAN-GUO LIU

To motivate this, let us consider a smooth function f(t). We sample this function with step sizek and obtain a sequence a = ai∞i=0 with ai = f(ik).

Using numerical approximations ([17]) for the fractional calculus, we find the following sequencefor fractional integral Jγ , 0 < γ ≤ 1:

(cγ)j =kγ

γΓ(γ)((j + 1)γ − jγ).

Then, Jγf ≈ cγ ∗ a. cγ is completely monotone.These sequences cγ do not form a convolution semi-group. However, each sequence generates a

group. Let c(α)γ be the group generated by cγ , with c

(γ)γ = cγ . We hope that c(α)

γ is a reasonableconvolution group to define discrete fractional calculus. Below, we focus on the case γ = 1.

In general, the sequence may not be from sampling of f and we may not have the concept of timestep k. This then motivates the following discrete fractional calculus for a general given sequence:

7.1. A definition of discrete fractional calculus. Consider the sequence

c(α) := c(α)1 ,∀α ∈ R, (71)

whose generating function is F (z) = (1− z)−α. Note that c(1) = (1, 1, 1, . . .) and c(α), 0 ≤ α ≤ 1 arecompletely monotone.

Definition 7. For a sequence a = (a0, a1, . . .), we define the discrete fractional operators Jdα : RN →RN as

Jdαa = c(α) ∗ a. (72)

By the discussion in [18], c(α) form a discrete convolution group, and it follows that Jdα : α ∈ Rform a group.

7.2. Consistency with the continuous convolution group. Even though we define the discreteconvolution group without worrying its relationship with the Riemann-Liouville fractional calculus,we show that they are actually consistent. Given a function time-continuous function f(t), we picka time step k > 0 and define the sequence a with ai = f(ik) (i = 0, 1, 2, . . .). We consider

Tαf = kαJdαa. (73)

We now show that for t > 0 (Tαf)n converges to Jαf(t) as k = t/n→ 0+ when f is smooth. To beconvenient, we’ll only consider |α| ≤ 1. We will start with some lemmas.

Lemma 20. The m-th term of c(α) has the following asymptotic behavior as m→∞:

c(α)m = [zm](1− z)−α ∼ mα−1

Γ(α)

(1 +

α(α− 1)

2m+O(

1

m2)

), (74)

for α 6= 0,−1,−2, . . ..

Here, [zm]F (z) means the coefficient of zm in the Taylor series of F (z) about center 0. One canrefer to [8] for the proof.

Lemma 21. For |α| < 1, let Sn =∑ni=0 c

(α)i . Then, as m→∞, we have:

Sm =mα

Γ(1 + α)

(1 +O(

1

m)

), (75)

Rm =

m∑i=0

(m− i)c(α)i =

m1+α

Γ(2 + α)

(1 +O(

1

m)

). (76)

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 31

Proof. α = 0 is trivial. Suppose α 6= 0. Sn is the convolution between c(α) and c(1) and S =c(α+1) by the group property. Hence, the generating function for S is (1− z)−1−α. Similarly, sincec(2) = (1, 2, 3, . . .), R = c(α+2). Applying Lemma 20 yields the claims.

Now we state the consistency result:

Theorem 8. Suppose f ∈ C2[0,∞). For any t > 0, define k = t/n. Then, |(Tαf)n− (Jαf)(t)| → 0as n→∞ for |α| ≤ 1.

Proof. Below, we only show the consistency and we are not trying to find the best estimate for theconvergence rate.α = 0, (T0f)n = f(t) and the claim is trivial.Case 1: α > 0If α = 1, we have c(1) = k(1, 1, 1, 1, 1, . . .) is the sequence for integration. We find that Tαf =∑ni=0 kf(t− ik). This is O(k) approximation for the integral.Consider 0 < α < 1. Fix t > 0 and k = t/n with n 1. Let 1 K n. By Lemma 20, we have

(Tαf)n = kαK−1∑i=0

c(α)i f((n− i)k) + kα

n∑i=K

iα−1

Γ(α)f((n− i)k) + kα

n∑i=K

O(1

i2−α).

The last term is easily estimated: kα∑ni=K O( 1

i2−α ) = O(Kα−1kα).

Consider the first term. We have f((n− i)k) = f(t)− f ′(ξ)ik and f(t− s) = f(t)− f ′(ξ)s. Then,it follows by Lemma 20 and Lemma 21∣∣∣kα K−1∑

i=0

c(α)i f((n−i)k)− 1

Γ(α)

∫ (K−1)k

0

f(t−s)sα−1ds∣∣∣ ≤ |f(t)|

∣∣∣kα K−1∑i=0

c(α)i −

1

Γ(α)

∫ (K−1)k

0

sα−1ds∣∣∣

+ sup |f ′|Kkα+1K−1∑i=0

c(α)i + C

∫ (K−1)k

0

sup |f ′|sαds ≤ C(Kα−1kα +K1+αk1+α).

Lastly a very rough estimate is∣∣∣∣∣kαn∑

i=K

iα−1

Γ(α)f((n− i)k)− 1

Γ(α)

∫ t

(K−1)k

f(t− s)sα−1ds

∣∣∣∣∣ ≤ C(Kk)α−2k.

Choosing K ∼ k−1/2, all terms tend to zero as k → 0.Case 2: −1 ≤ α < 0. γ = |α|.If α = −1, the sequence is (1,−1, 0, 0, . . .). (T−1f)n = f ′(nk) + O(k) = f ′(t) + O(k) by the

standard finite difference.In the case that f(t) ∈ H1(0, T )∩C2[0,∞), f(0) = 0. (T−1f)i = f ′(ik)− f ′(0)δi0 +O(k) for any

i ≤ n. The consistency for −1 < α < 0 follows then from the group property

(Tαf)n = (T−γf)n = [T1−γ(T−1f)]n = [T1−γf′]n − f ′(0)c(1−γ)

n k1−γ +O(k).

We still have O(k) as the error because ‖k1+αc(1+α)‖1 ≤ 2k1+α by Theorem 1, and ‖(k1+αc(1+α)) ∗O(k)‖∞ ≤ Ck. By the result for integral, we find that T1−γ(f ′)−

∫ t0(t− s)−γf ′(s)ds→ 0 as k → 0.

Since f(0) = 0, the latter is equal to (J−γf)(t). The consistency is shown.Consider a general smooth function f that we do not have f(0) = 0. (T−1f)0 = f(0)/k. The

first term is very singular and the above group argument fails. We now verify directly that theconsistency is also true. Consider that α ∈ (−1, 0) and γ = |α|. By definition,

(T−γf)n =f(t)

kγ+

1

kγ

n∑i=1

c(−γ)i f((n− i)k).

32 LEI LI AND JIAN-GUO LIU

The continuous Riemann-Liouville fraction derivative equals

(J−γf)(t) = f(0)1

Γ(1− γ)t−γ +

1

Γ(1− γ)

∫ t

0

f ′(s)

(t− s)γds

=f(t− k/b)

kγ+

1

Γ(1− γ)

[∫ t

t−k/b

f ′(s)

(t− s)γds− γ

∫ t−k/b

0

f(s)

(t− s)γ+1ds

],

where b is chosen such that bγ = Γ(1− γ) ≥ 1.It is not hard to see

f(t)

kγ− f(t− k/b)

kγ− 1

Γ(1− γ)

∫ t

t−k/b

f ′(s)

(t− s)γds = O(k1−γ).

We need to estimate ∣∣∣ 1

kγ

n∑i=1

c(−γ)i f((n− i)k)− 1

Γ(−γ)

∫ t

k/b

f(t− s)sγ+1

ds∣∣∣. (77)

We first observe that expression (77) is of order O(k) and O((k/b)1−γ) if f(s) = 1 and f(s) = t−srespectively.

Consider that f = 1. By Lemma 21 and noting bγ = −γΓ(−γ), we have

k−γn∑i=1

c(−γ)i = k−γ

n∑i=0

c(−γ)i − k−γ = k−γ

(n−γ

Γ(1− γ)− 1

)+O

(1

(nk)γn

)=

1

Γ(−γ)

∫ t

k/b

1

sγ+1ds+O

(1

(nk)γn

).

For f(s) = t− s, we need to estimate |k−γ∑ni=1 c

(−γ)i (ik)− 1

Γ(−γ)

∫ tk/b

s−γds|. By Lemma 21 again,

we find

k−γn∑i=1

c(−γ)i (n− i)k − 1

Γ(−γ)

∫ t

k/b

t− ssγ+1

ds = O((k/b)1−γ).

By what has been just computed k−γ∑ni=1 c

(−γ)i nk − 1

Γ(−γ)

∫ tk/b

tsγ+1 ds = O(k) and thus∣∣∣∣∣k−γ

n∑i=1

c(−γ)i ik − 1

Γ(−γ)

∫ t

k/b

s−γds

∣∣∣∣∣ = O((k/b)1−γ).

Using these two facts, we can therefore assume f(t) = f ′(t) = 0 in Equation (77) with introducingerror O(k1−γ).

For 1 ≤ i ≤ K − 1, |f((n− i)k)| ≤ C(ik)2. Since c(−γ)i is all negative for i ≥ 1 and by Lemma 21:

|k−γK−1∑i=1

c(−γ)i

1

2f ′′(ξ)(ik)2| ≤ k−γKk2C1

K−1∑i=1

|ic(−γ)i | ≤ CKk2−γK1−γ ≤ C(Kk)2−γ .

Also, for k/b ≤ s ≤ (K − 1)k, |f(t− s)| ≤ Cs2, and hence∣∣∣∣∣∫ (K−1)k

k/b

f(t− s)sγ+1

ds

∣∣∣∣∣ ≤ C(Kk)2−γ .

By Lemma 20, ∣∣∣∣∣k−γn∑

i=K

(c(−γ)i − i−1−γ

Γ(−γ)

)f((n− i)k)

∣∣∣∣∣ ≤ CK−1−γ .

ON CONVOLUTION GROUPS AND FRACTIONAL CALCULUS 33

Hence,∣∣∣∣∣k−γn∑

i=K

c(−γ)i f((n− i)k)− 1

Γ(−γ)

∫ t

(K−1)k

f(t− s)sγ+1

ds

∣∣∣∣∣ ≤CK−1−γ +

∣∣∣∣∣k−γn∑

i=K

i−1−γ

Γ(−γ)f((n− i)k)− 1

Γ(−γ)

∫ t

(K−1)k

f(t− s)sγ+1

ds

∣∣∣∣∣ .The last term, by a very rough estimate, could be bounded by (Kk)−2−γk. Hence, if K = kε−

1+γ2+γ ,

the last term, (Kk)2−γ and K−1−γ all tend to zero as k → 0.

Remark 10. In the case α = −1 and f(0) 6= 0, for which the alternative group argument in the proof

fails, (Tαf)0 = f(0)k . This actually approximates the singular term in J−1f , δ(t)f(0).

Acknowledgements

L. Li is grateful to Xianghong Chen and Xiaoqian Xu for discussion on Sobolev spaces. The workof J.-G Liu is partially supported by KI-Net NSF RNMS grant No. 1107291 and NSF DMS grantNo. 1514826.

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Department of Mathematics, Duke University, Durham, NC 27708, USA.

E-mail address: leili@math.duke.edu

Departments of Physics and Mathematics, Duke University, Durham, NC 27708, USA.E-mail address: jliu@phy.duke.edu