Remarks on Convolutions and Fractional Derivative of Distributions

This paper begins to present relations among the convolutional definitions given by Fisher and Li, and further shows that the following fractional Taylor’s expansion holds based on convolution d dxλ θ(x)φ(x) = ∞ ∑ k=0 φ(k)(0) xk−λ + Γ(k − λ + 1) if λ ≥ 0, with demonstration of several examples. As an application, we solve the Poisson’s integral equation below ∫ π/2 0 f (x cosω) sin2λ+1 ωdω = θ(x)g(x) by fractional derivative of distributions and the Taylor’s expansion obtained.

Note that if f and g are locally integrable functions satisfying either of the conditions in (a) or (b) in Definition 1.2, then Definition 1.2 is in agreement with Definition 1.1. It also follows that if the convolution f * g exists by Definition 1.1 or 1.2, then the following equations hold: where all the derivatives above are in the distributional sense.
Both Definitions 1.1 and 1.2 are very restrictive and can only be used for a small class of distributions. In order to extend the convolution to a larger class of distributions, (Jones, 1973) introduced the following definition: Definition 1.3 Let f and g be distributions in D ′ and let τ(x) be an infinitely differentiable function satisfying the following conditions: for n = 1, 2, · · · . Then the convolution f * g is defined as the limit of the sequence { f n * g n }, provided the limit h exists in the sense that lim n→∞ ( f n * g n , ϕ) = (h, ϕ) for all testing functions ϕ ∈ D.
It can be proven that if a convolution exists by Definition 1.1 or 1.2 then it exists by Definition 1.3 and defines the same distribution. Therefore, Definition 1.3 generalizes Definitions 1.1 and 1.2. However, there are still many convolutions which cannot be given by Definition 1.3. In order to fix this, (Fisher, 1987) presented the following definition: Definition 1.4 Let f and g be distributions and let for n = 1, 2, · · · , where τ is defined as in Definition 1.3. Let f n (x) = f (x)τ n (x) for n = 1, 2, · · · . Then the noncommutative neutrix convolution f ⊙ g is defined as the neutrix limit of the sequence { f n * g}, provided the limit h exists in the sense that for all ϕ ∈ D, where N is the neutrix, (van der Corput, 1959-60) (use the neutrix to abandon unwanted infinite quantities from asymptotic expressions), having domain N ′ = {1, 2, · · · } and range the real numbers, with negligible functions that are finite linear sums of functions n λ ln r−1 n, ln r n, (λ > 0, r = 1, 2, · · · ) and all functions of n that converge to zero in the normal sense as n tends to infinity.
The convolution f n * g in this definition is again in the sense of Definition 1.2 as the support of f n is contained in the interval [−n − n −n , n + n −n ]. It is also proven that if a convolution exists by Definition 1.1 or 1.2 then the noncommutative neutrix convolution exists and defines the same distribution (Fisher, 1987).
To overcome the convolutional noncommutativity above, (Fisher & Li, 1993) introduced the following commutative neutrix convolution of distributions f and g by f ⊗ g to distinguish it from the noncommutative neutrix convolution in Definition 1.4: Definition 1.5 Let f and g be distributions and let τ n be defined as in Definition 1.4. Let f n (x) = f (x)τ n (x) and g n (x) = g(x)τ n (x) for n = 1, 2, · · · . Then the commutative neutrix convolution f ⊗ g is defined as the neutrix limit of the sequence { f n * g n }, provided the limit h exists in the sense that for all ϕ ∈ D, where N is the neutrix given above.
While Definition 1.5 defines the commutative neutrix convolution, its computational complexity hinders the calculation processes due to the factor τ n (t)τ n (x − t) appearing in the convolution f n * g n . To address this, (Li, Clarkson, & Patel, in press) recently introduced the following definition: Definition 1.6 Let f and g be distributions and let τ n be defined as in Definition 1.4. Let f n (x) = f (x)τ n (x) and g n (x) = g(x)τ n (x) for n = 1, 2, · · · . Then the commutative neutrix convolution f * g of f and g is defined as the neutrix limit of the sequence 1/2{ f n * g + f * g n }, provided the limit h exists in the sense that for all ϕ ∈ D. If the normal limit exists, then it is simply called the commutative convolution. Clearly, this definition generalizes Definitions 1.1 and 1.2.
Notes on the neutrix limit: (Fisher, 1982), with his coauthors (Fisher & Kuribayashi, 1987;Fisher & Taş, 2005;Fisher & Ozcag, 2012;Fisher & Al-Sirehy, 2015;Lazarova, Jolevska-Tuneska, Akturk, & Ozcag, 2016;Ozcag, Lazarova, & Jolevska-Tuneska, 2016;Fisher, Ozcag, & Al-Sirehy, 2017), has actively used Temples' δ-sequence and the concept of neutrix limit to deduce numerous products, powers, convolutions, and compositions of distributions. The technique of neglecting appropriately defined infinite quantities and resulting in a finite value extracted from the divergent integral, is usually referred to as the Hadamard finite part. In fact, Fisher's method in the computation can be regarded as a particular application of the neutrix calculus. This is a general principle for the discarding of unwanted infinite quantities from asymptotic expansions and has been exploited in the context of distribution by Fisher in connection with the problem of distributional multiplication, convolution and composition.
On the other hand, fractional calculus, first mentioned in the letter from Leibniz to L'Hôpital dated 30 September 1695, can be regarded as a branch of analysis which deals with integral and differential equations often with weakly singular kernels. A lot of contributions to the theory of fractional calculus up to the middle of the 20th century were made by many famous mathematicians including Laplace, Fourier, Abel, Liouville, Riemann, Grünwald, Letnikov, Heaviside, Weyl, Erdélyi and others. After 1970, there was a clear movement from theoretical research of fractional calculus to its applications in various fields. Up to now, fractional calculus has been found in almost every realm of science and engineering. As far as we know, it is one of the best tools to characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws, allometric scaling laws, and so on.
As outlined in the abstract, the goal of this paper is to discuss relations among different convolutional definitions and define fractional derivatives and integrals based on Definition 1.6. We further present the fractional Taylor's expansion for the distribution d λ dx λ θ(x)ϕ(x) as well as its applications to solving several Poisson's integral equations and a one-term differential equation in distribution.

Remarks on the Convolutions
It seems true that Definition 1.5 is equivalent to Definition 1.6 from many examples calculated in (Li et al., in press) and the following argument: Our first result is stated in Theorem 2.1 below, where we prove the opposite by a counterexample.
Theorem 2.1 Definition 1.5 is not equivalent to Definition 1.6.
Proof. By Definition 1.5, we have and It follows that lim Making the substitution u = x − t, we get For x > 0, choose n such that n −n < x, then It follows that for x > 0, ) .
For x < 0, choose n such that −n −n > x, then It follows that for x < 0, ) . Therefore, By Definition 1.6, we have We first notice that xτ n (x) * 1 evaluates to zero, as the resulting integrand is an odd function, that is, As for x * τ n (x), by substituting Again, it is clear that I 2 evaluates to zero. For I 1 , we have With the use of the neutrix limit, Thus, x * 1 = 0 by Definition 1.6. This completes the proof of Theorem 2.1. 2 Remark 1 It is interesting to point out that if we add the neutrix limit in Definition 1.3 then the convolution x * 1 is also zero (otherwise, it diverges). In fact, Evidently, By the mean value theorem, τ 2 (y)dy using integration by parts.

Fractional Derivative of Distributions
Let D ′ (R + ) be the subspace of D ′ with support contained in R + . It follows from (Gel'fand & Shilov, 1964;Li, 2015) that is an entire function of λ on the complex plane, and For the functional Φ λ = x λ−1 + Γ(λ) , the derivative formula is simpler than that for x λ + . In fact, Let λ and µ be arbitrary complex numbers. Then it is easy to show by equation (4), without any help of analytic continuation mentioned in all current books. Journal of Mathematics Research Vol. 10, No. 1;2018 Let λ be an arbitrary complex number and H be a subspace of D ′ given by Clearly, H contains D ′ (R + ) as a proper subspace since Definition 1.6 generalizes Definition 1.2.
Let g(x) be the distribution in H. We define the primitive of order λ of g as the convolution given by Definition 1.6 in the distributional sense: Note that the convolution on the right-hand side is well defined since g is in H.
Thus equation (6) with various λ will not only give the fractional derivatives, but also the fractional integrals of g(x) ∈ H when λ Z, and it reduces to integer-order derivatives or integrals when λ ∈ Z. We shall define the convolution as the fractional derivative of the distribution g(x) with order λ, writing it as for Reλ ≥ 0. Similarly, d λ dx λ g is interpreted as the fractional integral of order λ if Reλ < 0. We are now ready to show the following fractional Taylor's expansion in the distributional sense with applications presented in several examples.
In particular, d Similarly, we get for 0 < λ < 1 by using the formula In particular, d Example 3 Let λ > 0 and Proof. Clearly,
If m = 1, then 0 < λ < 1 and If m ≥ 2, then we also have The following theorem can be found in (Li et al., 2017).
Theorem 3.2 Let g(x) be given in D ′ (R + ) and f (x) be unknown in D ′ (R + ). Then the generalized Abel's integral equation given by where α is any real number. In particular, if −m < α < −m + 1 for m ∈ Z + then As an application of fractional derivative in distribution, we present the following theorem which converts Poisson's integral equation to the generalized Abel's integral equation. Hence we are able to solve it by Theorem 3.2.
Proof. Performing the substitution x = r cos ω we convert equation (9) into which can be changed to Making the substitutions which has the solution by Theorem 3.2. This completes the proof of Theorem 3.3. 2 Remark 3 We would like to add that all the derivatives in Theorem 3.3 are in the distributional sense and they may not exist classically, as the fractional derivatives of distributions are more general. Here is an example to illustrate this.

Conclusion
In this paper, we mainly provide a counterexample to show that two definitions for defining distributional convolution are not equivalent, and further study fractional calculus of distributions based on our newly defined convolution. We present a fractional Taylor's expansion with several applications to solving Poisson's integral equations and fractional differential equation in the generalized sense, which cannot be achieved classically. A challenge problem is how to solve the generalized Abel's integral equation for all α ∈ R where g ∈ D ′ (R) is given and f is unknown. The authors welcome and appreciate any discussion from interested readers.