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Fractional Calculus and its Applications in Physics

Original Research ARTICLE

Front. Phys., 24 October 2018 | https://doi.org/10.3389/fphy.2018.00118

Sinc-Fractional Operator on Shannon Wavelet Space

  • Engineering School, DEIM, Tuscia University, Viterbo, Italy

In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.

1. Introduction

In recent years, fractional calculus has been growing fast both in theory and applications to many different fields. Several classical and fundamental problems have been revised by using fractional methods, thus showing unexpected new results [13], while more and more new problems were shaped to fit the theoretical models of fractional calculus [47].

In fractional calculus is based on two universally accepted principles: the first one is that the definition of fractional derivative is not unique, thus giving raise to a neverending controversial debate on the best fractional operator. The second principle is that, although the missing uniqueness of the fractional operator, fractional calculus is an essential tool for a deeper and more comprehensive investigation of complex, non-linear, local, or non-local problems.

Therefore according to the suitable choice of the fractional differential operator, there follows a corresponding model of analysis so that the physical model and the corresponding physical interpretation of the results it strongly depends on the chosen fractional operator.

In some recent papers [815] the classical Lie symmetry analysis has been combined with the Riemman-Liouville fractional derivative to solve time fractional partial differential equations. In these papers, Lie point symmetries have been used to convert a fractional partial differential equation into a non-linear ordinary differential equation, that can be solved by suitable methods. Some fractional operators have been used also to study non-differentiable functions [see e.g., [16] some of them are more suitable for the analysis of non-differentiable sets, or fractal sets like the Cantor fractal set [47] Some fractional operators have been specially defined to analyze complex functions [1719]. For instance the chaotic decay to zero of the complex ζ-Riemann function was easily shown by using a suitable fractional derivative [19].

Among the many interesting definitions of fractional operators, some Authors have recenlty proposed a fractional differential operator based on the sinc-function [20]. This function is very popular in the signal analysis, also because it is a localized function with slow decay. Moreover, it is the fundamental basic function for the definition of the so-called Shannon wavelet theory, i.e., the multiscale analysis on Shannon wavelets [2126].

This paper will focus on the definition of a fractional derivative by the Shannon wavelets. These functions belong to a special family of wavelets which have a sharp compact support in the frequency space, so that their Fourier transform are box-functions in frequencies. This is a great advantage because, the frequency domain of a signal can be easily decomposed in terms of scaled box-functions.

Wavelet theory has been growing very fast so that there has been also a wide spreading of wavelets for the solution of theoretical and applied problems. However, alike the various definition of fractional operators there exist also many different families of wavelets and this missing uniqueness it might be considered as a drawback because of the arbitrary choice. Nevertheless all families of wavelets enjoy two fundamental properties their localization in time (or frequency) and the multiscale decomposition. Due to their localization they can be used to detect, and single out, localized singularities and/or peaks, while the multiscale property enable to decompose the approximation space into separate scales [27]. Thanks to these properties wavelets have been used to solve non-linear problems and moreover they are the most suitable tool for the analysis of multiscale problems.

The sinc-fractional operator will be generalized in order to compute the fractional derivative of the L2(ℝ)-functions belonging to the Hilbert space defined by the Shannon wavelet. In doing so, we will be able to compute the fractional derivative of these functions by knowing only their wavelet coefficients. Moreover, with this approach we will be able to decompose the fractional derivative at different scales, thus showing the influence of a given scale in multiscale physical problems.

The organization of this paper is as follows: Preliminary remarks on fractional operators are given in section 2. In section 3 the sinc fractional derivative, as given by Yang et al. [20] is described. Section 4 gives the basic properties on the multiscale approximation defined on Shannon wavelet. The differential properties of the functions belonging to the Hilbert space based on Shannon wavelet are given in section 5, together with the explicit form of the integer order derivatives (see also [24, 26]). Section 6 deals with the sinc-fractional derivative on the Hilbert space based on Shannon wavelets, i.e., sinc-fractional derivative of functions which can be represented as Shannon wavelet series.

2. Preliminary Remarks

In this section some of the most popular definition of fractional derivatives [2830] are given.

Let us start with the Riemann-Liouville derivative.

Definition 1. The Riemann-Liouville integral of fractional order ν ≥ 0 of a function f(x), is defined as

(Jνf)(t)={1Γ(ν)0t(tτ)ν1f(τ)dτ,ν>0,f(t),     ν=0.

The Riemann-Liouville fractional operator Jα has the following properties:

(a) Jα(Jβf(t))=Jβ(Jαf(t)),(b) Jα(Jβf(t))=Jα+βf(t),(c) Jαtν                  =Γ(ν+1)Γ(α+ν+1)tν+α,  α,β0,ν>1(d) Jνeλt                =1νΓ(ν)eλttν,  ν>0 ,(e) Jνc                     =cνΓ(ν)tν,  ν>0 .

From this definition there follows the corresponding derivative according to the following:

Definition 2. Riemann-Liouville fractional derivative of order α > 0 is defined as

DRLαf(t)=dndtnJnαf(t),     n,  n1<αn.    (2.1)

The main problem with this derivative is the unvanishing value for a constant function, therefore it was proposed by Caputo the following [28, 29].

Let f(x) ∈ Cn be a n-differentiable function, α a positive value, then

Definition 3. The α-order Caputo fractional derivative is defined as

DCαf(x)={dnf(x)dxn,0<α,1Γ(nα)0xf(n)(τ)(xτ)αn+1dτ, t>0,0n1<α<n.

where n is an integer, x > 0, and fCn.

It can be easily shown that:

(a) JαDCαf(x)=f(x)k=0n1f(k)(0+)xkk!,    t>0.
(b) DCαJαf(x)=f(x).
(c) DCαtn={0,for n0 and α<n,Γ(n+1)Γ(nα+1)tnα, otherwise.
(d) DCαDCβf(x)=DCαDCβf(x) .

3. Sinc-Fractional Derivative

Riemann-Liouville (RL) and Caputo (C) derivatives are the most popular derivatives and have been used in many applications (see e.g., [2, 3, 16, 18, 25, 26, 29, 3141]), nevertheless they both suffer for some unavoidable drawbacks. In particular, the RL-derivative is unvanishing when f(x) ≠ constant while the C-derivative is defined on a singular kernel. Because of that, in recent years many efforts were devoted to find some more flexible non-singular derivatives. Moreover, due to the fact that the fractional derivative is not univocally defined, there have been proposed many alternative interesting new definitions.

Indeed the more general fractional derivative with a given kernel K(x, α), which generalizes the C-derivative is:

Dαf(x)={dnf(x)dxn,    0<α,0xf(n)(τ)K(xτ,α)dτ, x>0,  0n1<α<n.    (3.1)

The kernel should be defined in a such a way that at least the two conditions

limα0K(xτ,α)=1,    limα1K(xτ,α)=δ(xτ)    (3.2)

hold true, moreover, in order to be a non-singular kernel, it should be also

limxτK(xτ,α)0, α .    (3.3)

Although there are several definitions of derivatives they all depend on a kernel. In particular, it can be easily seen that the C-derivative [42], the Caputo-Fabrizio (CF) derivative [34], and the Atangana-Baleanu (AB) derivative [43] are some special cases of (3.1) corresponding respectively to the kernels:

(C)  K(xτ,α)=1Γ(nα)(xτ)nα1
(CF)  K(xτ,α)=M(α)1αeα1α(xτ)    (3.4)
(AB)  K(xτ,α)=B(α)1αEα(α1α(xτ)),

where the Mittag-Leffler function is taken as

Eα(x)  def__ k=0xαkΓ(αk+1).

It can be easily shown that all kernels (3.4) fulfill (3.2) while only (CF) and (AB) fulfill also the condition (3.3).

3.1. The Yang-Gao-Terneiro Machado-Baleanu Fractional Derivative [20]

With respect to the integration variable τ all kernels (3.4) have a decay to zero, in a such way that for a bounded f(n)(x) the integral (3.1)2 converges.

Among the non-singular kernels with decay to zero a fractional derivative based on a sinc-function kernel was recently defined by Yang, Gao, Terneiro Machado, and Baleanu (YGTMB) [20].

The sinc-function, defined as Yang et al.[20]

sincx  def__  sinπxπx ,    (3.5)

owns a quite large amount of nice properties, so that it became a fundamental tools in applied science and signal analysis. In particular, it was shown (see e.g., [20]) that, for a given x

limα01αsinc(xα)=δ(x)    (3.6)

being δ(x) the Dirac-delta function

δ(x)={0,x01,x=0    .

More in general from (3.6) it is

limα01αsinc(xτα)=δ(xτ) .    (3.7)

By using the sinc-function, we have the following definition of the sinc fractional derivative [20].

Definition 4 (Yang-Gao-Tenreiro Machado-Baleanu). The YGTMB fractional derivative is defined as Yang et al. [20]

DYGTMBαf(x)  def__ αP(α)1αaxsincα(xτ)1αf(n)(x)dτ,0n1                                                                                                                    <α<n .    (3.8)

We can see that also this kernel

(S)    K(xτ,α)=αP(α)1αsincα(xτ)1α    (3.9)

belongs to the class of kernels (3.1). It can be also shown that this kernel fulfills the conditions (3.2),(3.3) (see [20]) being

limα0αP(α)1αsincα(xτ)1α=1, limα1αP(α)1αsincα(xτ)1α=δ(xτ) ,

and the normalization constant factor P(α) is such that

limα0[K(x,α)P(α)]=limα1[K(x,α)P(α)]=1

In particular, there follows from (3.8)

DYGTMBαf(n)(x)={f(n1)(x)f(n1)(0),     α=0f(n)(x),      α=1(0n1<α<n).

3.1.1. Polynomial Approximation of the Kernel

The sinc kernel (3.9) can be written also as an infinite product as follows. Starting from the known product:

sinx=xk=1(1x2k2π2)

by taking into account (3.5) it is

sinc x=sin(πx)πx=1πx(πx)k=1(1(πx)2k2π2)

so that

sincx=k=1(1x2k2)

It should be noticed that in the interval [−1, 1] the sinc-function can be approximated by

sincxk=1n(1x2k2)

so that if we define as the error of approximation

ε(n)=|sincxk=1n(1x2k2)|

we have

max ε(1) ≤ 0.14, max ε(2) ≤ 0.08, max ε(3) ≤ 0.055, max ε(4) ≤ 0.04,

so that already with n = 1:

sinc x(1x2)

the error of approximation in [−1, 1] is less that 15%.

It should be noticed that with this approximation the YGTMB-derivative (3.8) becomes

DYGTMB*αf(x)=αP(α)1αax[1α2(1α)2(xτ)2]f(n)(x)dτ,                                                                n1<α<n

that is

DYGTMB*αf(x)=αP(α)1α[f(n1)(x)f(n1)(0)]α3P(α)(1α)3                                      ax(xτ)2f(n)(x)dτ, 0n1<α<n

By assuming as a normalization factor

P(α)=(1α)3α

we get

DYGTMB*αf(x)=(1α)2[f(n1)(x)f(n1)(0)]α3                                      ax(xτ)2f(n)(x)dτ, 0n1<α<n

so that the fractional derivative can be seen as the interpolation between the function and its derivative (as shown e.g., in Cattani [25, 26]).

3.2. Sinc Fractional Derivative With Unbounded Domain

Let us consider the integral of sinc function over the unbounded domain [−∞, ∞]. By a direct computation it can be shown that

sinπxπxdx=1, 11sinπxπxdx1.17

so that the sinc-function is a function mainly localized around the origin. In fact, the sinc function is known as a function with a decay to zero, therefore we can extend the definition (3.8) over the unbounded domain ℝ so that we can define the sinc fractional derivative as the YGTMB fractional derivative on the unbounded domain ℝ, that is

Definition 5 (sinc fractional derivative). The sinc fractional derivative DSα of a function f(x) is defined as

DSαf(x)  def__ αP(α)1αsincα(xτ)1αf(n)(x)dτ,             0n1    <    α    <    n    (3.10)

where the normalization factor P(α) is chosen to fulfill conditions (3.2), (3.3) and the kernel is

K(xτ,α)=αP(α)1αsincα(xτ)1α .

In particular, we can also assume

α1α=2β

so that

α=2β12β, β=log2α1α

and the derivative (3.10) can be written as

DSβf(x)  def__ 2βsinc(2βτ2βx)f(n)(τ)dτ, 2βnn+1 .    (3.11)

4. Shannon Wavelets

The sinc-function plays a fundamental role also in wavelet theory. In fact, the basic functions (scaling and wavelet) of the so-called Shannon wavelets (see e.g., [2126]) can be defined by the sinc (3.5). In this section, some remarks on Shannon wavelets and connection coefficients are shortly summarized.

4.1. Preliminary Remarks

Shannon wavelet theory (see e.g., [2124]) is based on the scaling function φ(x), also known as sinc function, and the wavelet function ψ(x) respectively defined as

{φ(x)=sincx  def__ sinπxπx=eπixeπix2πix .ψ(x)=sin2π(x12)sinπ(x12)π(x12)          =e2iπx(i+eiπx+e3iπx+ie4iπx)2π(x12) .    (4.1)

The second function can be expressed in terms of the first, as

ψ(x)=2φ(2x1)φ(x12)    (4.2)

The families of translated and dilated Shannon scaling functions [2124], are

φkn(x)=2n/2φ(2nxk)=2n/2sinπ(2nxk)π(2nxk)             =2n/2eπi(2nxk)eπi(2nxk)2πi(2nxk) ,             =2n/22πi(2nxk)s=0πsiss![1(1)s](2nxk)s             =2n/22πi(2nxk)s=0πsiss!(1eπs)(2nxk)s             =2n/21s=1πs1is1s![1(1)s](2nxk)s1.    (4.3)

By a direct computation it can be easily shown that this series can be also written as

φkn(x)=2n/2s=0(1)sπ2s(2s+1)!(2nxk)2s    (4.4)

that is

φkn(x)=2n/2s=0(1)sπ2s(2s+1)!j=02s(2sj)(2nx)j(k)2sj    (4.5)

In the special case when k = 0, from (4.4) we have

φ0n(x)=2n/2s=0(1)sπ2s(2s+1)!22nsx2s    (4.6)

while for the translated instances at the zero scale n = 0 we obtain from (4.4)

φk(x)  def__ φ(xk)=s=0(1)sπ2s(2s+1)!(xk)2s    (4.7)

Analogously, the translated and dilated instances of the Shannon wavelets are

ψkn(x)=2n/2sin2π(2nxk12)sinπ(2nxk12)π(2nxk12) ,             =2n/22π(2nxk12)r=12i1+rerπi(2nxk)i1rerπi(2nxk)     (4.8)

or, by taking into account (4.2)

ψkn(x)=2φkn+1(x)φkn(x12)    (4.9)

and Equation (4.3), it is

ψkn(x)=2n/2s=1πs1is1s![1(1)s](2nxk)s1              2n/21s=1πs1is1s![1(1)s](2n(x12)k)s1.

From (4.9), by taking into account (4.4), it is

ψkn(x)=2n/2s=0(1)sπ2s(2s+1)!{23/2(2n+1xk)2s                  [(2nxk)2n1]2s}    (4.10)

so that at the zero scale n = 0 it is

ψk(x) def__ ψk0(x)=ψ(xk)=s=0(1)sπ2s(2s+1)!{23/2(2xk)2s[(xk)2s12]2s

and, at the origin k = 0

ψn(x) def__ ψ0n(x)=2n/2s=0(1)sπ2s(2s+1)!{23/2(2n+1x)2s(2nx2n1)2s}.

By assuming,

φ00(x)=φ(x) , ψ00(x)=ψ(x) ,φk0(x)=φk(x)=φ(xk),ψk0(x)=ψk(x)=ψ(xk),

and taking into account (4.4),(4.10) the fundamental functions φ(x), ψ(x), can be expressed as the power series

{φ(x)=s=0(1)sπ2s(2s+1)!x2sψ(x)=s=0(1)sπ2s(2s+1)![22s+3/2x2s(x12)2s] .    (4.11)

4.2. Properties of the Shannon Wavelet

Shannon wavelets enjoy some interesting properties. In particular, when they are evaluated at some special points they assume some very simple expressions. For instance, according to (4.3), it is

φk(h)=φh(k)=φ(hk)=φ(kh)=δkh, (h, k),    (4.12)

so that

φk(h)=δkh={0,hk,(h, k)1,h=k,(h, k)

Analogously we have [24]

ψkn(h)=(1)2nhk21+n/2(2n+1h2k1)π, (2n+1h2k10)ψkn(x)=0,  x=2n(k+12±13), (n, k)                         limx2n(h+12)ψkn(x)=2n/2δhk,    (4.13)

being,

ψk0(0)=(1)k+12(2k+1)π

and since k ∈ ℤ, 2k + 1 ≠ 0.

It can be shown (see e.g., [25]) that both scaling and wavelet functions are bounded, being:

max[φk(xM)]=1, xM=k,    (4.14)
max[ψkn(xM)]=2n/233π, xM={2n(k+16)2n13(18k+7),    (4.15)

and

limx±φkn(x)=0, limx±ψkn(x)=0.

4.3. Shannon Wavelets in Fourier Domain

In order to define the multiscale analysis, based on Shannon wavelets, we need to define the Hilbert space of functions that can be reconstructed by them. The Shannon scaling function owns a very simple expression in the Fourier domain, therefore it would be easier to define the scalar product in Fourier domain. To this purpose we define the Fourier transform of the function f(x) ∈ L2(ℝ), and its inverse transform as

f^(ω)= f(x)^  def__12πf(x)eiωxdx, f(x)=f^(ω)eiωxdω .

The Fourier transform of (4.1) give us [23]

{φ(ω)=12πχ(ω+3π)={1/(2π),πω<π0,elsewhereψ^(ω)=12πeiω/2[χ(2ω)+χ(2ω)]     (4.16)

with

χ(ω)={1,2πω<4π0,elsewhere .

The Fourier transform fulfills many interesting properties and among them the following:

f(ax)^=1a f^(ωa), f(xb)^=eibωf^(ω),  dndxnf(x)^=(iω)n f^(ω).    (4.17)

So that for the dilated and translated instances of scaling/wavelet function, in the frequency domain, are

{φ^kn(ω)=2n/22πeiωk/2nχ(ω/2n+3π)ψ^kn(ω)=2n/22πeiω(k+1/2)/2n[χ(ω/2n1)+χ(ω/2n1)].    (4.18)

For the integer order derivatives of scaling and wavelet, according to (4.17), it is

ddxφkn(x)^=(iω)φkn(ω),                  ddxψkn(x)^=(iω)ψkn(ω)    (4.19)

and, thanks to (4.18), we get

{ddxφkn(x)^=(iω)2n/22πeiωk/2nχ(ω/2n+3π),ddxψkn(x)^=(iω)2n/22πeiω(k+1/2)/2n[χ(ω/2n1)                              +χ(ω/2n1)].    (4.20)

The simple form of these derivative will help us to easily define also the fractional derivatives of these functions. Moreover, as we will see in the next section they form a basis for the L2(ℝ)-functions.

4.4. Wavelet Analysis and Synthesis

Both families of Shannon scaling and wavelet are L2(ℝ)-functions, therefore for each f(x) ∈ L2(ℝ) and g(x) ∈ L2(ℝ), the inner product is defined as

f,gdef__f(x)g(x)¯dx    (4.21)

where the bar stands for the complex conjugate. By taking into account the Parseval theorem

f(x)g(x)¯ dx=2πf^(ω)g^(ω)¯dω,

it is

f,gdef__f(x)g(x)¯dx=2πf^(ω)g^(ω)¯dω=2πf^,g^,    (4.22)

Shannon wavelets fulfill the following orthogonality properties (for the proof see e.g., [23, 24])

ψkn(x),ψhm(x)=δnmδhk,    φk0(x),φh0(x)=δkh,φk0(x),ψhm(x)=0,  m0 ,    (4.23)

δnm, δhk being the Kroenecker symbols.

Let BL2(ℝ) the set of functions f(x) in L2(ℝ) such that the integrals

{αkdef__f(x),φk(x)(4.22)__f(x)φk0(x)¯dxβkn def__f(x),ψkn(x)(4.22)__f(x)ψkn(x)¯dx ,    (4.24)

exist with finite values, then it can be shown [23, 24, 27, 44], that the series

f(x)=h=αhφh(x)+n=0k=βknψkn(x),    (4.25)

converges to f(x). So that each function f(x) ∈ BL2(ℝ) can be expressed as the wavelet series (4.25), and it is fully characterized by the wavelet coefficient αh ,βkn.

According to (4.22) the coefficients can be also computed in the Fourier domain [24] so that, together with (4.24) we can alternatively use the integrals

{αk=ππf^(ω)eiωkdω ,βkn=2n/2[2nπ2n+1πf^(ω)eiω(k+1/2)/2ndω                      +2n+1π2nπf^(ω)eiω(k+1/2)/2ndω] .    (4.26)

In the frequency domain, Equation (4.25) gives [24]

f^(ω)=12πχ(ω+3π)h=αheiωh          +12πχ(ω/2n1)n=0k=2n/2βkneiω(k+1/2)/2n          12πχ(ω/2n1)n=0k=2n/2βkneiω(k+1/2)/2n .

When the upper bound for the series of (4.25), is finite, then we have the approximation

f(x)h=KKαhφh(x)+n=0Nk=SSβknψkn(x).    (4.27)

The error of the approximation has been estimated in Cattani [24, 26].

5. Connection Coefficients and Derivatives

Let us assume that a function f(x) ∈ B, so that f(x) is a function belonging to the Hilbert space based on Shannon wavelets and thus being represented in the form of (4.25). In this section we will give the explicit form of the n-order integer derivative f(n)(x) and the sinc fractional order derivative DSαf(x). In order to get these derivatives we need to compute the ℓ-th integer order derivatives of the Shannon family (scaling and wavelet functions) φh(x),ψkn(x) and the sinc-fractional derivative. The Equations (4.20) already give us the expression of the ℓ-order derivative in the Fourier domain. In the following sections we will give the explict form of these derivatives also in the space domain.

5.1. Integer Order Derivatives of the Shannon Wavelets

It can be shown that the integer order derivatives of the Shannon family can be expressed as orthogonal wavelet series [23, 24, 26] as follows:

Definition 6. The integer n-order derivative of the Shannon scaling and wavelet functions are

{ddxφh(x)=k=λhk()φk(x) ,ddxψhm(x)=n=0k=γ()hkmnψkn(x) ,    (5.1)

being

λkh()ddxφk0(x),φh0(x), γ()khnmddxψkn(x),ψhm(x) ,    (5.2)

the connection coefficients [21, 23, 4550].

It should be noticed that the connection coefficients are not symmetric. In fact it is

ddxφk0(x),φh0(x)=ddxφk0(x),φh0(x)φk0(x),ddxφh0(x) ,

and by taking into account (4.23), there follows that

λkh()=λhk()  hk

Analogously we have for the coefficients

γ()khnm=γ()hknm

The connection coefficients can be easily computed so that it can be shown [21, 23, 24]

Theorem 1. The connection coefficients (5.2)1 of the Shannon scaling functions φk(x) are

λkh()={(1)kh+i2πs=1!πss![i(kh)]s+1[(1)s1],khiπ+12π(+1)[1+(1)],k=h ,    (5.3)

when ℓ ≥ 1. When ℓ = 0, it is

λkh(0)=δkh .

For the proof see e.g., [23].

Analogously, by defining the sign-function μ(x) = sign(x), it can be shown that

Theorem 2. The connection coefficients (5.2)2 of the Shannon wavelets ψkn(x) are

{γ()khnm=μ(hk)δnm{s=1+1(1)[1+μ(hk)](2s+1)/2!isπs(s+1)!|hk|s(1)s2(h+k)2ns1              ×{2+1[(1)4h+s+(1)4k+]2s[(1)3k+h++(1)3h+k+s]}},khγ()khnm=δnm[iπ2n1+1(2+11)(1+(1))],k=h    (5.4)

for ℓ ≥ 1, and

γ(0)khnm=δkhδnm    (5.5)

ℓ = 0 respectively.

For the proof see [23].

As a consequence of Equations (5.3),(5.8) the ℓ-order derivative of the basic functions (4.11) are

{ddxφ(x)=k=λ0k()φk(x),ddxψ(x)=n=0k=γ()0h0nψhn(x)                   =h=γ()0h00ψh0(x),    (5.6)

with

λ0k()={(1)k+i2πs=1!πss!(ik)s+1[(1)s1] ,k0iπ+12π(+1)[1+(1)],k=0,    (5.7)

and

{γ()0h00=μ(h){s=1+1(1)[1+μ(h)](2s+1)/2                      ×!isπs(s+1)!|h|s(1)s2h2s1×{2+1[(1)4h+s                      +(1)]2s[(1)h++(1)3h+s]}},h0γ()0h00=[iπ21+1(2+11)(1+(1))],h=0    (5.8)

In particular it is

λ() def__λ00()=iπ+12π(+1)[1+(1)]    (5.9)

It can be easily shown that λ(ℓ) = 0 for odd ℓ so that we have

λ()={(1)sπ2s2s+1,=2s 0                      ,=2s+1n (s=1,2,)    (5.10)

For instance according to (5.6), (5.7) a good approximation of the 2nd order derivative of φ(x) is

d2dx2φ(x)k=22λ0k()φk(x)=12φ2(x)+2φ1(x)13π2φ(x)                         +2φ1(x)12φ2(x) .

Also for higher derivatives with high amplitude, the approximation is quite good. For instance according to (5.6), (5.7) for the 7-th derivative of φ(x) a quite good approximation is obtained with 15 terms

ddxφ(x)k=77λ0k()φk(x)

5.2. Properties of Connection Coefficients

The connection coefficients own many interesting properties like e.g., the following for the scaling functions

Theorem 3. The connection coefficients (5.3) are defined recursively by

λkh(+1)={+1khλkh()(1)khiπ+1kh[(1)+1],khiπ+1+2λkh()+(i)+1π+1+2,k=h,    (5.11)

Proof : see [26].

Analogously for the coefficients γ.

Theorem 4. The connection coefficients (5.8) are recursively given by the matrix at the lowest scale level:

γ()khnn=2(n1)γ()kh11 .    (5.12)

Proof : see [26].

Moreover we can easily check that

γ(2+1)khnn=γ(2+1)hknn, γ(2)khnn=γ(2)hknn .

5.3. Taylor Series

By using the connection coefficients, and taking into account that the basic functions, according to (5.1), are C-functions, it is easy to show the following theorem:

Theorem 5. Let f(x) ∈ BL2(ℝ) the ℓ ≥ 1 order derivative is given by

f()(x)=h,k=αhλhk()φk(x)+n,m=0k,s=βknγ()skmnψsm(x)    (5.13)

where the coefficients αh, βkn are given by (4.24) (or (4.26)) and the connection coefficients are given by (5.3), (5.8).

Proof : The proof easily follows from Equations (4.25), (5.1).

Theorem 6. If f(x) ∈ BψL2(ℝ) and f(x) ∈ CS the Taylor series of f(x) in x0 is

f(x)=f(x0)+r=1[h,k=αhλhk(r)φk(x0)          +n=0k,s=2r(n1)βknγ(r)sk11ψsn(x0)](xx0)rr!    (5.14)

being αh and βkn given by (4.24), (4.26).

Proof : From (4.25), the ℓ-order derivative (ℓ ≤ S) is

f()(x)=h=αhddxφh(x)+n=0k=βknddxψkn(x)           (5.1)__h=αhk=λhk()φk(x)+n=0k=βknm=                      s=γ()skmnψsm(x),             =h,k=αhλhk()φk(x)+n,m=0k,s=βknγ()skmnψsm(x) ,

so that by taking into account (5.12) the proof follows.

By a suitable choice of the initial point x0 Equation (5.14) can be simplified. For instance, at the integers, x0 = j, (j ∈ ℤ), according to Equations (4.12), (5.12) it is

f(x)f(j)+r=1S[h=αhλhj(r)+n=0                k,s=2r(n1)+1+n/2(2n+1h2s1)πβknγ(r)sk11ψsn(h)](xj)rr!

In particular, for x0 = j = 0, Equation (5.14) gives

f(x)=f(0)+r=1[h,k=αhλhk(r)φk(0)+n=0               k,s=2r(n1)βknγ(r)sk11ψsn(0)]xrr!          =f(0)+r=1[h=αhλh0(r)+n=0               k,s=2r(n1)βknγ(r)sk11ψsn(0)]xrr!    (5.15)

and since

ψsn(0)=(1)s21+n/2(2s1)π,  (2k10)

we get

f(x)=f(0)+r=1[h=αhλh0(r)+n=0k,s=              (1)s+12n(r+1/2)+1r(2s+1)πβknγ(r)sk11]xrr!    (5.16)

with λh0(r) given by (5.7) and γ        sk(r)11 by (5.8) respectively. So that each function f(x) ∈ BL2(ℝ), can be easily expressed as a power series, when the finite values of the wavelet coefficients αh, βkn are given, according to (4.24),(4.26).

There follows, in particular, the Taylor power series for the basic functions φ(x), ψ(x):

{φ(x)=1+r=1(h=λh0(r))xrr!ψ(x)=2π+r=1(n=0k,s=(1)s+12n(r+1/2)+1r(2s+1)πδknγ(r)sk11)xrr!    (5.17)

being ψ(0)=-2π, according to (4.1).

For a fixed r the series

Λr def__h=λh0(r), r1,h=0

is converging, as can be easily shown by using Equation (5.7). In particular it is

λh0(1)=(1)hh, λh0(2)=2(1)hh2, λh0(3)=(1)h(π2h6h3),λh0(4)=(1)h(4π2h224h4), λh0(5)=(1)h(π4h20π2h3+120h5)λh0(6)=(1)h(6π4h2120π2h4+720h6), 

Moreover, since for odd r it is Λ(r) = 0 while for even r it is

Λ2r=2h=0λh0(r), r1,h=0

so that φ(x) can be written as the power series

φ(x)=r=0Λrr!xr,  (Λ0 def__1)

The first (approximated) values of the coefficients Λ are:

Λ0=1, ,Λ1=0.69, ,Λ2=1.64, ,Λ3=1.43,Λ4=9.74, ,Λ5=6.19

In particular, the Taylor series for the wavelet function ψ(x) can be also easily computed as follows:

ψ(x)=ψ(0)+=1(dψ(x)dx)x=0x!          (5.1),(5.2)__2π+=1(k=γ()0k00ψk0(0))x!         (4.13)__2π+=1(k=γ()0k00(1)k+12(2k+1)π)x!

that is

ψ(x)=2π+=1(k=(1)k+12(2k+1)πγ()0k00)x!    (5.18)

6. Sinc-Fractional Derivatives for the Functions f(x) ∈ BL2(ℝ)

The sinc fractional derivative (3.10) is defined by a sinc kernel over an infinite domain. Although the sinc-function is the basic function for Shannon wavelet, this kernel is not a Shannon scaling function for the reason that the sinc function depends on the fractional (non-integer) order of derivative. On the other hand as shown by the Equation (5.13) the n-integer order derivative can be written as a linear combination of φk(x), ψkm(x). Therefore, in order to give an explicit form to (3.10) as a function of Shannon wavelet and connection coefficients, we need to compute the scalar products of Shannon scaling and wavelet with sinc-function.

6.1. Scalar Products of the Shannon Scaling and Wavelet Functions With Sinc-Function

In this section we consider the scalar product of the sinc function with the Shannon scaling and wavelet functions and corresponding derivatives. We need these products to compute the sinc fractional derivatives.

6.1.1. Scalar Product of the Shannon Scaling Function With Sinc-Function

Let us assume a, b ∈ ℝ and show the following theorem:

Theorem 7. The scalar product ot the scaling functions φk(τ) with the sinc-function is

sinc(aτb),φk(τ)={2πasinc(b+k),      a12πa2sinc(b+k)a,    a<1.    (6.1)

Proof : It is by definition

sinc(aτb),φk(τ)=sinc(aτb)φk(τ)dτ .

According to (4.22) this product can be easily done in the Fourier domain,

sinc(aτb),φk(τ)=2πsinc(aτ^b),φk(τ)^

from where by using the properties (4.17) it is

sinc(aτ ^b)=1asinc^(ωa)eibω, sinc^(ωa) (4.13)__ 12πχ(ωa+3π)

so that by taking into account (4.18)1

sinc(aτb),φk(τ)=2π1a12πeibωχ(ωa+3π),eikωχ(ω+3π) .

The integral can be easily computed, being

      eibωχ(ωa+3π),eikωχ(ω+3π)=ei(b+k)ωχ(ωa+3π)χ(ω+3π)dω .

There follows that, if 1a1 it is

ei(b+k)ωχ(ωa+3π)χ(ω+3π)dω=ei(b+k)ωχ(ω+3π)dω                                   =ππei(b+k)ωdω=2πsinc(b+k) .

While for 1a>1 it is

ei(b+k)ωχ(ωa+3π)χ(ω+3π)dω=ei(b+k)ωχ(ωa+3π)dω                        =πaπaei(b+k)ωdω=2πasinc(b+k)a .

From where there easily follows the result (6.1).

In particular, according to (4.17), it is

sincα(x^τ)1α=sincαα1(τ^x) (4.17)__ eiαα1xωsinc(αα1τ)

that is

sincα(x^τ)1α=α1αeiαα1xωsinc^(α1αω) .

Since we have

sinc(τ) ^(4.16)__ 12πχ(ω+3π)

there follows

sincα(x^τ)1α=12πα1αeiαα1xωχ(α1αω+3π)

so that, by taking

a=αα1, b=αα1x

from (6.1) we get

      sinc(αα1ταα1x),φk(τ)={2π(α1)αsinc(αα1x+k),α12π(α1)2α2sinc(x+kα1α),α<1 .    (6.2)

Analogously we can give an explicit form to the scalar product of the integer n-order derivative.

Theorem 8. The scalar product of the n-th order derivative φk(n)(τ) with the sinc-function is

sinc(aτb),φk(n)(τ)={ππ(iω)nei(b+k)ωdω,a1πaπa(iω)nei(b+k)ωdω,a<1    (6.3)

Proof : It is

sinc(aτb),φk(n)(τ)=sinc(aτb)φk(n)(τ)dτ .

According to (4.22) this product can be easily done in the Fourier domain,

sinc(aτb),φk(n)(τ)=2πsinc(aτ ^b),φk(n)(τ)^

from where by using the properties (4.17) it is

sinc(aτ^ b)=12πaeibωχ(ωa+3π) ,         φk(n)(τ)^=(iω)nφk(τ)^=(iω)neikωχ(ω+3π)

so that by taking into account (4.18)1

     sinc(aτb),φk(n)(τ)=2π1a12πeibωχ(ωa+3π),(iω)neikωχ(ω+3π) .

The integral can be easily computed, being

     eibωχ(ωa+3π),eikωχ(ω+3π)=(iω)nei(b+k)ωχ(ωa+3π)χ(ω+3π)dω .

There follows that, if 1a1 it is

(iω)nei(b+k)ωχ(ωa+3π)χ(ω+3π)dω                    =(iω)nei(b+k)ωχ(ω+3π)dω                          =ππ(iω)nei(b+k)ωdω .

While for 1a>1 it is

(iω)nei(b+k)ωχ(ωa+3π)χ(ω+3π)dω                    =(iω)nei(b+k)ωχ(ωa+3π)dω                         =πaπa(iω)nei(b+k)ωdω .

From where there easily follows the result (6.1).

In particular, for the first derivative it is

ππ(iω)ei(b+k)ωdω=2πb+k[sinc(b+k)cos(b+k)π]

and

πaπa(iω)nei(b+k)ωdω=2πa(b+k)[sincb+kaacosb+kaπ]

so that

        sinc(aτb),φk(τ)={2πb+k[sinc(b+k)cos(b+k)π],a12πa(b+k)[sincb+kaacosb+kaπ],a<1 .    (6.4)

In general the scalar product of the n-order derivative (with n > 1) is given by the lengthly computation of the integrals (6.3). In the next section we will see that this computation can be avoided by using the connection coefficients.

6.1.2. Scalar Product of the Shannon Wavelets With Sinc Function

Analogously, for the derivative of the wavelet function it can be easily shown that

Theorem 9. Let a, b ∈ ℝ, the scalar product ot the wavelet functions ψkn(τ) with the sinc-function is

sinc(aτb),ψkn(τ)=Γkn(τ,a,b)2n/2+1aπ(2n+1b2k1)××{0,a<1sin(12a(2b2n(1+2k)))π+cos(2nb+k)π,2n<a<2n+1sin(2n+1b+2k)π+cos(2nb+k)π,2n+1a.    (6.5)

Proof : It is

sinc(aτb),ψkn(τ)=sinc(aτb)ψkn(τ)dτ .

According to (4.22) this product can be easily done in the Fourier domain,

sinc(aτb),ψkn(τ)=2πsinc(aτ^ b),ψkn^

from where by using the properties (4.17), it is

sinc(aτ^ b)=1a12πχ(ωa+3π)eibω

so that by taking into account (4.18)2

sinc(aτb),ψkn(τ)=2n/22aπeibωχ(ωa+3π),eiω(k+1/2)/2n                                                  [χ(ω/2n1)+χ(ω/2n1)]

that is

sinc(aτb),ψkn(τ)=2n/22aπeiω(k+1/22nb)/2nχ(ωa+3π),                                                  [χ(ω/2n1)+χ(ω/2n1)].

Let us notice that the value of the scalar product (and then of the integral) depends on the non-vanishing values of the characteristic function. On the other hands the characteristic function χ depends on the values a, n, k. In fact the non-vanishing values of the characteristic functions are

χ(ωa+3π)=1,         ifaπ<ω<aπχ(ω/2n1)=1,        if  2nπ<ω<2n(2π)χ(ω/2n1)=1,     if2n(2π)<ω<2nπ .

There follow three cases:

1. aπ < π. In this case a < 1, the characteristic functions have some disjoint intervals and the scalar product vanishes

eiω(k+1/22nb)/2nχ(ωa+3π),[χ(ω/2n1)+χ(ω/2n1)]=0.

2. 2nπ < aπ < 2n(2π). Here we have 2n < a < 2n+1 the integral becomes

                eiω(k+1/22nb)/2nχ(ωa+3π),[χ(ω/2n1)+χ(ω/2n1)]=                    =aπ2nπeiω(k+1/22nb)/2ndω+2nπaπeiω(k+1/22nb)/2ndω=2n+22n+1b2k1[sin(12a(2b2n(1+2k)))π+cos(2nb+k)π] .

3. 2n(2π) ≤ aπ. We have 2n+1a so that the integral is

        eiω(k+1/22nb)/2nχ(ωa+3π),[χ(ω/2n1)+χ(ω/2n1)]=     =2n+1π2nπeiω(k+1/22nb)/2ndω+2nπ2n+1πeiω(k+1/22nb)/2ndω=2n+22n+1b2k1[sin(2n+1b+2k)π+cos(2nb+k)π] .

From where we obtain (6.5).

6.2. Sinc-Fractional Derivative of Functions f(x) ∈ BL2(ℝ)

In order to define the sinc-fractional derivative for the functions f(x) ∈ B, according to the reconstruction formula (4.25) we need to compute the sinc-fractional derivative of the scaling and wavelet functions. These derivatives are given by the following theorems.

Theorem 10. The sinc-fractional derivative (3.10) of the scaling function φk(x) is

DSαφh(x)=2πP(α)k=λhk(n)×                        {sinc(αα1x+k),                α1α1αsinc(x+α1αk),    α<1    (6.6)

Proof : Starting from the definition (3.10) of the sinc-derivative it is

DSαφh(x) (3.10)__ αP(α)1αsincα(xτ)1αdndτnφh(τ)dτ,    (6.7)

According to (4.22), the derivatives (6.7), can be written also as scalar product,

DSαφh(x)=αP(α)1αsincα(xτ)1α, dndτnφh(τ),    (6.8)

From here by using the integer order derivatives (5.1) we

DSαφh(x)=αP(α)1αk=λhk(n)sincα(xτ)1α, φk(τ),    (6.9)

So that the computation of the sinc-fractional derivative of a function, that can be expressed as wavelet series, is reduced to the computation of the scalar product:

sincα(xτ)1α, φk(τ), 0n1<α<n    (6.10)

which is given by (6.1) with

a=αα1, b=αα1x .

It can be easily seen that these inequalities imply

a1  α1, a<1  α<1 .

From these inequalities, by taking into account (6.1),(6.9), there easily follows (6.6).

Analogously, we have for the Shannon wavelet fractional derivatives the following

Theorem 11. The sinc-fractional derivative (3.10) of the Shannon wavelets ψhm(x) is

                              DSαψhm(x)=αP(α)1αs=0                  k=γ(n)hkms2s/2+1(α1)απ(2s+1αα12k1)××{0,a<1sin(12αα1(2αα1x2s(1+2k)))π+cos(2sαα1x+k)π,2s<a<2s+1sin(2s+1b+2k)π+cos(2sb+k)π,2s+1a    (6.11)

Proof : From the definition (3.10) it is

DSαψhm(x) (3.10)__ αP(α)1αsincα(xτ)1αdndτnψhm(τ)dτ,    (6.12)

According to (4.22), this derivative can be written also as scalar product,

DSαψhm(x)=αP(α)1αsincα(xτ)1α, dndτnψhm(τ),    (6.13)

so that by taking into account (5.1) which gives the integer order derivatives it is

DSαψhm(x)=αP(α)1αs=0k=γ(n)hkmssincα(xτ)1α, ψks(τ),    (6.14)

and using (6.5) we get (6.11).

Equations (6.6) and (6.11) enable us to compute explicitly the sinc-fractional derivative of any function belonging to the Hilbert space BL2(ℝ). In fact, let f(x) ∈ B a function such that it can be represented as the wavelet series (4.25). Its sinc-fractional derivative can be computed according to

Theorem 12. The sinc-fractional derivative of the wavelet representation (4.25) of function f(x) ∈ BL2(ℝ), is given by

                              DSνf(x)=2πP(ν)h=αhk=λhk(n)                              ×{sinc(νν1x+k),ν1ν1νsinc(x+ν1νk),ν<1                                    +νP(ν)1νm=0k=βhms=0                          k=γ(n)hkms2s/2+1(ν1)νπ(2s+1νν12k1)×
×{0,ν<1sin(12νν1(2νν1x2s(1+2k)))π+cos(2sνν1x+k)π,2s<ν<2s+1sin(2s+1νν1x+2k)π+cos(2sνν1x+k)π,2s+1ν    (6.15)

with 0 ≤ n−1 < ν < n.

Proof : Let us start from Equation (3.10), and the representation (4.25), because of the linearity of the operator we have

DSνf(x)=h=αhDSνφh(x)+m=0k=βhmDSνψhm(x)

where the wavelet coefficients αh, βhm are given by (4.24) [or (4.26)]. From here, by using (6.6) and (6.11), we get (6.15).

In particular, with n = 1 we have

Theorem 13. The sinc-fractional derivative of the wavelet representation (4.25) of function f(x) ∈ BL2(ℝ), with order 0 < ν < 1, is

DSνf(x)=2πP(ν)1ννh=αhk=λhk(1)sinc(x+ν1νk),                                                                                                           0<ν<1

Proof : Follows directly from Equation (6.15).

6.3. Example: Fractional Derivative of the Gaussian Function

In order to show the efficiency of the proposed method for the computation of a fractional derivative, let us consider the function ex2. A good approximation of this function, in terms of Shannon wavelet expansion (4.25), can be obtained as

ex2h=11αhφh(x)+n=00h=11βhnψhn(x)            α1φ1(x)+α0φ(x)+α1φ1(x)+            +β10ψ10(x)+β00ψ00(x)+β10ψ10(x)

where

α1=α1=0.123, α0=0.30,ψ10=ψ10=0.004, ψ00=0.001 .

If we neglect also the detailed coefficients βkn the approximate Shannon wavelet representation is

ex20.123 φ1(x)+0.30 φ(x)+0.123 φ1(x) .

From (6.16) we have

DSνex22πP(ν)1ννh=11αhk=11λhk(1)sinc(x+ν1νk),                                                                                                         0<ν<1 .

The matrix λhk(1), according to (5.3) is

λ11(1)=λ00(1)=λ11(1)=0 , λ01(1)=λ10(1)=λ10(1)=λ01(1)=1 ,                                                                                                        λ11(1)=λ11(1)=12

so that by simplifying we get

DSνex22πP(ν)1νν(α012α1)[sinc(xν1ν)                                             sinc(x+ν1ν)],0<ν<1

that is

DSνex20.47πP(ν)1νν[sinc(xν1ν)                                                  sinc(x+ν1ν)],0<ν<1 .

Conclusion

Sinc function is playing a fundamental role in mathematics and physics. Due to the many properties of this function it deserves a special role in applications. In recent years some Authors have proposed [20] a fractional derivative based on this function. Moreover a wavelet theory based on the sinc function has been settled thus extending the many features of the Sinc. In this paper the sinc-fractional derivative has been extended to the Shannon wavelet space, in order to give the explicit analytical form of the fractional derivatives of functions belonging to the wavelet space. It has been shown that the sinc-fractional derivative is the most natural and suitable choice of fractional operator when dealing with functions that can be represented as Shannon wavelet series.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Conflict of Interest Statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: fractional calculus, shannon wavelet, sinc-function, operational matrix, connection coefficients

Citation: Cattani C (2018) Sinc-Fractional Operator on Shannon Wavelet Space. Front. Phys. 6:118. doi: 10.3389/fphy.2018.00118

Received: 30 August 2018; Accepted: 28 September 2018;
Published: 24 October 2018.

Edited by:

Dumitru Baleanu, University of Craiova, Romania

Reviewed by:

Xiao-Jun Yang, China University of Mining and Technology, China
Jordan Yankov Hristov, University of Chemical Technology and Metallurgy, Bulgaria
Abdullahi Yusuf, Federal University, Nigeria

Copyright © 2018 Cattani. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Carlo Cattani, cattani@unitus.it