# Fractional Calculus and Shannon Wavelet

Authors: Carlo Cattani
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for anyL2(ℝ) function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.


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## Body

## 1. Introduction

Shannon wavelet theory [1, 2] is based on a family of orthogonal functions having many interesting properties. They enjoy the many advantages of wavelets [3, 4]; moreover, being analytical functions they are infinitely differentiable. Thus, enabling us to define the so-called connection coefficients [5–7] for any order derivative. Connection coefficients are an expedient tool for the projection of differential operators, useful for computing the wavelet solution of integrodifferential equations [8–13].Wavelets are localized functions, in time and/or frequency, which are the basis for energy-bounded functions and in particular forL2(ℝ)-functions. So that localized pulse problems [14, 15] can be easily approached and analyzed. Moreover, wavelet allows the multiscale decomposition of problems, thus emphasizing the contribution of each scale. By defining a suitable inner product on the orthogonal family of scaling/wavelet functions, any L2(ℝ)-function can be approximated at a fixed scale, by a truncated series having, as basis, the scaling functions and the wavelet functions. The wavelet coefficients of these series represent the contribution of each scale.Shannon wavelets are related to the harmonic wavelets [3, 5, 8], being the real part thereof, and to the well-known sinc function, which is the basic function in signal analysis. It should be also noticed that, as compared with other wavelet families, the main advantage of Shannon wavelets is that they are analytical functions, thus being infinitely differentiable. Moreover, they are sharply bounded in the frequency domain, so that, by taking into account the Parseval identity, any computation can be easily performed by their Fourier transforms.The theory of connection coefficients was initially given [10, 13] for the compactly supported wavelet families, such as the Daubechies wavelets [4]. The computation of these connection coefficients was based on the recursive equations of the wavelet theory and the explicit forms of these coefficients were given only up to the second order derivatives. The connection coefficients are the wavelet coefficients of the derivatives of the wavelet basis. These coefficients are a fundamental tool for the approximation of differential operators, with respect to the wavelet basis.In some recent papers, the connection coefficients for Shannon wavelets have been explicitly computed up to any order derivative with a finite analytical form. This is due to the analytical form of Shannon wavelets and the discovery by Cattani of a suitable series expansion for the connection coefficients [2, 6, 7].In the following, we will define the wavelet representation of fractional derivative, so that the fractional derivative of anL2(ℝ)-function can be easily computed by knowing the connection coefficients. The fractional derivatives of the Shannon scaling/wavelet basis are defined and the error of the approximation will be explicitly computed. Moreover, a comparison with the classical definition of Grünwald formula [16, 17] is given, by showing the major performance of wavelets, in terms of rate of convergence.In particular, Section2 gives some preliminary remarks, definitions, and properties about Shannon wavelets. Their corresponding connection coefficients are discussed in Section 3. This Section deals with some properties of connection coefficients, functional equalities, and error of approximation. Fractional derivatives of the Shannon scaling function and wavelets are given in Section 4. In this section, it is also shown that the fractional derivative is a semigroup. The error of the approximation is explicitly computed and compared with classical definitions of the fractional derivative, and in particular with the Grünwald formula.


## 2. Preliminary Remarks

In this section, some remarks on Shannon wavelets and connection coefficients are given (see also [7]).Shannon wavelet theory (see e.g. [1, 2, 6, 7, 9]) is based on the scaling function φ(x), also known as sinc function, and the wavelet function ψ(x), respectively, defined as
(2.1)φ(x)=sincx≝sin⁡πxπx=eπix-e-πix2πix,ψ(x)=sin⁡2π(x-(1/2))-sin⁡π(x-(1/2))π(x-(1/2))=e-2iπx(-i+eiπx+e3iπx+ie4iπx)2π(x-(1/2)).
The corresponding families of translated and dilated instances wavelet [1, 2, 6, 7, 9], on which is based the multiscale analysis [4], are
(2.2)φkn(x)=2n/2φ(2nx-k)=2n/2sin⁡π(2nx-k)π(2nx-k)=2n/2eπi(2nx-k)-e-πi(2nx-k)2πi(2nx-k),ψkn(x)=2n/2sin⁡2π(2nx-k-(1/2))-sin⁡π(2nx-k-(1/2))π(2nx-k-(1/2))=2n/22π(2nx-k-(1/2))∑s=12‍i1+sesπi(2nx-k)-i1-se-sπi(2nx-k),
being, in particular,
(2.3)φ00(x)=φ(x),ψ00(x)=ψ(x),φk0(x)=φk(x)=φ(x-k),ψk0(x)=ψk(x)=ψ(x-k).
Let
(2.4)f^(ω)=f(x)^≝12π∫-∞∞‍f(x)e-iωxdx,f(x)=∫-∞∞‍f^(ω)eiωxdω
be the Fourier transform of the function f(x)∈L2(ℝ), and its inverse transform, respectively. The Fourier transform of  (2.1) give us [2]
(2.5)φ^(ω)=12πχ(ω+3π)={12π,-π≤ω<π0,elsewhereψ^(ω)=12πeiω/2[χ(2ω)+χ(-2ω)],
with
(2.6)χ(ω)={1,2π≤ω<4π0,elsewhere.Analogously for the dilated and translated instances of scaling/wavelet function, in the frequency domain, it is(2.7)φ^kn(ω)=2-n/22πeiωk/2nχ(ω2n+3π)ψ^kn(ω)=2-n/22πeiω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)].Both families of Shannon scaling and wavelet areL2(ℝ)-functions therefore, for each f(x)∈L2(ℝ) and g(x)∈L2(ℝ), the inner product is defined as
(2.8)〈f,g〉≝∫-∞∞‍f(x)g(x)¯dx=2π∫-∞∞‍f^(ω)g^(ω)¯dω=2π〈f^,g^〉,
where the bar stands for the complex conjugate.Shannon wavelets fulfill the following orthogonality properties (for the proof see e.g., [2, 7]):
(2.9)〈ψkn(x),ψhm(x)〉=δnmδhk,〈φk0(x),φh0(x)〉=δkh,〈φk0(x),ψhm(x)〉=0,m≥0,δnm,δhk being the Kronecker symbols.


### 2.1. Properties of the Shannon Wavelet

According to  (2.2), Shannon wavelets can be easily computed at some special points, being in particular
(2.10)φk(h)=φh(k)=φ(h-k)=φ(k-h)=δkh,(h,k∈Z),
so that
(2.11)φk(x)={0,x=h≠k,(h,k∈Z)1,x=h=k,(h,k∈Z).
It is also [7]
(2.12)ψkn(h)=(-1)2nh-k21+n/2(2n+1h-2k-1)π,(2n+1h-2k-1≠0)ψkn(x)=0,x=2-n(k+12±13),(n∈N,k∈Z)lim⁡x→2-n(h+(1/2))ψkn(x)=-2n/2δhk.In the following, we will be interested on the maximum values of these functions which can be easily computed. The maximum value of the scaling functionφk(x) can be found at the integers x=k(2.13)max⁡[φk(xM)]=1,xM=k,
and the max values of ψkn(x) are
(2.14)max⁡[ψkn(xM)]=2n/233π,xM={-2-n(k+16)2-n-13(18k+7).Both families of scaling and wavelet functions belong toL2(ℝ), thus having a bounded range and (slow) decay to zero
(2.15)lim⁡x→±∞φkn(x)=0,lim⁡x→±∞ψkn(x)=0.Letℬ⊂L2(ℝ) the set of functions f(x) in L2(ℝ) such that the integrals
(2.16)αk≝〈f(x),φk(x)〉=(2.8)∫-∞∞‍f(x)φk0(x)¯dxβkn≝〈f(x),ψkn(x)〉=(2.8)∫-∞∞‍f(x)ψkn(x)¯dx
exist with finite values, then it can be shown [2–4, 7] that the series
(2.17)f(x)=∑h=-∞∞‍αhφh(x)+∑n=0∞‍∑k=-∞∞‍βknψkn(x)
converges to f(x).According to  (2.8), the coefficients can be also computed in the Fourier domain [7] so that
(2.18)αk=∫-ππ‍f^(ω)eiωkdω,βkn=2-n/2[∫2nπ2n+1π‍f^(ω)eiω(k+1/2)/2ndω+∫-2n+1π-2nπ‍f^(ω)eiω(k+1/2)/2ndω].
In the frequency domain, (2.17) gives [7]
(2.19)f^(ω)=12πχ(ω+3π)∑h=-∞∞‍αheiωh+12πχ(ω2n-1)∑n=0∞‍∑k=-∞∞‍2-n/2βkneiω(k+1/2)/2n+12πχ(-ω2n-1)∑n=0∞‍∑k=-∞∞‍2-n/2βkneiω(k+1/2)/2n.
When the upper bound for the series of  (2.17) is finite, then we have the approximation
(2.20)f(x)≅∑h=-KK‍αhφh(x)+∑n=0N‍∑k=-SS‍βknψkn(x).
The error of the approximation has been estimated in [7].


### 2.2. Reconstruction of the Derivatives

In order to represent the differential operators in wavelet bases, we have to compute the wavelet decomposition of the derivatives. It can be shown [2, 7] that the derivatives of the Shannon wavelets are orthogonal functions:
(2.21)dldxlφh(x)=∑k=-∞∞‍λhk(l)φk(x),dldxlψhm(x)=∑n=0∞‍∑k=-∞∞‍γ(l)hkmnψkn(x),
being
(2.22)λkh(l)≝〈dldxlφk0(x),φh0(x)〉,γ(l)khmn≝〈dldxlψkn(x),ψhm(x)〉,
the connection coefficients [2, 5, 6, 8–13].The computation of connection coefficients can be easily performed in the Fourier domain, thanks to the equality  (2.8)
(2.23)λkh(l)=2π〈dldxlφk(x)^,φh(x)^〉,γ(l)khmn=2π〈dldxlψkn(x)^,ψhm(x)^〉.In fact, in the Fourier domain, theℓ-order derivative of the (scaling) wavelet functions are simply
(2.24)dldxlφkn(x)^=(iω)lφ^kn(ω),dldxlψkn(x)^=(iω)lψ^kn(ω),
and, according to (2.7),
(2.25)dldxlφkn(x)^=(iω)l2-n/22πeiωk/2nχ(ω2n+3π),dldxlψkn(x)^=(iω)l2-n/22πeiω(k+(1/2))/2n[χ(ω2n-1)+χ(-ω2n-1)].It has been shown [2, 6, 7] that the any order connection coefficients (2.22)1 of the Shannon scaling functions φk(x) are
(2.26)λkh(l)={(-1)k-hil2π∑s=1l‍l!πss![i(k-h)]l-s+1[(-1)s-1],k≠hilπl+12π(l+1)[1+(-1)l],k=h,
or, by defining
(2.27)μ(m)=sign⁡(m)={1,m>0-1,m<00,m=0,
shortly as,
(2.28)λkh(l)=ilπl2(l+1)[1+(-1)l](1-|μ(k-h)|)+(-1)k-h|μ(k-h)|il2π∑s=1l‍l!πss![i(k-h)]l-s+1[(-1)s-1],
when ℓ≥1, and for ℓ=0,
(2.29)λkh(0)=δkh.For the proof see [2].Analogously for the connection coefficients (2.22)2 we have that the any order connection coefficients of the Shannon scaling wavelets ψkn(x) are
(2.30)γ(l)khnm=μ(h-k)δnm{∑s=1l+1‍(-1)[1+μ(h-k)](2l-s+1)/2l!il-sπl-s(l-s+1)!|h-k|s(-1)-s-2(h+k)2nl-s-1(h-k)δnm∑s=1l+1×{2l+1[(-1)4h+s+(-1)4k+l]-2s[(-1)3k+h+l+(-1)3h+k+s]}∑‍},k≠hγ(l)khnm=δnm[ilπl2nl-1l+1(2l+1-1)1+((-1)l)],k=h,
or, shortly
(2.31)γ(l)khnm=δnm{il(1-|μ(h-k)|)πl2nl-1l+1(2l+1-1)(1+(-1)l)πl2nl+1+μ(h-k)∑s=1l+1‍(-1)[1+μ(h-k)](2l-s+1)/2l!il-sπl-s(l-s+1)!|h-k|s(-1)-s-2(h+k)2nl-s-1πl2nl+1×{2l+1[(-1)4h+s+(-1)4k+l]-2s[(-1)3k+h+l+(-1)3h+k+s]}πl2nl-1l+1},
for ℓ≥1, and
(2.32)γ(0)khnm=δkhδnm,ℓ=0, respectively.For the proof see [2].


## 2.1. Properties of the Shannon Wavelet

According to  (2.2), Shannon wavelets can be easily computed at some special points, being in particular
(2.10)φk(h)=φh(k)=φ(h-k)=φ(k-h)=δkh,(h,k∈Z),
so that
(2.11)φk(x)={0,x=h≠k,(h,k∈Z)1,x=h=k,(h,k∈Z).
It is also [7]
(2.12)ψkn(h)=(-1)2nh-k21+n/2(2n+1h-2k-1)π,(2n+1h-2k-1≠0)ψkn(x)=0,x=2-n(k+12±13),(n∈N,k∈Z)lim⁡x→2-n(h+(1/2))ψkn(x)=-2n/2δhk.In the following, we will be interested on the maximum values of these functions which can be easily computed. The maximum value of the scaling functionφk(x) can be found at the integers x=k(2.13)max⁡[φk(xM)]=1,xM=k,
and the max values of ψkn(x) are
(2.14)max⁡[ψkn(xM)]=2n/233π,xM={-2-n(k+16)2-n-13(18k+7).Both families of scaling and wavelet functions belong toL2(ℝ), thus having a bounded range and (slow) decay to zero
(2.15)lim⁡x→±∞φkn(x)=0,lim⁡x→±∞ψkn(x)=0.Letℬ⊂L2(ℝ) the set of functions f(x) in L2(ℝ) such that the integrals
(2.16)αk≝〈f(x),φk(x)〉=(2.8)∫-∞∞‍f(x)φk0(x)¯dxβkn≝〈f(x),ψkn(x)〉=(2.8)∫-∞∞‍f(x)ψkn(x)¯dx
exist with finite values, then it can be shown [2–4, 7] that the series
(2.17)f(x)=∑h=-∞∞‍αhφh(x)+∑n=0∞‍∑k=-∞∞‍βknψkn(x)
converges to f(x).According to  (2.8), the coefficients can be also computed in the Fourier domain [7] so that
(2.18)αk=∫-ππ‍f^(ω)eiωkdω,βkn=2-n/2[∫2nπ2n+1π‍f^(ω)eiω(k+1/2)/2ndω+∫-2n+1π-2nπ‍f^(ω)eiω(k+1/2)/2ndω].
In the frequency domain, (2.17) gives [7]
(2.19)f^(ω)=12πχ(ω+3π)∑h=-∞∞‍αheiωh+12πχ(ω2n-1)∑n=0∞‍∑k=-∞∞‍2-n/2βkneiω(k+1/2)/2n+12πχ(-ω2n-1)∑n=0∞‍∑k=-∞∞‍2-n/2βkneiω(k+1/2)/2n.
When the upper bound for the series of  (2.17) is finite, then we have the approximation
(2.20)f(x)≅∑h=-KK‍αhφh(x)+∑n=0N‍∑k=-SS‍βknψkn(x).
The error of the approximation has been estimated in [7].


## 2.2. Reconstruction of the Derivatives

In order to represent the differential operators in wavelet bases, we have to compute the wavelet decomposition of the derivatives. It can be shown [2, 7] that the derivatives of the Shannon wavelets are orthogonal functions:
(2.21)dldxlφh(x)=∑k=-∞∞‍λhk(l)φk(x),dldxlψhm(x)=∑n=0∞‍∑k=-∞∞‍γ(l)hkmnψkn(x),
being
(2.22)λkh(l)≝〈dldxlφk0(x),φh0(x)〉,γ(l)khmn≝〈dldxlψkn(x),ψhm(x)〉,
the connection coefficients [2, 5, 6, 8–13].The computation of connection coefficients can be easily performed in the Fourier domain, thanks to the equality  (2.8)
(2.23)λkh(l)=2π〈dldxlφk(x)^,φh(x)^〉,γ(l)khmn=2π〈dldxlψkn(x)^,ψhm(x)^〉.In fact, in the Fourier domain, theℓ-order derivative of the (scaling) wavelet functions are simply
(2.24)dldxlφkn(x)^=(iω)lφ^kn(ω),dldxlψkn(x)^=(iω)lψ^kn(ω),
and, according to (2.7),
(2.25)dldxlφkn(x)^=(iω)l2-n/22πeiωk/2nχ(ω2n+3π),dldxlψkn(x)^=(iω)l2-n/22πeiω(k+(1/2))/2n[χ(ω2n-1)+χ(-ω2n-1)].It has been shown [2, 6, 7] that the any order connection coefficients (2.22)1 of the Shannon scaling functions φk(x) are
(2.26)λkh(l)={(-1)k-hil2π∑s=1l‍l!πss![i(k-h)]l-s+1[(-1)s-1],k≠hilπl+12π(l+1)[1+(-1)l],k=h,
or, by defining
(2.27)μ(m)=sign⁡(m)={1,m>0-1,m<00,m=0,
shortly as,
(2.28)λkh(l)=ilπl2(l+1)[1+(-1)l](1-|μ(k-h)|)+(-1)k-h|μ(k-h)|il2π∑s=1l‍l!πss![i(k-h)]l-s+1[(-1)s-1],
when ℓ≥1, and for ℓ=0,
(2.29)λkh(0)=δkh.For the proof see [2].Analogously for the connection coefficients (2.22)2 we have that the any order connection coefficients of the Shannon scaling wavelets ψkn(x) are
(2.30)γ(l)khnm=μ(h-k)δnm{∑s=1l+1‍(-1)[1+μ(h-k)](2l-s+1)/2l!il-sπl-s(l-s+1)!|h-k|s(-1)-s-2(h+k)2nl-s-1(h-k)δnm∑s=1l+1×{2l+1[(-1)4h+s+(-1)4k+l]-2s[(-1)3k+h+l+(-1)3h+k+s]}∑‍},k≠hγ(l)khnm=δnm[ilπl2nl-1l+1(2l+1-1)1+((-1)l)],k=h,
or, shortly
(2.31)γ(l)khnm=δnm{il(1-|μ(h-k)|)πl2nl-1l+1(2l+1-1)(1+(-1)l)πl2nl+1+μ(h-k)∑s=1l+1‍(-1)[1+μ(h-k)](2l-s+1)/2l!il-sπl-s(l-s+1)!|h-k|s(-1)-s-2(h+k)2nl-s-1πl2nl+1×{2l+1[(-1)4h+s+(-1)4k+l]-2s[(-1)3k+h+l+(-1)3h+k+s]}πl2nl-1l+1},
for ℓ≥1, and
(2.32)γ(0)khnm=δkhδnm,ℓ=0, respectively.For the proof see [2].


## 3. Remarks on Connection Coefficients

### 3.1. Recursiveness

The connection coefficients fulfill some recursive formula as follows.Theorem 3.1.
The connection coefficients (2.26) are recursively given by
(3.1)λkh(l+1)={l+1k-hλkh(l)-(-1)k-hilπl+1k-h[(-1)l+1],k≠hiπl+1l+2λkh(l)+(-i)l+1πl+1l+2,k=h,Proof.
Let us show first whenk=h. From the definition  (2.26), it is
(3.2)λkk(l+1)=il+1πl+22π(l+2)[1+(-1)l+1]=iπ(l+1)(l+2)ilπl+12π(l+1)[1+(-1)l+1+(-1)l-(-1)l]=iπ(l+1)(l+2)ilπl+12π(l+1)[1+(-1)l+2(-1)l+1],
from where (3.1)2 follows. Analogously with simple computation we obtain (3.1)1.Shorty and with some caution, (3.1) can be written as
(3.3)λkh(l+1)=(1-δkh)[l+1k-hλkh(l)-(-1)k-hilπl+1k-h[(-1)l+1]]+δkh[iπl+1l+2λkh(l)+(-i)l+1πl+1l+2],
that is,
(3.4)λkh(l+1)=[(1-δkh)l+1k-h+δkhiπl+1l+2]λkh(l)-(1-δkh)(-1)k-hilπl+1k-h[(-1)l+1]+δkh(-i)l+1πl+1l+2.It is not so easy to find out a similar property also for theγ-coefficients as a function of ℓ however, there is a simple rule for the recursiveness of the scale (upper) indexes, as follows.Theorem 3.2.
The connection coefficients (2.30) are recursively given by the matrix at the lowest scale level:
(3.5)γ(l)khnn=2l(n-1)γ(l)kh11.Proof.
As can be seen from (2.30) parameter n appears only in the factor
(3.6)2nl-1,
so that (3.5) follows from the identity
(3.7)2nl-1=2l(n-1)2l-1.Moreover, it can be shown also that(3.8)γ(2l+1)khnn=-γ(2l+1)hknn,γ(2l)hknn=γ(2l)hknn.


### 3.2. Taylor Series

By using the connection coefficients, it is easy to show the following theorem.Theorem 3.3.
Iff(x)∈Bψ⊂L2(ℝ) and f(x)∈𝒞S the Taylor series of f(x) in x0 is
(3.9)f(x)=f(x0)+∑r=1S‍[∑h,k=-∞∞‍αhλhk(r)φk(x0)+∑n=0∞‍∑k,s=-∞∞‍2r(n-1)βknγ(r)sk11ψsn(x0)](x-x0)rr!+RS(x,x0),
being αh and βkn given by (2.16), (2.18) and RS(x,x0) the error.Proof.
From (2.17), the ℓ-order derivative (ℓ≤S) is
(3.10)f(l)(x)=∑h=-∞∞‍αhdldxlφh(x)+∑n=0∞‍∑k=-∞∞‍βkndldxlψkn(x),=(2.21)∑h=-∞∞‍αh∑k=-∞∞‍λhk(l)φk(x)+∑n=0∞‍∑k=-∞∞‍βkn∑m=-∞∞‍∑s=-∞∞‍γ(l)skmnψsm(x),=∑h,k=-∞∞‍αhλhk(l)φk(x)+∑n,m=0∞‍∑k,s=-∞∞‍βknγ(l)skmnψsm(x),
so that by taking into account  (3.5) the proof follows.In particular, by a suitable choice of the initial pointx0, (3.9) can be simplified. For instance, at the integers, x0=h,(h∈ℤ), according to (2.10), (2.12) and (3.5), it is
(3.11)f(x)≅f(h)+∑r=1S‍[∑h=-∞∞‍αhλhh(r)+∑n=0∞‍∑k,s=-∞∞‍(-1)2nh-s2r(n-1)+1+n/2(2n+1h-2s-1)πβknγ(r)sk11](x-h)rr!.


### 3.3. Functional Equations

The connection coefficients fulfill some identities as follows.Theorem 3.4.
For anyk∈ℤ and ℓ∈ℕ, it is
(3.12)(iω)le-iωk=∑h=-∞∞‍λkh(l)e-iωh,-π≤ω≤π,
or
(3.13)(iω)l=∑h=-∞∞‍λkh(l)e-iω(h-k),-π≤ω≤π,∀k∈Z.Proof.
From (2.21), by a Fourier transform of both sides and taking into account  (2.24), we get
(3.14)(iω)lφ^k(ω)=∑h=-∞∞‍λkh(l)φ^h(ω)(iω)le-iωkχ(ω+3π)=(2.7)∑h=-∞∞‍λkh(l)e-iωhχ(ω+3π),
from where the identity (3.12) follows.In particular, by assuming, without restrictions,k=0, we have the following (see Figure 1).Approximation of(iω)ℓ (plain) by the r.h.s of (3.15) at different scale: (a) ℑ[(iω)3],k=5,|hmax|=5; (b) ℑ[(iω)3],k=5,|hmax|=10; (c) ℜ[(iω)2],k=7,|hmax|=5; (d) ℜ[(iω)2],k=7,|hmax|=8.
(a)
(b)
(c)
(d)Corollary 3.5.
For anyℓ∈ℕ it is
(3.15)(iω)l=∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π,so thatλ0h(ℓ) are the Fourier coefficients of the power (iω)ℓ.Analogously, from (2.21)2, we have the following.Theorem 3.6.
For anyk∈ℤ and ℓ,n∈ℕ it is
(3.16)(iω)le-iω(k+1/2)/2n=∑h=-∞∞‍γ(l)khnne-iω(h+1/2)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π],
or
(3.17)(iω)l=∑h=-∞∞‍γ(l)khnne-iω(h-k)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].In particular, withk=0, and taking into account  (3.5), we have the following.Corollary 3.7.
For anyℓ,n∈ℕ it is
(3.18)(iω)l=2l(n-1)∑h=-∞∞‍γ(l)0h11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].As a consequence of the previous theorems we have the following.Theorem 3.8.
For anyℓ,n∈ℕ it is
(3.19)(iω)l={∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π2l(n-1)∑h=-∞∞‍γ(l)0h11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].There we have the following.Corollary 3.9.
The Fourier transform of the derivatives of a function is(3.20)dldxlf(x)=^f^(ω)×{∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π,2l(n-1)∑h=-∞∞‍γ(l)kh11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].If we expresseiω as a Taylor series we have
(3.21)eiω=∑l=0∞‍(iω)ll!,
so that eiω with -π≤ω≤π is the solution of the functional equation
(3.22)X=∑l=0∞‍∑h=-∞∞‍1l!λ0h(l)X-h.Moreover, the theorem of moments(3.23)∫R‍xlf(x)dx=ildf^(ω)dωl
can be written as
(3.24)∫R‍xlf(x)dx=ilf^(ω)×{∑h=-∞∞‍λkh(l)e-iωh,-π≤ω≤π∑h=-∞∞‍γ(l)khnne-iω(h-k)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].


### 3.4. Error of the Approximation by Connection Coefficients

For a fixed scale of approximation in (2.21), it is possible to estimate the error as follows. It should be noticed that the approximation depends on a the upper bound of the limits in the sums.Theorem 3.10 (error of the approximation of scaling functions derivatives).
The error of the approximation in (2.21)1 is given by
(3.25)|dldxlφh(x)-∑k=-NN‍λhk(l)φk(x)|≤|λh(-N-1)(l)+λh(N+1)(l)|.Proof.
The error of the approximation (2.21)1 is defined as
(3.26)dldxlφh(x)-∑k=-NN‍λhk(l)φk(x)=∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x).
Concerning the r.h.s, and according to  (2.13), it is
(3.27)∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x)≤max⁡x∈R[∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x)]=λh(-N-1)(l)φ-N-1(x)+λh(N+1)(l)φN+1(x)≤λh(-N-1)(l)+λh(N+1)(l).Theorem 3.11 (error of the approximation of wavelet functions derivatives).
The error of the approximation in (2.21)2 is given by
(3.28)|dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)|≤|2l(m-1)+m233π[γ(l)h(-S-1)11+γ(l)h(S+1)11]|.Proof.
The error of the approximation is(3.29)dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)=∑n=N+1∞‍[∑k=-∞-S-1‍γ(l)hkmnψkn(x)+∑k=S+1∞‍γ(l)hkmnψkn(x)].
If m<N, the r.h.s. according to  (2.30) is zero; therefore, we assume that m>N so that the last equation becomes
(3.30)dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)=[∑k=-∞-S-1‍γ(l)hkmmψkn(x)+∑k=S+1∞‍γ(l)hkmmψkn(x)]=(3.5)2l(m-1)[∑k=-∞-S-1‍γ(l)hk11+∑k=S+1∞‍γ(l)hk11]ψkm(x)≤2l(m-1)max⁡{[∑k=-∞-S-1‍γ(l)hk11+∑k=S+1∞‍γ(l)hk11]ψkm(x)}=2l(m-1)[γ(l)h(-S-1)11ψ(-S-1)m(x)+γ(l)h(S+1)11ψ(S+1)m(x)]≤2l(m-1)[γ(l)h(-S-1)11max⁡ψ(-S-1)m(x)+γ(l)h(S+1)11max⁡ψ(S+1)m(x)]=(2.14)2l(m-1)2m/233π[γ(l)h(-S-1)11+γ(l)h(S+1)11].


## 3.1. Recursiveness

The connection coefficients fulfill some recursive formula as follows.Theorem 3.1.
The connection coefficients (2.26) are recursively given by
(3.1)λkh(l+1)={l+1k-hλkh(l)-(-1)k-hilπl+1k-h[(-1)l+1],k≠hiπl+1l+2λkh(l)+(-i)l+1πl+1l+2,k=h,Proof.
Let us show first whenk=h. From the definition  (2.26), it is
(3.2)λkk(l+1)=il+1πl+22π(l+2)[1+(-1)l+1]=iπ(l+1)(l+2)ilπl+12π(l+1)[1+(-1)l+1+(-1)l-(-1)l]=iπ(l+1)(l+2)ilπl+12π(l+1)[1+(-1)l+2(-1)l+1],
from where (3.1)2 follows. Analogously with simple computation we obtain (3.1)1.Shorty and with some caution, (3.1) can be written as
(3.3)λkh(l+1)=(1-δkh)[l+1k-hλkh(l)-(-1)k-hilπl+1k-h[(-1)l+1]]+δkh[iπl+1l+2λkh(l)+(-i)l+1πl+1l+2],
that is,
(3.4)λkh(l+1)=[(1-δkh)l+1k-h+δkhiπl+1l+2]λkh(l)-(1-δkh)(-1)k-hilπl+1k-h[(-1)l+1]+δkh(-i)l+1πl+1l+2.It is not so easy to find out a similar property also for theγ-coefficients as a function of ℓ however, there is a simple rule for the recursiveness of the scale (upper) indexes, as follows.Theorem 3.2.
The connection coefficients (2.30) are recursively given by the matrix at the lowest scale level:
(3.5)γ(l)khnn=2l(n-1)γ(l)kh11.Proof.
As can be seen from (2.30) parameter n appears only in the factor
(3.6)2nl-1,
so that (3.5) follows from the identity
(3.7)2nl-1=2l(n-1)2l-1.Moreover, it can be shown also that(3.8)γ(2l+1)khnn=-γ(2l+1)hknn,γ(2l)hknn=γ(2l)hknn.


## 3.2. Taylor Series

By using the connection coefficients, it is easy to show the following theorem.Theorem 3.3.
Iff(x)∈Bψ⊂L2(ℝ) and f(x)∈𝒞S the Taylor series of f(x) in x0 is
(3.9)f(x)=f(x0)+∑r=1S‍[∑h,k=-∞∞‍αhλhk(r)φk(x0)+∑n=0∞‍∑k,s=-∞∞‍2r(n-1)βknγ(r)sk11ψsn(x0)](x-x0)rr!+RS(x,x0),
being αh and βkn given by (2.16), (2.18) and RS(x,x0) the error.Proof.
From (2.17), the ℓ-order derivative (ℓ≤S) is
(3.10)f(l)(x)=∑h=-∞∞‍αhdldxlφh(x)+∑n=0∞‍∑k=-∞∞‍βkndldxlψkn(x),=(2.21)∑h=-∞∞‍αh∑k=-∞∞‍λhk(l)φk(x)+∑n=0∞‍∑k=-∞∞‍βkn∑m=-∞∞‍∑s=-∞∞‍γ(l)skmnψsm(x),=∑h,k=-∞∞‍αhλhk(l)φk(x)+∑n,m=0∞‍∑k,s=-∞∞‍βknγ(l)skmnψsm(x),
so that by taking into account  (3.5) the proof follows.In particular, by a suitable choice of the initial pointx0, (3.9) can be simplified. For instance, at the integers, x0=h,(h∈ℤ), according to (2.10), (2.12) and (3.5), it is
(3.11)f(x)≅f(h)+∑r=1S‍[∑h=-∞∞‍αhλhh(r)+∑n=0∞‍∑k,s=-∞∞‍(-1)2nh-s2r(n-1)+1+n/2(2n+1h-2s-1)πβknγ(r)sk11](x-h)rr!.


## 3.3. Functional Equations

The connection coefficients fulfill some identities as follows.Theorem 3.4.
For anyk∈ℤ and ℓ∈ℕ, it is
(3.12)(iω)le-iωk=∑h=-∞∞‍λkh(l)e-iωh,-π≤ω≤π,
or
(3.13)(iω)l=∑h=-∞∞‍λkh(l)e-iω(h-k),-π≤ω≤π,∀k∈Z.Proof.
From (2.21), by a Fourier transform of both sides and taking into account  (2.24), we get
(3.14)(iω)lφ^k(ω)=∑h=-∞∞‍λkh(l)φ^h(ω)(iω)le-iωkχ(ω+3π)=(2.7)∑h=-∞∞‍λkh(l)e-iωhχ(ω+3π),
from where the identity (3.12) follows.In particular, by assuming, without restrictions,k=0, we have the following (see Figure 1).Approximation of(iω)ℓ (plain) by the r.h.s of (3.15) at different scale: (a) ℑ[(iω)3],k=5,|hmax|=5; (b) ℑ[(iω)3],k=5,|hmax|=10; (c) ℜ[(iω)2],k=7,|hmax|=5; (d) ℜ[(iω)2],k=7,|hmax|=8.
(a)
(b)
(c)
(d)Corollary 3.5.
For anyℓ∈ℕ it is
(3.15)(iω)l=∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π,so thatλ0h(ℓ) are the Fourier coefficients of the power (iω)ℓ.Analogously, from (2.21)2, we have the following.Theorem 3.6.
For anyk∈ℤ and ℓ,n∈ℕ it is
(3.16)(iω)le-iω(k+1/2)/2n=∑h=-∞∞‍γ(l)khnne-iω(h+1/2)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π],
or
(3.17)(iω)l=∑h=-∞∞‍γ(l)khnne-iω(h-k)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].In particular, withk=0, and taking into account  (3.5), we have the following.Corollary 3.7.
For anyℓ,n∈ℕ it is
(3.18)(iω)l=2l(n-1)∑h=-∞∞‍γ(l)0h11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].As a consequence of the previous theorems we have the following.Theorem 3.8.
For anyℓ,n∈ℕ it is
(3.19)(iω)l={∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π2l(n-1)∑h=-∞∞‍γ(l)0h11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].There we have the following.Corollary 3.9.
The Fourier transform of the derivatives of a function is(3.20)dldxlf(x)=^f^(ω)×{∑h=-∞∞‍λ0h(l)e-iωh,-π≤ω≤π,2l(n-1)∑h=-∞∞‍γ(l)kh11e-iωh/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].If we expresseiω as a Taylor series we have
(3.21)eiω=∑l=0∞‍(iω)ll!,
so that eiω with -π≤ω≤π is the solution of the functional equation
(3.22)X=∑l=0∞‍∑h=-∞∞‍1l!λ0h(l)X-h.Moreover, the theorem of moments(3.23)∫R‍xlf(x)dx=ildf^(ω)dωl
can be written as
(3.24)∫R‍xlf(x)dx=ilf^(ω)×{∑h=-∞∞‍λkh(l)e-iωh,-π≤ω≤π∑h=-∞∞‍γ(l)khnne-iω(h-k)/2n,ω∈[-2n+1π,-2nπ]∪[2nπ,2n+1π].


## 3.4. Error of the Approximation by Connection Coefficients

For a fixed scale of approximation in (2.21), it is possible to estimate the error as follows. It should be noticed that the approximation depends on a the upper bound of the limits in the sums.Theorem 3.10 (error of the approximation of scaling functions derivatives).
The error of the approximation in (2.21)1 is given by
(3.25)|dldxlφh(x)-∑k=-NN‍λhk(l)φk(x)|≤|λh(-N-1)(l)+λh(N+1)(l)|.Proof.
The error of the approximation (2.21)1 is defined as
(3.26)dldxlφh(x)-∑k=-NN‍λhk(l)φk(x)=∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x).
Concerning the r.h.s, and according to  (2.13), it is
(3.27)∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x)≤max⁡x∈R[∑k=-∞-N-1‍λhk(l)φk(x)+∑k=N+1∞‍λhk(l)φk(x)]=λh(-N-1)(l)φ-N-1(x)+λh(N+1)(l)φN+1(x)≤λh(-N-1)(l)+λh(N+1)(l).Theorem 3.11 (error of the approximation of wavelet functions derivatives).
The error of the approximation in (2.21)2 is given by
(3.28)|dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)|≤|2l(m-1)+m233π[γ(l)h(-S-1)11+γ(l)h(S+1)11]|.Proof.
The error of the approximation is(3.29)dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)=∑n=N+1∞‍[∑k=-∞-S-1‍γ(l)hkmnψkn(x)+∑k=S+1∞‍γ(l)hkmnψkn(x)].
If m<N, the r.h.s. according to  (2.30) is zero; therefore, we assume that m>N so that the last equation becomes
(3.30)dldxlψhm(x)-∑n=0N‍∑k=-SS‍γ(l)hkmnψkn(x)=[∑k=-∞-S-1‍γ(l)hkmmψkn(x)+∑k=S+1∞‍γ(l)hkmmψkn(x)]=(3.5)2l(m-1)[∑k=-∞-S-1‍γ(l)hk11+∑k=S+1∞‍γ(l)hk11]ψkm(x)≤2l(m-1)max⁡{[∑k=-∞-S-1‍γ(l)hk11+∑k=S+1∞‍γ(l)hk11]ψkm(x)}=2l(m-1)[γ(l)h(-S-1)11ψ(-S-1)m(x)+γ(l)h(S+1)11ψ(S+1)m(x)]≤2l(m-1)[γ(l)h(-S-1)11max⁡ψ(-S-1)m(x)+γ(l)h(S+1)11max⁡ψ(S+1)m(x)]=(2.14)2l(m-1)2m/233π[γ(l)h(-S-1)11+γ(l)h(S+1)11].


## 4. Fractional Derivatives of the Wavelet Basis

The simplest way to define the fractional derivative is based on the assumption that the noninteger derivative of the exponential function formally coincides with the derivative with integer order so that(4.1)dνdxνeax=aνeaxν∈Q.
For negative values of ν, this formula still holds true and it represents the integration.It is known that the fractional derivative cannot be analytically computed except for some special functions, such as (see e.g., [16–18]) the following:
(4.2)dνdxνeax=aνeax,dνdxνcos⁡ax=aνcos⁡(ax+π2ν),dνdxνsin⁡ax=aνsin⁡(ax+π2ν).
From these, classical examples, we can see that the fractional derivative can be also interpreted as an interpolating function between derivatives with integer order, so that
(4.3)dνdxνf(x)=(1-ν)f(x)+νddxf(x),0≤ν≤1.More in general, letf(x) be a single-valued real function, then the Riemann-Liouville fractional order derivative is defined as [16]
(4.4)dνdxνf(x)≝1Γ(1-ν)ddx∫0x‍f(ξ)(x-ξ)νdξ,(0<ν<1,x>0),Γ(ν) being the gamma function.Other equivalent representations were given by Caputo (for a differentiable function)(4.5)dνdxνf(x)≝1Γ(1-ν)∫0x‍f′(ξ)(x-ξ)νdξ,0<ν<1,
and by Grünwald (see e.g., [17, 18])(4.6)dνdxνf(x)=lim⁡N→∞1Γ(-ν)(xN)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)x],(0<ν<1,x>0).
However, a drawback in the Grünwald definition, as well as in the Riemann-Liouville, is that it cannot be computed for negative values of the variable (x<0).


### 4.1. Fractional Derivative of the Shannon Scaling Function

Let us assume that the fractional order derivative is defined by a linear interpolation of the integer order derivatives, so that the fractional derivative of the scaling-wavelet basis(4.7)dl+νdxl+νφh(x),dl+νdxl+νψhm(x).
with
(4.8)0≤ν≤1,
can be defined as
(4.9)dl+νdxl+νφh(x)≝(1-ν)dldxlφh(x)+νdl+1dxl+1φh(x),dl+νdxl+νψhm(x)≝(1-ν)dldxlψhm(x)+νdl+1dxl+1ψhm(x).Let us show the following.Theorem 4.1.
The fractional derivative of the Shannon scaling functions is(4.10)dl+νdxl+νφh(x)≝∑k=-∞∞‍λhk(l+ν)φk(x)={∑k=-∞∞‍[(1-ν)λhk(l)+νλhk(l+1)]φk(x),l>0∑k=-∞∞‍[(1-ν)δhk+νλhk(1)]φk(x),l=0.Proof.
From (4.9), by taking into account (2.21), it is
(4.11)dl+νdxl+νφh(x)≝(1-ν)∑k=-∞∞‍λhk(l)φk(x)+ν∑k=-∞∞‍λhk(l+1)φk(x)=(3.1)∑k=-∞∞‍[(1-ν)λhk(l)+νλhk(l+1)]φk(x),
and, when ℓ=0,
(4.12)dνdxνφh(x)=∑k=-∞∞‍[(1-ν)δhk+νλhk(1)]φk(x).With this definition, the fractional order derivative of the scaling functions is a commutative operator according to the following.Theorem 4.2.
The operator (4.10) is a semigroup, so that
(4.13)dμdxμdνdxνφh(x)=dνdxνdμdxμφh(x)=dμ+νdxμ+νφh(x).Proof.
Without loss of generality, let us show that(4.14)dμdxμdνdxνφ(x)=dνdxνdμdxμφ(x).
According to (4.10)2, it is
(4.15)dνdxνφ0(x)=∑k=-∞∞‍[(1-ν)δ0k+νλ0k(1)]φk(x),
that is
(4.16)dνdxνφ(x)=(1-ν)φ(x)+ν[λ00(1)φ(x)+∑k≠0k=-∞∞λ0k(1)φk(x)],
and, taking into account (2.26), by explicit computation we have
(4.17)dνdxνφ(x)=(1-ν)φ(x)+ν∑k≠0k=-∞∞(-1)kkφk(x).
By deriving, with respect to μ, we have
(4.18)dμdxμdνdxνφ(x)=(1-ν)dμdxμφ(x)+ν∑k≠0k=-∞∞(-1)kkdμdxμφk(x)=(4.17)(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]=(26)+ν∑k≠0k=-∞∞(-1)kkdμdxμφk(x),
that is, according to (2.26),
(4.19)dμdxμdνdxνφ(x)=(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]+ν∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍[(1-μ)δsk+μλsk(1)]φs(x)=(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]+ν(1-μ)∑k≠0k=-∞∞(-1)kkφk(x)+νμ∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍λsk(1)φs(x).
From where,
(4.20)dμdxμdνdxνφ(x)=(1-ν)(1-μ)φ(x)+[(1-ν)μ+ν(1-μ)]∑k≠0k=-∞∞(-1)kkφk(x)+νμ∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍λsk(1)φs(x),
the proof follows due to the symmetry of the change μ→ν.It can be easily seen that together with  (4.17) also the following equations hold:
(4.21)dνdxνφ1(x)=(1-ν)φ1(x)+ν∑k≠0k=-∞∞(-1)kk-1φk(x)dνdxνφ-1(x)=(1-ν)φ-1(x)+ν∑k≠0k=-∞∞(-1)k1+kφk(x),
and, in general,
(4.22)dνdxνφh(x)=(1-ν)φh(x)+ν∑k≠0k=-∞∞(-1)kk-hφk(x).Moreover, whenμ+ν=1, then we can see that the definition  (2.26) reduces to the ordinary derivative, according to the following.Theorem 4.3.
Whenμ+ν=1, then
(4.23)dμdxμdνdxνφh(x)=dμ+νdxμ+νφh(x)=ddxφh(x).Proof.
If we restrict toφ(x), according to the definition  (2.26), it is
(4.24)dμdxμdνdxνφ(x)=∑k=-∞∞‍[(1-(μ+ν))δ0k+(μ+ν)λ0k(1)]φk(x),
and since (μ+ν)=1 we have
(4.25)dμdxμdνdxνφ(x)=ddxφ(x)=∑k=-∞∞‍λ0k(1)φk(x),According to the definition (4.10), the fractional derivative is an interpolation between integer order derivative (see Figure 2).Figure 2
Fractional derivative of the scaling functions(dν/dxν)φ(x) with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


### 4.2. Error of the Approximation of  (4.10)

In the definition (4.10), the fractional derivative depends on a fixed bound N of the infinite series. In this section, it will be shown that the rate of convergence of the series, on the r.h.s of  (4.10), is quite fast; already with low values of N, the approximation is quite good (Figure 3).Fractional derivative of the scaling functions(d3/10/dx3/10)φ(x) with upper limit N=1,…,10 (a) and N=10,…,50 (b).
(a)
(b)


#### 4.2.1. Rate of Convergence

If we compare the fractional derivative(dν/dxν)φh(x) given by (4.10) with the Grünwald definition  (4.6), we can see that the approximation by connection coefficients is good (see Figure 4), with a lower number of terms. Moreover, the definition based on connection coefficients can be extended also to negative values of the variable.Fractional derivative of the scaling functions(dν/dxν)φh(x) by Grünwald approximation (4.6) (shaded) and connection coefficients interpolation (4.10)2 (plain): (a) ν=1/10,h=0 with upper limit N=1 (connection coefficients) and N=4 (Grünwald); (b) ν=1/10,h=1 with upper limit N=1 (connection coefficients) and N=1 (Grünwald); (c) ν=1/20,h=1 with upper limit N=2 (connection coefficients) and N=8 (Grünwald); (d) ν=9/10,h=1 with upper limit N=10 (connection coefficients) and N=50 (Grünwald).
(a)
(b)
(c)
(d)Since we have defined the fractional derivative on an infinite seriesN→∞, as well as the Grünwald formula, we can explicitly compute the error of the approximation as the difference between the approximated value at N+1 and the corresponding value of the infinite series at N. For instance, with respect to  (4.10), it is
(4.26)εNν=max⁡x∈R|∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)|,
while for the Grünwald formula  (4.6) we have
(4.27)ϵNν=max⁡x>0|1Γ(-ν)(xN+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)x]--1Γ(-ν)(xN)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)x]|,
Let us show the following.Theorem 4.4.
Forℓ=0, the approximation error of (4.10)2 is given by
(4.28)εNν=2ν|(-1)N+1h(N+1)2-h2|.Proof.
By taking into account  (4.22), it is
(4.29)∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)=ν[(-1)N+1-(N+1)-hφ-(N+1)(x)φk(x)+(-1)N+1(N+1)-hφ(N+1)(x)]<(2.13)ν[(-1)N+1-(N+1)-h+(-1)N+1(N+1)-h]=2ν(-1)N+1h(N+1)2-h2.Analogously, the following can be shown.Theorem 4.5.
Forx>0, the approximation error of (4.6)2 is given by
(4.30)ϵNν=NνΓ(-ν)Γ(N-ν)Γ(N+1).Proof.
At the integerx=1, it is
(4.31)1Γ(-ν)(1N+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)(1N)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<1Γ(-ν)Nν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)Nν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<(2.13)NνΓ(-ν)Γ(N-ν)Γ(N+1).


### 4.3. Fractional Derivative of the Shannon Wavelet

Analogously to  (4.10), the following can be proved.Theorem 4.6.
The fractional derivative of the Shannon wavelet functions is(4.32)dl+νdxl+νψhm(x)≝∑k=-∞∞‍γ(l+ν)hkmmψkm(x)={2l(m-1)[∑k=-∞∞‍(1-ν)γ(l)hk11+ν2m-1γ(l+1)hk11]ψkm(x),l>0[∑k=-∞∞‍(1-ν)δhk+ν2m-1γ(1)hk11]ψkm(x),l=0.Proof.
From (4.9), by taking into account (2.21)
(4.33)dl+νdxl+νψhm(x)=(1-ν)∑n=0∞‍∑k=-∞∞‍γ(l)hkmnψkn(x)+ν∑n=0∞‍∑k=-∞∞‍γ(l+1)hkmnψkn(x)=(3.5)[(1-ν)∑n=0∞‍∑k=-∞∞‍δmn2l(n-1)γ(l)hk11+ν∑n=0∞‍∑k=-∞∞‍δmn2(l+1)(n-1)γ(l+1)hk11]ψkn(x)=[(1-ν)∑k=-∞∞‍2l(m-1)γ(l)hk11+ν∑k=-∞∞‍2(l+1)(m-1)γ(l+1)hk11]ψkm(x)=2l(m-1)[∑k=-∞∞‍(1-ν)γ(l)hk11+ν2m-1γ(l+1)hk11]ψkm(x).Analogously to the fractional derivative of the scaling function, also for the wavelet function, the fractional order derivatives are enveloped by the integer order derivatives (Figure5).Figure 5
Fractional derivative of the wavelet functions(dν/dxν)ψ00(x) with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


### 4.4. Fractional Derivative of anL2(ℝ) Function

Letf(x)∈ℬ⊂L2(ℝ) be a function such that  (2.17) holds, then its fractional derivative can be computed as
(4.34)dνdxνf(x)=∑h=-∞∞‍αhdνdxνφh(x)+∑n=0∞‍∑k=-∞∞‍βkndνdxνψkn(x),
where the fractional derivatives of the scaling functions φh(x) and wavelets ψkn(x) are given by  (4.10) and  (4.32), respectively.For instance, a good approximation ofy=e-x2 is (Figure 6)
(4.35)e-x2≅0.97φ(x)+0.39[φ-1(x)+φ1(x)].
The fractional derivative is
(4.36)dνdxνe-x2≅0.97dνdxνφ(x)+0.39dνdxν[φ-1(x)+φ1(x)],
so that by using  (4.17) and (4.21) we have
(4.37)dνdxνe-x2≅0.97[(1-ν)φ(x)+ν∑k≠0k=-∞∞(-1)kkφk(x)]+0.39(1-ν)[φ-1(x)+φ1(x)]+0.39ν[∑k≠0k=-∞∞(-1)k+1kφk(x)+∑k≠0k=-∞∞(-1)kk-1φk(x)],Figure 6
Fractional derivative of the functiony=e-x2 with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


## 4.1. Fractional Derivative of the Shannon Scaling Function

Let us assume that the fractional order derivative is defined by a linear interpolation of the integer order derivatives, so that the fractional derivative of the scaling-wavelet basis(4.7)dl+νdxl+νφh(x),dl+νdxl+νψhm(x).
with
(4.8)0≤ν≤1,
can be defined as
(4.9)dl+νdxl+νφh(x)≝(1-ν)dldxlφh(x)+νdl+1dxl+1φh(x),dl+νdxl+νψhm(x)≝(1-ν)dldxlψhm(x)+νdl+1dxl+1ψhm(x).Let us show the following.Theorem 4.1.
The fractional derivative of the Shannon scaling functions is(4.10)dl+νdxl+νφh(x)≝∑k=-∞∞‍λhk(l+ν)φk(x)={∑k=-∞∞‍[(1-ν)λhk(l)+νλhk(l+1)]φk(x),l>0∑k=-∞∞‍[(1-ν)δhk+νλhk(1)]φk(x),l=0.Proof.
From (4.9), by taking into account (2.21), it is
(4.11)dl+νdxl+νφh(x)≝(1-ν)∑k=-∞∞‍λhk(l)φk(x)+ν∑k=-∞∞‍λhk(l+1)φk(x)=(3.1)∑k=-∞∞‍[(1-ν)λhk(l)+νλhk(l+1)]φk(x),
and, when ℓ=0,
(4.12)dνdxνφh(x)=∑k=-∞∞‍[(1-ν)δhk+νλhk(1)]φk(x).With this definition, the fractional order derivative of the scaling functions is a commutative operator according to the following.Theorem 4.2.
The operator (4.10) is a semigroup, so that
(4.13)dμdxμdνdxνφh(x)=dνdxνdμdxμφh(x)=dμ+νdxμ+νφh(x).Proof.
Without loss of generality, let us show that(4.14)dμdxμdνdxνφ(x)=dνdxνdμdxμφ(x).
According to (4.10)2, it is
(4.15)dνdxνφ0(x)=∑k=-∞∞‍[(1-ν)δ0k+νλ0k(1)]φk(x),
that is
(4.16)dνdxνφ(x)=(1-ν)φ(x)+ν[λ00(1)φ(x)+∑k≠0k=-∞∞λ0k(1)φk(x)],
and, taking into account (2.26), by explicit computation we have
(4.17)dνdxνφ(x)=(1-ν)φ(x)+ν∑k≠0k=-∞∞(-1)kkφk(x).
By deriving, with respect to μ, we have
(4.18)dμdxμdνdxνφ(x)=(1-ν)dμdxμφ(x)+ν∑k≠0k=-∞∞(-1)kkdμdxμφk(x)=(4.17)(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]=(26)+ν∑k≠0k=-∞∞(-1)kkdμdxμφk(x),
that is, according to (2.26),
(4.19)dμdxμdνdxνφ(x)=(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]+ν∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍[(1-μ)δsk+μλsk(1)]φs(x)=(1-ν)[(1-μ)φ(x)+μ∑k≠0k=-∞∞(-1)kkφk(x)]+ν(1-μ)∑k≠0k=-∞∞(-1)kkφk(x)+νμ∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍λsk(1)φs(x).
From where,
(4.20)dμdxμdνdxνφ(x)=(1-ν)(1-μ)φ(x)+[(1-ν)μ+ν(1-μ)]∑k≠0k=-∞∞(-1)kkφk(x)+νμ∑k≠0k=-∞∞(-1)kk∑s=-∞∞‍λsk(1)φs(x),
the proof follows due to the symmetry of the change μ→ν.It can be easily seen that together with  (4.17) also the following equations hold:
(4.21)dνdxνφ1(x)=(1-ν)φ1(x)+ν∑k≠0k=-∞∞(-1)kk-1φk(x)dνdxνφ-1(x)=(1-ν)φ-1(x)+ν∑k≠0k=-∞∞(-1)k1+kφk(x),
and, in general,
(4.22)dνdxνφh(x)=(1-ν)φh(x)+ν∑k≠0k=-∞∞(-1)kk-hφk(x).Moreover, whenμ+ν=1, then we can see that the definition  (2.26) reduces to the ordinary derivative, according to the following.Theorem 4.3.
Whenμ+ν=1, then
(4.23)dμdxμdνdxνφh(x)=dμ+νdxμ+νφh(x)=ddxφh(x).Proof.
If we restrict toφ(x), according to the definition  (2.26), it is
(4.24)dμdxμdνdxνφ(x)=∑k=-∞∞‍[(1-(μ+ν))δ0k+(μ+ν)λ0k(1)]φk(x),
and since (μ+ν)=1 we have
(4.25)dμdxμdνdxνφ(x)=ddxφ(x)=∑k=-∞∞‍λ0k(1)φk(x),According to the definition (4.10), the fractional derivative is an interpolation between integer order derivative (see Figure 2).Figure 2
Fractional derivative of the scaling functions(dν/dxν)φ(x) with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


## 4.2. Error of the Approximation of  (4.10)

In the definition (4.10), the fractional derivative depends on a fixed bound N of the infinite series. In this section, it will be shown that the rate of convergence of the series, on the r.h.s of  (4.10), is quite fast; already with low values of N, the approximation is quite good (Figure 3).Fractional derivative of the scaling functions(d3/10/dx3/10)φ(x) with upper limit N=1,…,10 (a) and N=10,…,50 (b).
(a)
(b)


### 4.2.1. Rate of Convergence

If we compare the fractional derivative(dν/dxν)φh(x) given by (4.10) with the Grünwald definition  (4.6), we can see that the approximation by connection coefficients is good (see Figure 4), with a lower number of terms. Moreover, the definition based on connection coefficients can be extended also to negative values of the variable.Fractional derivative of the scaling functions(dν/dxν)φh(x) by Grünwald approximation (4.6) (shaded) and connection coefficients interpolation (4.10)2 (plain): (a) ν=1/10,h=0 with upper limit N=1 (connection coefficients) and N=4 (Grünwald); (b) ν=1/10,h=1 with upper limit N=1 (connection coefficients) and N=1 (Grünwald); (c) ν=1/20,h=1 with upper limit N=2 (connection coefficients) and N=8 (Grünwald); (d) ν=9/10,h=1 with upper limit N=10 (connection coefficients) and N=50 (Grünwald).
(a)
(b)
(c)
(d)Since we have defined the fractional derivative on an infinite seriesN→∞, as well as the Grünwald formula, we can explicitly compute the error of the approximation as the difference between the approximated value at N+1 and the corresponding value of the infinite series at N. For instance, with respect to  (4.10), it is
(4.26)εNν=max⁡x∈R|∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)|,
while for the Grünwald formula  (4.6) we have
(4.27)ϵNν=max⁡x>0|1Γ(-ν)(xN+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)x]--1Γ(-ν)(xN)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)x]|,
Let us show the following.Theorem 4.4.
Forℓ=0, the approximation error of (4.10)2 is given by
(4.28)εNν=2ν|(-1)N+1h(N+1)2-h2|.Proof.
By taking into account  (4.22), it is
(4.29)∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)=ν[(-1)N+1-(N+1)-hφ-(N+1)(x)φk(x)+(-1)N+1(N+1)-hφ(N+1)(x)]<(2.13)ν[(-1)N+1-(N+1)-h+(-1)N+1(N+1)-h]=2ν(-1)N+1h(N+1)2-h2.Analogously, the following can be shown.Theorem 4.5.
Forx>0, the approximation error of (4.6)2 is given by
(4.30)ϵNν=NνΓ(-ν)Γ(N-ν)Γ(N+1).Proof.
At the integerx=1, it is
(4.31)1Γ(-ν)(1N+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)(1N)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<1Γ(-ν)Nν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)Nν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<(2.13)NνΓ(-ν)Γ(N-ν)Γ(N+1).


## 4.2.1. Rate of Convergence

If we compare the fractional derivative(dν/dxν)φh(x) given by (4.10) with the Grünwald definition  (4.6), we can see that the approximation by connection coefficients is good (see Figure 4), with a lower number of terms. Moreover, the definition based on connection coefficients can be extended also to negative values of the variable.Fractional derivative of the scaling functions(dν/dxν)φh(x) by Grünwald approximation (4.6) (shaded) and connection coefficients interpolation (4.10)2 (plain): (a) ν=1/10,h=0 with upper limit N=1 (connection coefficients) and N=4 (Grünwald); (b) ν=1/10,h=1 with upper limit N=1 (connection coefficients) and N=1 (Grünwald); (c) ν=1/20,h=1 with upper limit N=2 (connection coefficients) and N=8 (Grünwald); (d) ν=9/10,h=1 with upper limit N=10 (connection coefficients) and N=50 (Grünwald).
(a)
(b)
(c)
(d)Since we have defined the fractional derivative on an infinite seriesN→∞, as well as the Grünwald formula, we can explicitly compute the error of the approximation as the difference between the approximated value at N+1 and the corresponding value of the infinite series at N. For instance, with respect to  (4.10), it is
(4.26)εNν=max⁡x∈R|∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)|,
while for the Grünwald formula  (4.6) we have
(4.27)ϵNν=max⁡x>0|1Γ(-ν)(xN+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)x]--1Γ(-ν)(xN)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)x]|,
Let us show the following.Theorem 4.4.
Forℓ=0, the approximation error of (4.10)2 is given by
(4.28)εNν=2ν|(-1)N+1h(N+1)2-h2|.Proof.
By taking into account  (4.22), it is
(4.29)∑k=-(N+1)N+1‍λhk(l+ν)φk(x)-∑k=-NN‍λhk(l+ν)φk(x)=ν[(-1)N+1-(N+1)-hφ-(N+1)(x)φk(x)+(-1)N+1(N+1)-hφ(N+1)(x)]<(2.13)ν[(-1)N+1-(N+1)-h+(-1)N+1(N+1)-h]=2ν(-1)N+1h(N+1)2-h2.Analogously, the following can be shown.Theorem 4.5.
Forx>0, the approximation error of (4.6)2 is given by
(4.30)ϵNν=NνΓ(-ν)Γ(N-ν)Γ(N+1).Proof.
At the integerx=1, it is
(4.31)1Γ(-ν)(1N+1)-ν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)(1N)-ν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<1Γ(-ν)Nν∑k=0N‍Γ(k-ν)Γ(k+1)f[(1-kN+1)]-1Γ(-ν)Nν∑k=0N-1‍Γ(k-ν)Γ(k+1)f[(1-kN)]<(2.13)NνΓ(-ν)Γ(N-ν)Γ(N+1).


## 4.3. Fractional Derivative of the Shannon Wavelet

Analogously to  (4.10), the following can be proved.Theorem 4.6.
The fractional derivative of the Shannon wavelet functions is(4.32)dl+νdxl+νψhm(x)≝∑k=-∞∞‍γ(l+ν)hkmmψkm(x)={2l(m-1)[∑k=-∞∞‍(1-ν)γ(l)hk11+ν2m-1γ(l+1)hk11]ψkm(x),l>0[∑k=-∞∞‍(1-ν)δhk+ν2m-1γ(1)hk11]ψkm(x),l=0.Proof.
From (4.9), by taking into account (2.21)
(4.33)dl+νdxl+νψhm(x)=(1-ν)∑n=0∞‍∑k=-∞∞‍γ(l)hkmnψkn(x)+ν∑n=0∞‍∑k=-∞∞‍γ(l+1)hkmnψkn(x)=(3.5)[(1-ν)∑n=0∞‍∑k=-∞∞‍δmn2l(n-1)γ(l)hk11+ν∑n=0∞‍∑k=-∞∞‍δmn2(l+1)(n-1)γ(l+1)hk11]ψkn(x)=[(1-ν)∑k=-∞∞‍2l(m-1)γ(l)hk11+ν∑k=-∞∞‍2(l+1)(m-1)γ(l+1)hk11]ψkm(x)=2l(m-1)[∑k=-∞∞‍(1-ν)γ(l)hk11+ν2m-1γ(l+1)hk11]ψkm(x).Analogously to the fractional derivative of the scaling function, also for the wavelet function, the fractional order derivatives are enveloped by the integer order derivatives (Figure5).Figure 5
Fractional derivative of the wavelet functions(dν/dxν)ψ00(x) with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


## 4.4. Fractional Derivative of anL2(ℝ) Function

Letf(x)∈ℬ⊂L2(ℝ) be a function such that  (2.17) holds, then its fractional derivative can be computed as
(4.34)dνdxνf(x)=∑h=-∞∞‍αhdνdxνφh(x)+∑n=0∞‍∑k=-∞∞‍βkndνdxνψkn(x),
where the fractional derivatives of the scaling functions φh(x) and wavelets ψkn(x) are given by  (4.10) and  (4.32), respectively.For instance, a good approximation ofy=e-x2 is (Figure 6)
(4.35)e-x2≅0.97φ(x)+0.39[φ-1(x)+φ1(x)].
The fractional derivative is
(4.36)dνdxνe-x2≅0.97dνdxνφ(x)+0.39dνdxν[φ-1(x)+φ1(x)],
so that by using  (4.17) and (4.21) we have
(4.37)dνdxνe-x2≅0.97[(1-ν)φ(x)+ν∑k≠0k=-∞∞(-1)kkφk(x)]+0.39(1-ν)[φ-1(x)+φ1(x)]+0.39ν[∑k≠0k=-∞∞(-1)k+1kφk(x)+∑k≠0k=-∞∞(-1)kk-1φk(x)],Figure 6
Fractional derivative of the functiony=e-x2 with upper limit N=4 at different values of ν=0,1/5,2/5,3/5,4/5,1.


## 5. Conclusion

In this paper, fractional calculus has been revised by using Shannon wavelets. Fractional derivatives of the Shannon scaling/wavelet functions, based on connection coefficients, are explicitly computed and the approximation error is estimated. In the comparison with the classical Grünwald formula of fractional derivative, Shannon wavelets and connection coefficients make a better approximation and rate of convergence.



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*Source: 502812-2012-08-07.xml*