Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 502812, 26 pages
doi:10.1155/2012/502812
Research Article
Fractional Calculus and Shannon Wavelet
Carlo Cattani
Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Carlo Cattani, ccattani@unisa.it
Received 18 February 2012; Accepted 13 May 2012
Academic Editor: Cristian Toma
Copyright q 2012 Carlo Cattani. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 any L2 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.
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 socalled 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 for L2 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
2
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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 an L2 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, Section 2 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 ϕx,
also known as sinc function, and the wavelet function ψx, respectively, defined as
def
ϕx sinc x sin πx eπix − e−πix
,
πx
2πix
sin 2πx − 1/2 − sin πx − 1/2
πx − 1/2
e−2 i π x −i ei π x e3 i π x i e4 i π x
.
2πx − 1/2
ψx 2.1
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3
The corresponding families of translated and dilated instances wavelet 1, 2, 6, 7, 9, on which
is based the multiscale analysis 4, are
ϕnk x 2n/2 ϕ2n x − k 2n/2
eπi2 x−k − e−πi2 x−k
,
2πi2n x − k
n
2n/2
ψkn x 2n/2
sin π2n x − k
π2n x − k
n
sin 2π2n x − k − 1/2 − sin π2n x − k − 1/2
π2n x − k − 1/2
2.2
2
2n/2
n
n
i1s esπi2 x−k − i1−s e−sπi2 x−k ,
n
2π2 x − k − 1/2 s1
being, in particular,
ϕ00 x ϕx,
ψ00 x ψx,
ϕ0k x ϕk x ϕx − k,
ψk0 x ψk x ψx − k.
2.3
Let
1
def
fω
fx
2π
∞
−∞
fxe−iωx dx,
fx ∞
−∞
iωx
dω
fωe
2.4
be the Fourier transform of the function fx ∈ L2 R, and its inverse transform, respectively.
The Fourier transform of 2.1 give us 2
⎧
⎨ 1 ,
1
χω 3π 2π
ϕω
⎩0,
2π
ψω
−π ≤ ω < π
elsewhere
2.5
1 iω/2 e
χ2ω χ−2ω ,
2π
with
χω 1,
0,
2π ≤ ω < 4π
elsewhere.
2.6
Analogously for the dilated and translated instances of scaling/wavelet function, in
the frequency domain, it is
2−n/2 iωk/2n ω
e
χ n 3π
2π
2
−n/2
ω
ω
2
n
iωk1/2/2n
e
ψk ω χ n−1 χ − n−1 .
2π
2
2
ϕnk ω 2.7
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Both families of Shannon scaling and wavelet are L2 R-functions therefore, for each
fx ∈ L2 R and gx ∈ L2 R, the inner product is defined as
f, g
def
∞
−∞
fxgxdx 2π
∞
−∞
g ωdω 2π f,
g ,
fω
2.8
where the bar stands for the complex conjugate.
Shannon wavelets fulfill the following orthogonality properties for the proof see e.g.,
2, 7:
ψkn x, ψhm x δnm δhk ,
ϕ0k x, ϕ0h x δkh ,
ϕ0k x, ψhm x 0,
m ≥ 0,
2.9
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
ϕk h ϕh k ϕh − k ϕk − h δkh ,
h, k ∈ Z,
2.10
so that
ϕk x 0,
1,
x h/
k,
h, k ∈ Z
x h k, h, k ∈ Z.
2.11
It is also 7
n
ψkn h −12
ψkn x 0,
h−k
21n/2
2n1 h − 2k − 1 ,
/0
− 2k − 1 π
1 1
−n
k ±
x2
, n ∈ N, k ∈ Z
2 3
2n1 h
ψ n x
lim
x → 2−n h1/2 k
2.12
−2n/2 δhk .
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 function ϕk x can be
found at the integers x k
max ϕk xM 1,
xM k,
2.13
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5
and the max values of ψkn x are
√
3 3
,
max ψkn xM 2n/2
π
xM
⎧
1
⎪
−n
⎪
k
⎨−2
6
−n−1
⎪
2
⎪
⎩
18k 7.
3
2.14
Both families of scaling and wavelet functions belong to L2 R, thus having a bounded
range and slow decay to zero
lim ϕn x
x → ±∞ k
0,
lim ψ n x
x → ±∞ k
0.
2.15
Let B ⊂ L2 R the set of functions fx in L2 R such that the integrals
2.8
αk fx, ϕk x def
def
βkn 2.8
fx, ψkn x
∞
−∞
∞
−∞
fxϕ0k xdx
2.16
fxψkn xdx
exist with finite values, then it can be shown 2–4, 7 that the series
fx ∞
αh ϕh x ∞ ∞
βkn ψkn x
2.17
n0 k−∞
h−∞
converges to fx.
According to 2.8, the coefficients can be also computed in the Fourier domain 7 so
that
αk βkn
−n/2
2
2n1 π
π
−π
fω
eiωk dω,
iωk1/2/2n
dω fωe
−2n π
2n π
−2n1 π
iωk1/2/2n
dω .
fωe
2.18
In the frequency domain, 2.17 gives 7
∞
1
χω 3π
αh ei ωh
fω
2π
h−∞
∞ ∞
ω 1
n
χ n−1
2−n/2 βkn eiωk1/2/2
2π
2
n0 k−∞
∞ ∞
ω 1
n
χ − n−1
2−n/2 βkn eiωk1/2/2 .
2π
2
n0 k−∞
2.19
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When the upper bound for the series of 2.17 is finite, then we have the approximation
fx ∼
K
αh ϕh x N S
βkn ψkn x.
2.20
n0 k−S
h−K
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:
∞
d
ϕ
λhk ϕk x,
x
h
dx
k−∞
∞ ∞
mn
d
m
ψ
γ hk ψkn x,
x
h
dx
n0 k−∞
2.21
being
def
λkh d
0
0
ϕ x, ϕh x ,
dx
k
γ
def
mn
kh
d
n
m
ψ x, ψh x ,
dx
k
2.22
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
λkh
2π
d
ϕk x, ϕ
h x ,
dx
γ
mn
kh
2π
d
n
m
ψ x, ψ
x .
h
dx
k
2.23
In fact, in the Fourier domain, the -order derivative of the scaling wavelet functions
are simply
d
ϕn x iω
ϕnk ω,
dx
k
d
ψ n x iω
ψkn ω,
dx
k
2.24
and, according to 2.7,
ω
−n/2
d
2
iωk/2n
n
e
ϕ
χ
3π
,
x
iω
2π
2n
dx
k
−n/2
d
ω
ω
2
iωk1/2/2n
n
e
ψ x iω
χ n−1 χ − n−1 .
2π
2
2
dx
k
2.25
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7
It has been shown 2, 6, 7 that the any order connection coefficients 2.221 of the
Shannon scaling functions ϕk x are
λkh
⎧
⎪
!π s
⎪
k−h i
⎪
h
−1s − 1 , k /
⎨−1
2π s1 s!ik − h
−s1
⎪
i
π 1 ⎪
⎪
⎩
1 −1
,
k h,
2π
1
2.26
or, by defining
⎧
⎪
⎪
⎨1,
μm signm −1,
⎪
⎪
⎩0,
m>0
m<0
m 0,
2.27
shortly as,
λkh i
π 1 −1
1 − μk − h
2
1
−1
i
!π s
μk − h
−1s − 1 ,
−s1
2π s1 s!ik − h
2.28
k−h when ≥ 1, and for 0,
0
λkh δkh .
2.29
For the proof see 2.
Analogously for the connection coefficients 2.222 we have that the any order
connection coefficients of the Shannon scaling wavelets ψkn x are
γ
nm
kh
μh − kδ
γ
nm
kh
δ
nm
i
π
nm
1
−11μh−k2
−s1/2
!i
−s π −s
−s−2hk n
−s−1
2
s −1
−
s
1!|h
−
k|
s1
!
4hs
4k
3kh
3hks
1
s
− 2 −1
, k
× 2
−1
−1
−1
/h
n
−1 2
1
1
2
,
− 1 1 −1
k h,
2.30
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or, shortly
γ
nm
kh
δ
nm
π 2n
−1 1
i
1 − μh − k
2 − 1 1 −1
1
1
−11μh−k2
−s1/2
!i
−s π −s
−s−2hk n
−s−1
2
s −1
−
s
1!
−
k|
|h
s1
!
4hs
4k
3kh
3hks
1
s
− 2 −1
,
× 2
−1
−1
−1
μh − k
2.31
for ≥ 1, and
nm
γ 0kh δkh δnm ,
2.32
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
1
λkh
⎧
1 1 ⎪
k−h i π
⎪
⎪
λ
−
1
,
−1
−1
⎨ k − h kh
k−h
1 −i
1 π 1
⎪
⎪
⎪
,
⎩iπ 2 λkh 2
k/
h
k h,
3.1
Proof. Let us show first when k h. From the definition 2.26, it is
1
λkk
i
1 π 2 1 −1
1
2π
2
iπ
1 i
π 1 1 −1
1 −1
− −1
2 2π
1
iπ
1 i
π 1 1 −1
2−1
1 ,
2 2π
1
from where 3.12 follows. Analogously with simple computation we obtain 3.11 .
3.2
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9
Shorty and with some caution, 3.1 can be written as
1
λkh
1 i
π 1 λkh − −1k−h
1 − δkh −1
1
k−h
k−h
1 −i
1 π 1
λ ,
δkh iπ
2 kh
2
3.3
that is,
1
λkh
1
1 δkh iπ
1 − δkh λ
k−h
2 kh
i
π 1 −i
1 π 1
.
− 1 − δkh −1k−h
−1
1 δkh
k−h
2
3.4
It is not so easy to find out a similar property also for the γ-coefficients as a function of
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:
nn
11
γ kh 2
n−1 γ kh .
3.5
Proof. As can be seen from 2.30 parameter n appears only in the factor
2n
−1 ,
3.6
2n
−1 2
n−1 2
−1 .
3.7
so that 3.5 follows from the identity
Moreover, it can be shown also that
nn
nn
γ 2
1kh −γ 2
1hk ,
nn
nn
γ 2
hk γ 2
hk .
3.2. Taylor Series
By using the connection coefficients, it is easy to show the following theorem.
3.8
10
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Theorem 3.3. If fx ∈ Bψ ⊂ L2 R and fx ∈ CS the Taylor series of fx in x0 is
fx fx0 S
∞
∞
∞
x − x0 r
r
rn−1 n r11
n
RS x, x0 ,
αh λhk ϕk x0 2
βk γ sk ψs x0 r!
n0 k,s−∞
r1 h,k−∞
3.9
being αh and βkn given by 2.16, 2.18 and RS x, x0 the error.
Proof. From 2.17, the -order derivative ≤ S is
∞
f x ∞ ∞
d
d
ϕ
βkn ψkn x,
x
h
dx
dx
n0 k−∞
αh
h−∞
2.21
∞
αh
h−∞
∞
∞
λhk ϕk x k−∞
∞
αh λhk ϕk x h,k−∞
∞ ∞
βkn
n0 k−∞
∞
∞
∞ mn
γ sk ψsm x,
3.10
m−∞s−∞
mn
βkn γ sk ψsm x,
n,m0 k,s−∞
so that by taking into account 3.5 the proof follows.
In particular, by a suitable choice of the initial point x0 , 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
fx ∼
fh S
∞
r
αh λhh
r1 h−∞
∞ ∞
−1
2n h−s
n0 k,s−∞
2rn−11n/2
x − hr
n r11
sk
.
βk γ
n1
r!
2 h − 2s − 1 π
3.11
3.3. Functional Equations
The connection coefficients fulfill some identities as follows.
Theorem 3.4. For any k ∈ Z and ∈ N, it is
iω
e−iωk ∞
λkh e−iωh ,
−π ≤ ω ≤ π,
3.12
h−∞
or
iω
∞
λkh e−iωh−k ,
h−∞
−π ≤ ω ≤ π, ∀k ∈ Z.
3.13
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11
Proof. From 2.21, by a Fourier transform of both sides and taking into account 2.24, we get
∞
iω
ϕk ω λkh ϕh ω
h−∞
−iωk
iω e
2.7
χω 3π 3.14
∞
λkh e−iωh χω
h−∞
3π,
from where the identity 3.12 follows.
In particular, by assuming, without restrictions, k 0, we have the following see
Figure 1.
Corollary 3.5. For any ∈ N it is
iω
∞
λ0h e−iωh ,
−π ≤ ω ≤ π,
3.15
h−∞
so that λ0h are the Fourier coefficients of the power iω
.
Analogously, from 2.212 , we have the following.
Theorem 3.6. For any k ∈ Z and , n ∈ N it is
iω
e−iωk1/2/2 n
∞
nn
γ kh e−iωh1/2/2 ,
n
ω ∈ −2n1 π, −2n π ∪ 2n π , 2n1 π ,
3.16
h−∞
or
∞
iω
nn
γ kh e−iωh−k/2 ,
n
ω ∈ −2n1 π , −2n π ∪ 2n π , 2n1 π .
3.17
h−∞
In particular, with k 0, and taking into account 3.5, we have the following.
Corollary 3.7. For any , n ∈ N it is
iω
2
n−1
∞
11
γ 0h e−iωh/2 ,
n
ω ∈ −2n1 π , −2n π ∪ 2n π , 2n1 π .
3.18
h−∞
As a consequence of the previous theorems we have the following.
Theorem 3.8. For any , n ∈ N it is
iω ⎧
∞
⎪
⎪
λ0h e−iωh ,
⎪
⎨
−π ≤ ω ≤ π
∞
11
n
⎪
⎪
n−1
⎪
γ 0h e−iωh/2 ,
⎩2
ω ∈ −2n1 π, −2n π ∪ 2n π, 2n1 π .
h−∞
h−∞
3.19
12
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−π
1
−1
π
x
−π
a
−π
1
−1
π
x
b
π
x
−π
−1
c
−1
π
x
d
Figure 1: Approximation of iω
plain by the r.h.s of 3.15 at different scale: a iω3 , k 5, |hmax | 5; b iω3 , k 5, |hmax | 10; c iω2 , k 7, |hmax | 5; d iω2 , k 7, |hmax | 8.
There we have the following.
Corollary 3.9. The Fourier transform of the derivatives of a function is
⎧
∞
⎪
⎪
λ0h e−iωh ,
⎪
⎨
d
fx fω × h−∞ ∞
11
n
⎪
dx
⎪
n−1
⎪
γ kh e−iωh/2 ,
⎩2
h−∞
−π ≤ ω ≤ π
ω ∈ −2n1 π, −2n π ∪ 2n π, 2n1 π .
3.20
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13
If we express eiω as a Taylor series we have
eiω ∞
iω
!
0
,
3.21
so that eiω with −π ≤ ω ≤ π is the solution of the functional equation
X
∞
∞ 1 −h
λ0h X .
!
0 h−∞
3.22
Moreover, the theorem of moments
R
x
fxdx i
dfω
dω
3.23
can be written as
x fxdx i fω
×
R
⎧ ∞
⎪
⎪
⎪
λkh e−iωh ,
⎪
⎨
h−∞
∞
⎪
⎪
⎪
⎪
⎩
nn
−π ≤ ω ≤ π
γ kh e−iωh−k/2 ,
n
ω ∈ −2n1 π, −2n π ∪ 2n π, 2n1 π .
h−∞
3.24
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 2.211 is given by
N
d
λhk ϕk x ≤ λh−N−1 λhN1 .
ϕh x −
dx
k−N
3.25
Proof. The error of the approximation 2.211 is defined as
N
−N−1
∞
d
ϕ
λ
ϕ
λ
ϕ
λhk ϕk x.
−
x
x
x
h
k
k
hk
hk
dx
k−N
k−∞
kN1
3.26
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Concerning the r.h.s, and according to 2.13, it is
−N−1
∞
λhk ϕk x k−∞
≤ max
x∈R
λhk ϕk x
kN1
−N−1
λhk
k−∞
ϕk x ∞
λhk
kN1
3.27
ϕk x
λh−N−1 ϕ−N−1 x λhN1 ϕN1 x ≤ λh−N−1 λhN1 .
Theorem 3.11 error of the approximation of wavelet functions derivatives. The error of the
approximation in 2.212 is given by
√ N S
d
11
11
3
3
m
mn
n
m−1m/2
γ h−S−1 γ hS1 .
γ hk ψk x ≤ 2
ψh x −
dx
π
n0 k−S
3.28
Proof. The error of the approximation is
N ∞
−S−1
∞
S
d
m
mn
n
mn
n
mn
n
hk
hk
hk
ψ x −
γ
ψk x γ
ψk x γ
ψk x .
dx
h
n0 k−S
nN1 k−∞
kS1
3.29
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
N S
mn
d
m
ψ
γ hk ψkn x −
x
h
dx
n0 k−S
−S−1
γ
mm
hk
k−∞
3.5
2
m−1
ψkn x
−S−1
γ
≤2
γ
mm
hk
ψkn x
kS1
11
hk
k−∞
m−1
∞
max
∞
γ
11
hk
ψkm x
kS1
−S−1
γ
k−∞
11
hk
∞
γ
11
hk
!
ψkm x
kS1
11
11
m
m
2
m−1 γ h−S−1 ψ−S−1
x γ hS1 ψS1
x
11
11
m
m
≤ 2
m−1 γ h−S−1 max ψ−S−1
x γ hS1 max ψS1
x
2.14
√ 11
3 11
γ h−S−1 γ hS1 .
π
m−1 m/2 3
2
2
3.30
Mathematical Problems in Engineering
15
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
dν ax
e aν eax
dxν
4.1
ν ∈ 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:
dν
π cos ax aν cos ax ν ,
ν
dx
2
ν
d
π
sin ax aν sin ax ν .
dxν
2
dν ax
e aν eax ,
dxν
4.2
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
dν
d
fx 1 − νfx ν fx,
dxν
dx
0 ≤ ν ≤ 1.
4.3
More in general, let fx be a single-valued real function, then the Riemann-Liouville
fractional order derivative is defined as 16
dν
d
1
def
fx ν
dx
Γ1 − ν dx
x
0
fξ
dξ,
x − ξν
0 < ν < 1, x > 0,
4.4
Γν being the gamma function.
Other equivalent representations were given by Caputo for a differentiable function
dν
1
def
fx dxν
Γ1 − ν
x
0
f ξ
dξ,
x − ξν
0 < ν < 1,
4.5
and by Grünwald see e.g., 17, 18
Γk − ν dν
k
1 x −ν N−1
f 1−
x ,
fx lim
N → ∞ Γ−ν N
dxν
Γk 1
N
k0
0 < ν < 1, x > 0.
4.6
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.
16
Mathematical Problems in Engineering
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
d
ν
ϕh x,
dx
ν
d
ν m
ψ x.
dx
ν h
4.7
with
0 ≤ ν ≤ 1,
4.8
can be defined as
d
d
ν
d
1
def
ϕ
−
ν
ϕ
ϕh x,
ν
1
x
x
h
h
dx
ν
dx
dx
1
4.9
d
m
d
ν m
d
1 m
def
ψ
−
ν
ψ
ψ x.
ν
1
x
x
h
h
dx
ν
dx
dx
1 h
Let us show the following.
Theorem 4.1. The fractional derivative of the Shannon scaling functions is
⎧
∞ 1
⎪
⎪
ϕk x,
1 − νλhk νλhk
⎪
⎨
∞
d
ν
def ν
ϕ
λhk ϕk x k−∞
x
h
∞ ⎪
1
dx
ν
⎪
k−∞
⎪
ϕk x,
νλ
−
νδ
1
hk
⎩
hk
>0
0.
4.10
k−∞
Proof. From 4.9, by taking into account 2.21, it is
∞
∞
d
ν
def
1
ϕ
−
ν
λ
ϕ
λhk ϕk x
ν
1
x
x
h
k
hk
dx
ν
k−∞
k−∞
3.1
∞ 1
ϕk x,
1 − νλhk νλhk
4.11
k−∞
and, when 0,
∞ dν
1
ϕk x.
ϕ
νλ
−
νδ
x
1
h
hk
hk
dxν
k−∞
4.12
With this definition, the fractional order derivative of the scaling functions is a
commutative operator according to the following.
Mathematical Problems in Engineering
17
Theorem 4.2. The operator 4.10 is a semigroup, so that
dμ dν
dν dμ
dμν
ϕh x ϕh x ϕh x.
μ
μ
ν
ν
dx dx
dx dx
dxμν
4.13
Proof. Without loss of generality, let us show that
dμ dν
dν dμ
ϕx
ϕx.
dxμ dxν
dxν dxμ
4.14
∞ dν
1
ϕ
x
1 − νδ0k νλ0k ϕk x,
0
ν
dx
k−∞
4.15
According to 4.102 , it is
that is
⎡
⎤
⎢ 1
d
ϕx 1 − νϕx ν⎢
⎣λ00 ϕx ν
dx
ν
∞
⎥
1
λ0k ϕk x⎥
⎦,
4.16
k/
0
k−∞
and, taking into account 2.26, by explicit computation we have
∞
dν
−1k
ϕk x.
ϕx
−
νϕx
ν
1
dxν
k
k/
0
4.17
k−∞
By deriving, with respect to μ, we have
∞
dμ
dμ dν
−1k dμ
ϕx
−
ν
ϕx
ν
ϕk x
1
dxμ dxν
dxμ
k dxμ
k/
0
⎡
k−∞
⎤
∞
⎢
⎥
−1k
ϕk x⎥
1 − ν⎢
⎣ 1 − μ ϕx μ
⎦
k
k/
0
4.17
k−∞
ν
∞
−1k dμ
ϕk x,
k dxμ
k/
0
k−∞
4.18
18
Mathematical Problems in Engineering
that is, according to 2.26,
⎡
⎤
∞
⎢
⎥
dμ dν
−1k
⎢ 1 − μ ϕx μ
⎥
ϕ
ϕx
−
ν
1
x
k
μ
⎣
⎦
dx dxν
k
k/
0
k−∞
ν
∞
∞ −1k 1
1 − μ δsk μλsk ϕs x
k s−∞
k/
0
k−∞
⎡
⎤
4.19
∞
⎢
⎥
−1k
ϕk x⎥
1 − ν⎢
⎣ 1 − μ ϕx μ
⎦
k
k/
0
k−∞
∞
∞
∞
−1k
−1k 1
ϕk x νμ
ν 1−μ
λsk ϕs x.
k
k
s−∞
k/
0
k/
0
k−∞
k−∞
From where,
∞
dμ dν
−1k
ϕk x
ϕx
−
ν
1
−
μ
ϕx
−
νμ
ν
1
−
μ
1
1
dxμ dxν
k
k/
0
k−∞
νμ
∞
k/
0
k−∞
−1
k
k
∞
4.20
1
λsk ϕs x,
s−∞
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:
∞
dν
−1k
ϕk x
ϕ
−
νϕ
ν
x
1
x
1
1
dxν
k−1
k/
0
k−∞
∞
dν
−1k
ϕk x,
ϕ−1 x 1 − νϕ−1 x ν
ν
dx
1k
k/
0
4.21
k−∞
and, in general,
∞
dν
−1k
ϕk x.
ϕ
−
νϕ
ν
x
1
x
h
h
dxν
k−h
k
/0
4.22
k−∞
Moreover, when μ ν 1, then we can see that the definition 2.26 reduces to the
ordinary derivative, according to the following.
Mathematical Problems in Engineering
19
1
A =0
π
x
A =1
−1
Figure 2: Fractional derivative of the scaling functions dν /dxν ϕx with upper limit N 4 at different
values of ν 0, 1/5, 2/5, 3/5, 4/5, 1.
Theorem 4.3. When μ ν 1, then
dμ dν
dμν
d
ϕh x.
ϕ
ϕh x x
h
μ
μν
ν
dx dx
dx
dx
4.23
Proof. If we restrict to ϕx, according to the definition 2.26, it is
∞ 1 dμ dν
ϕx
μ
ν
λ0k ϕk x,
1
−
μ
ν
δ
0k
dxμ dxν
k−∞
4.24
and since μ ν 1 we have
∞
dμ dν
d
1
ϕx
ϕx
λ0k ϕk x,
dxμ dxν
dx
k−∞
4.25
According to the definition 4.10, the fractional derivative is an interpolation between
integer order derivative see Figure 2.
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.
20
Mathematical Problems in Engineering
10 < N < 50
1 < N < 10
−π
π
x
a
π
−π
x
b
Figure 3: Fractional derivative of the scaling functions d3/10 /dx3/10 ϕx with upper limit N 1, . . . , 10
a and N 10, . . . , 50 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.
Since we have defined the fractional derivative on an infinite series N → ∞, 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
ν
εN
N1
N
ν
ν
max
λhk ϕk x −
λhk ϕk x,
x∈R k−N1
k−N
4.26
while for the Grünwald formula 4.6 we have
ν
N
N
1 x −ν k
Γk − ν
f 1−
x
max
x>0 Γ−ν N 1
Γk 1
N1
k0
Γk − ν 1 x −ν N−1
k
−
f 1−
x ,
Γ−ν N
Γk
1
N
k0
Let us show the following.
4.27
Mathematical Problems in Engineering
21
1
1
π
2π
x
−1
π
2π
x
−1
a
b
1
1
π
2π
x
π
2π
x
−1
−1
c
d
Figure 4: Fractional derivative of the scaling functions dν /dxν ϕh x by Grünwald approximation 4.6
shaded and connection coefficients interpolation 4.102 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.
Theorem 4.4. For 0, the approximation error of 4.102 is given by
ν
εN
−1N1 h 2ν
.
N 12 − h2 4.28
22
Mathematical Problems in Engineering
Proof. By taking into account 4.22, it is
N1
ν
λhk ϕk x
k−N1
−
N
ν
λhk ϕk x
ν
k−N
−1N1
ϕ−N1 x
−N 1 − h
−1N1
ϕN1 x
N 1 − h
2.13
−1N1
−1N1
< ν
−N 1 − h N 1 − h
2ν−1N1 h
N 12 − h2
4.29
.
Analogously, the following can be shown.
Theorem 4.5. For x > 0, the approximation error of 4.62 is given by
ν
N
N ν ΓN − ν
.
Γ−ν ΓN 1
4.30
Proof. At the integer x 1, it is
−ν −ν N−1
N
Γk − ν 1
1
k
1
1
k
Γk − ν
f 1−
−
f 1−
Γ−ν N 1
Γk 1
N1
Γ−ν N
Γk 1
N
k0
k0
N
N−1
Γk − ν 1
k
1
k
Γk − ν
ν
ν
<
N
f 1−
−
N
f 1−
Γ−ν
Γk 1
N1
Γ−ν
Γk 1
N
k0
k0
2.13
<
N ν ΓN − ν
.
Γ−ν ΓN 1
4.31
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
∞
d
ν m
def νmm m
hk ψ x
ψh x γ
k
ν
dx
k−∞
⎧
∞
⎪
⎪
11
m−1 111
m−1
⎪
hk
hk
ψkm x,
ν 2 γ
1 − νγ
⎪
⎨2
k−∞
∞
⎪
⎪
m−1 111
⎪
hk
ψkm x,
1 − νδhk ν2 γ
⎪
⎩
k−∞
>0
0.
4.32
Mathematical Problems in Engineering
23
Proof. From 4.9, by taking into account 2.21
∞ ∞ ∞
∞
mn
d
ν m
mn
hk ψ n x ν
ψ
γ
γ 1hk ψkn x
−
ν
x
1
h
k
ν
dx
n0 k−∞
n0 k−∞
∞ ∞ ∞
∞
3.5
mn n−1 11
mn 1n−1 111
hk
hk
ψkn x
1 − ν
δ 2
γ
ν
δ 2
γ
n0 k−∞
∞
1 − ν
n0 k−∞
m−1 11
hk
2
γ
k−∞
2
m−1
∞
ν
∞
1m−1 111
hk
2
k−∞
1 − νγ
11
hk
ψkm x
m−1 111
hk
ν 2
γ
γ
ψkm x.
k−∞
4.33
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
Figure 5.
4.4. Fractional Derivative of an L2 R Function
Let fx ∈ B ⊂ L2 R be a function such that 2.17 holds, then its fractional derivative can be
computed as
∞
∞ ∞
ν
dν
dν
n d
fx
α
ϕ
β
ψ n x,
x
h
h
k
ν
ν k
dxν
dx
dx
n0 k−∞
h−∞
4.34
where the fractional derivatives of the scaling functions ϕh x and wavelets ψkn x are given
by 4.10 and 4.32, respectively.
2
For instance, a good approximation of y e−x is Figure 6
2
e−x ∼
0.97ϕx 0.39 ϕ−1 x ϕ1 x .
4.35
dν −x2 ∼
dν dν
e 0.97 ν ϕx 0.39 ν ϕ−1 x ϕ1 x ,
ν
dx
dx
dx
4.36
The fractional derivative is
24
Mathematical Problems in Engineering
1
A=0
A=1
π
x
−1
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.
1
A =0
3
−3
x
A=1
−1
Figure 6: Fractional derivative of the function y e−x with upper limit N 4 at different values of
ν 0, 1/5, 2/5, 3/5, 4/5, 1.
2
Mathematical Problems in Engineering
25
so that by using 4.17 and 4.21 we have
⎤
⎡
∞
⎥
⎢
dν −x2 ∼
−1k
⎥
⎢1 − νϕx ν
ϕ
e
0.97
x
k
⎦
⎣
dxν
k
k/
0
k−∞
0.391 − ν ϕ−1 x ϕ1 x
⎡
4.37
⎤
∞
∞
⎢
⎥
−1k1
−1k
0.39ν⎢
ϕk x ϕk x⎥
⎣
⎦,
k
k−1
k/
0
k/
0
k−∞
k−∞
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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