# Shannon  Wavelets for the Solution of Integrodifferential Equations

Authors: Carlo Cattani
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

Shannon wavelets are used to define a method for the solution of integrodifferential equations.
This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation
of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients.
The Shannon sampling theorem is considered in a more general approach suitable for analysing
functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet
reconstruction ofL2(ℝ) functions. Shannon wavelets are C∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients).


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## Body

## 1. Introduction

In recent years wavelets have been successfully applied to the wavelet representation of integro-differential operators, thus giving rise to the so-called wavelet solutions of PDE and integral equations. While wavelet solutions of PDEs can be easily find in a large specific literature, the wavelet representation of integro-differential operators cannot be considered completely achieved and only few papers discuss in depth this question with particular regards to methods for the integral equations. Some of them refer to the Haar wavelets [1–3] to the harmonic wavelets [4–9] and to the spline-Shannon wavelets [10–13]. These methods are mainly based on the Petrov-Galerkin method with a suitable choice of the collocation points [14]. Alternatively to the collocation method, there has been also proposed, for the solution of PDEs, the evaluation of the differential operators on the wavelet basis, thus defining the so-called connection coefficients [6, 15–21].Wavelets [22] are localized functions which are a useful tool in many different applications: signal analysis, data compression, operator analysis, PDE solving (see, e.g., [15, 23] and references therein), vibration analysis, and solid mechanics [23]. Very often wavelets have been used only as any other kind of orthogonal functions, without taking into consideration their fundamental properties. The main feature of wavelets is, in fact, their possibility to split objects into different scale components [22, 23] according to the multiscale resolution analysis. For the L2(ℝ) functions, that is, functions with decay to infinity, wavelets give the best approximation. When the function is localized in space, that is, the bottom length of the function is within a short interval (function with a compact support), such as pulses, any other reconstruction, but wavelets, leads towards undesirable problems such as the Gibbs phenomenon when the approximation is made in the Fourier basis. Wavelets are the most expedient basis for the analysis of impulse functions (pulses) [24, 25].Among the many families of wavelets, Shannon wavelets [17] offer some more specific advantages, which are often missing in the others. In fact, Shannon wavelets(1)
are analytically defined;(2)
are infinitely differentiable;(3)
are sharply bounded in the frequency domain, thus allowing a decomposition of frequencies in narrow bands;(4)
enjoy a generalization of the Shannon sampling theorem, which extend to all range of frequencies [17](5)
give rise to the connection coefficients which can be analytically defined [15–17] for any order derivatives, while for the other wavelet families they were computed only numerically and only for the lower order derivatives [18, 19, 21].The (Shannon wavelet) connection coefficients are obtained in [17] as a finite series (for any order derivatives). In Latto's method [18, 20, 21], instead, these coefficients were obtained only (for the Daubechies wavelets) by using the inclusion axiom but in approximated form and only for the first two-order derivatives. The knowledge of the derivatives of the basis enables us to approximate a function and its derivatives and it is an expedient tool for the projection of differential operators in the numerical computation of the solution of both partial and ordinary differential equations [6, 15, 23, 26].The wavelet reconstruction by using Shannon wavelets is also a fundamental step in the analysis of functions-operators. In fact, due to their definition Shannon wavelets are box functions in the frequency domain, thus allowing a sharp decorrelation of frequencies, which is an important feature in many physical-engineering applications. In fact, the reconstruction by Shannon wavelets ranges in multifrequency bands. Comparing with the Shannon sampling theorem where the frequency band is only one, the reconstruction by Shannon wavelets can be done for functions ranging in all frequency bands (see, e.g., [17]). The Shannon sampling theorem [27], which plays a fundamental role in signal analysis and applications, will be generalized, so that under suitable hypotheses a few set of values (samples) and a preliminary chosen Shannon wavelet basis enable us to completely represent, by the wavelet coefficients, the continuous signal and its frequencies.The Shannon wavelet solution of an integrodifferential equation (with functions localized in space and slow decay in frequency) will be computed by using the Petrov-Galerkin method and the connection coefficients. The wavelet coefficients enable to represent the solution in the frequency domain singling out the contribution to different frequencies.This paper is organized as follows. Section2 deals with some preliminary remarks and properties of Shannon wavelets also in frequency domain; the reconstruction of a function is given in Section 3 together with the generalization of the Shannon sampling theorem; the error of the wavelet approximation is computed. The wavelet reconstruction of the derivatives of the basis and the connection coefficients are given in Section 4. Section 5 deals with the Shannon wavelet solution of an integrodifferential equation and an example is given at last in Section 6.


## 2. Shannon Wavelets

Shannon wavelets theory (see, e.g., [16, 17, 28, 29]) is based on the scaling function φ(x) (also known as sinc function)(2.1)φ(x)=sincx=defsinπxπx=eπix-e-πix2πix,
and the corresponding wavelet [16, 17, 28, 29](2.2)ψ(x)=sinπ(x-1/2)-sin2π(x-1/2)π(x-1/2)=e-2iπx(-i+eiπx+e3iπx+ie4iπx)(π-2πx).From these functions a multiscale analysis [22] can be derived. The dilated and translated instances, depending on the scaling parameter n and space shift k, are(2.3)φkn(x)=2n/2φ(2nx-k)=2n/2sinπ(2nx-k)π(2nx-k)=2n/2eπi(2nx-k)-e-πi(2nx-k)2πi(2nx-k),(2.4)ψkn(x)=2n/2sinπ(2nx-k-1/2)-sin2π(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)
respectively.


### 2.1. Properties of the Shannon Scaling and Wavelet Functions

By a direct computation it can be easily seen that(2.5)φk0(h)=δkh,(h,k∈ℤ),
with δkh Kroneker symbol, so that(2.6)φk0(x)=0,x=h≠k(h,k∈ℤ),(2.7)ψkn(x)=0,x=2-n(k+12±13),(n∈ℕ,k∈ℤ).
It is also(2.8)limx→2-n(h+1/2)ψkn(x)=-2n/2δhk.Thus, according to (2.5), (2.8), for each fixed scale n, we can choose a set of points x:(2.9)x∈{h}∪{2-n(h+12±13)},(n∈ℕ,h∈ℤ),
where either the scaling functions or the wavelet vanishes, but it is important to notice that when the scaling function is zero, the wavelet is not and viceversa. As we shall see later, this property will simplify the numerical methods based on collocation point.Since they belong toL2(ℝ), both families of scaling and wavelet functions have a (slow) decay to zero; in fact, according to their definition (2.3), (2.4)(2.10)limx→±∞φkn(x)=0,limx→±∞ψkn(x)=0,
it can be also easily checked that for a fixed x0(2.11)φk+1n(x0)<φkn(x0),φk+1n(x0)φkn(x0)=2nx-k2nx-k+1<1,ψk+1n(x0)ψkn(x0)=2n+1x-2k-12n+1x-2k-3×2sin(π(2nx-k))-12sin(π(2nx-k))+1.
Since(2.12)limx→∞2n+1x-2k-12n+1x-2k-3=1,2sin(π(2nx-k))-1<2sin(π(2nx-k))+1,
it is(2.13)limx→∞ψk+1n(x)ψkn(x)<1.
Analogously we have(2.14)ψkn+1(x0)ψkn(x0)=2(2n+1x-2k-1)2n+2x-2k-1×cos(π(2n+1x-k))-sin(2π(2n+1x-k))cos(π(2nx-k))-sin(2π(2nx-k)),limx→2-n(k+1/2)ψk+1n+1(x)ψkn(x)=22(coskπ-sin2kπ)(2k-1)π=(-1)k22(2k-1)π,|(-1)k22(2k-1)π|<1.The maximum and minimum values of these functions can be easily computed. The maximum value of the scaling functionφk0(x) can be found in correspondence of x=k(2.15)max[φk0(xM)]=1,xM=k.
The min value of φk0(x) can be computed only numerically and it is(2.16)min[φk0(x)]≅φk0(xm)=sin2π2π,xm=k-1±2.The minimum of the waveletψkn(x) can be found in correspondence of the middle point of the zeroes (2.7) so that(2.17)min[ψkn(xm)]=-2n/2,xm=2-n-1(2k+1),
and the max values of ψkn(x) are(2.18)max[ψkn(xM)]=2n/233π,xM={-2-n(k+16),2-n-13(18k+7).


### 2.2. Shannon Wavelets Theory in the Fourier Domain

Let(2.19)f̂(ω)=f(x)̂=def12π∫-∞∞f(x)e-iωxdx
be the Fourier transform of the function f(x)∈L2(ℝ), and(2.20)f(x)=2π∫-∞∞f̂(ω)eiωxdω
its inverse transform.The Fourier transform of (2.1), (2.2) gives us(2.21)φ̂(ω)=12πχ(ω+3π)={12π,-π≤ω<π0,elsewhere,
and [17](2.22)ψ̂(ω)=12πe-iω[χ(2ω)+χ(-2ω)]
with(2.23)χ(ω)={1,2π≤ω<4π,0,elsewhere.
Analogously for the dilated and translated instances of scaling/wavelet function, in the frequency domain, it is(2.24)φ̂kn(ω)=2-n/22πe-iωk/2nχ(ω2n+3π),ψ̂kn(ω)=-2-n/22πe-iω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)].
It can be seen that(2.25)χ(ω+3π)[χ(ω2n-1)+χ(-ω2n-1)]=0
so that by using the function φ̂k0(ω) and ψ̂kn(ω) there is a decorrelation into different non-overlapping frequency bands.For eachf(x)∈L2(ℝ) and g(x)∈L2(ℝ), the inner product is defined as(2.26)〈f,g〉=def∫-∞∞f(x)g(x)̅dx,
which, according to the Parseval equality, can be expressed also as(2.27)〈f,g〉=def∫-∞∞f(x)g(x)̅dx=2π∫-∞∞f̂(ω)ĝ(ω)̅dω=2π〈f̂,ĝ〉,
where the bar stands for the complex conjugate.With respect to the inner product (2.26). The following can be shown. [16, 17]Theorem 2.1.
Shannon wavelets are orthonormal functions, in the sense that(2.28)〈ψkn(x),ψhm(x)〉=δnmδhk,
With δnm,δhk being the Kroenecker symbols.For the proof see [17]. Moreover we have [16, 17].Theorem 2.2.
The translated instances of the Shannon scaling functionsφkn(x), at the level n=0, are orthogonal, in the sense that
(2.29)〈φk0(x),φh0(x)〉=δkh,
being φk0(x)=defφ(x-k).See the proof in [17].The scalar product of the (Shannon) scaling functions with respect to the corresponding wavelets is characterized by the following [16, 17].Theorem 2.3.
The translated instances of the Shannon scaling functionsφkn(x), at the level n=0, are orthogonal to the Shannon wavelets, in the sense that
(2.30)〈φk0(x),ψhm(x)〉=0,m≥0,
being φk0(x)=defφ(x-k).Proof is in [17].


## 2.1. Properties of the Shannon Scaling and Wavelet Functions

By a direct computation it can be easily seen that(2.5)φk0(h)=δkh,(h,k∈ℤ),
with δkh Kroneker symbol, so that(2.6)φk0(x)=0,x=h≠k(h,k∈ℤ),(2.7)ψkn(x)=0,x=2-n(k+12±13),(n∈ℕ,k∈ℤ).
It is also(2.8)limx→2-n(h+1/2)ψkn(x)=-2n/2δhk.Thus, according to (2.5), (2.8), for each fixed scale n, we can choose a set of points x:(2.9)x∈{h}∪{2-n(h+12±13)},(n∈ℕ,h∈ℤ),
where either the scaling functions or the wavelet vanishes, but it is important to notice that when the scaling function is zero, the wavelet is not and viceversa. As we shall see later, this property will simplify the numerical methods based on collocation point.Since they belong toL2(ℝ), both families of scaling and wavelet functions have a (slow) decay to zero; in fact, according to their definition (2.3), (2.4)(2.10)limx→±∞φkn(x)=0,limx→±∞ψkn(x)=0,
it can be also easily checked that for a fixed x0(2.11)φk+1n(x0)<φkn(x0),φk+1n(x0)φkn(x0)=2nx-k2nx-k+1<1,ψk+1n(x0)ψkn(x0)=2n+1x-2k-12n+1x-2k-3×2sin(π(2nx-k))-12sin(π(2nx-k))+1.
Since(2.12)limx→∞2n+1x-2k-12n+1x-2k-3=1,2sin(π(2nx-k))-1<2sin(π(2nx-k))+1,
it is(2.13)limx→∞ψk+1n(x)ψkn(x)<1.
Analogously we have(2.14)ψkn+1(x0)ψkn(x0)=2(2n+1x-2k-1)2n+2x-2k-1×cos(π(2n+1x-k))-sin(2π(2n+1x-k))cos(π(2nx-k))-sin(2π(2nx-k)),limx→2-n(k+1/2)ψk+1n+1(x)ψkn(x)=22(coskπ-sin2kπ)(2k-1)π=(-1)k22(2k-1)π,|(-1)k22(2k-1)π|<1.The maximum and minimum values of these functions can be easily computed. The maximum value of the scaling functionφk0(x) can be found in correspondence of x=k(2.15)max[φk0(xM)]=1,xM=k.
The min value of φk0(x) can be computed only numerically and it is(2.16)min[φk0(x)]≅φk0(xm)=sin2π2π,xm=k-1±2.The minimum of the waveletψkn(x) can be found in correspondence of the middle point of the zeroes (2.7) so that(2.17)min[ψkn(xm)]=-2n/2,xm=2-n-1(2k+1),
and the max values of ψkn(x) are(2.18)max[ψkn(xM)]=2n/233π,xM={-2-n(k+16),2-n-13(18k+7).


## 2.2. Shannon Wavelets Theory in the Fourier Domain

Let(2.19)f̂(ω)=f(x)̂=def12π∫-∞∞f(x)e-iωxdx
be the Fourier transform of the function f(x)∈L2(ℝ), and(2.20)f(x)=2π∫-∞∞f̂(ω)eiωxdω
its inverse transform.The Fourier transform of (2.1), (2.2) gives us(2.21)φ̂(ω)=12πχ(ω+3π)={12π,-π≤ω<π0,elsewhere,
and [17](2.22)ψ̂(ω)=12πe-iω[χ(2ω)+χ(-2ω)]
with(2.23)χ(ω)={1,2π≤ω<4π,0,elsewhere.
Analogously for the dilated and translated instances of scaling/wavelet function, in the frequency domain, it is(2.24)φ̂kn(ω)=2-n/22πe-iωk/2nχ(ω2n+3π),ψ̂kn(ω)=-2-n/22πe-iω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)].
It can be seen that(2.25)χ(ω+3π)[χ(ω2n-1)+χ(-ω2n-1)]=0
so that by using the function φ̂k0(ω) and ψ̂kn(ω) there is a decorrelation into different non-overlapping frequency bands.For eachf(x)∈L2(ℝ) and g(x)∈L2(ℝ), the inner product is defined as(2.26)〈f,g〉=def∫-∞∞f(x)g(x)̅dx,
which, according to the Parseval equality, can be expressed also as(2.27)〈f,g〉=def∫-∞∞f(x)g(x)̅dx=2π∫-∞∞f̂(ω)ĝ(ω)̅dω=2π〈f̂,ĝ〉,
where the bar stands for the complex conjugate.With respect to the inner product (2.26). The following can be shown. [16, 17]Theorem 2.1.
Shannon wavelets are orthonormal functions, in the sense that(2.28)〈ψkn(x),ψhm(x)〉=δnmδhk,
With δnm,δhk being the Kroenecker symbols.For the proof see [17]. Moreover we have [16, 17].Theorem 2.2.
The translated instances of the Shannon scaling functionsφkn(x), at the level n=0, are orthogonal, in the sense that
(2.29)〈φk0(x),φh0(x)〉=δkh,
being φk0(x)=defφ(x-k).See the proof in [17].The scalar product of the (Shannon) scaling functions with respect to the corresponding wavelets is characterized by the following [16, 17].Theorem 2.3.
The translated instances of the Shannon scaling functionsφkn(x), at the level n=0, are orthogonal to the Shannon wavelets, in the sense that
(2.30)〈φk0(x),ψhm(x)〉=0,m≥0,
being φk0(x)=defφ(x-k).Proof is in [17].


## 3. Reconstruction of a Function by Shannon Wavelets

Letf(x)∈L2(ℝ) be a function such that for any value of the parameters n,k∈ℤ, it is(3.1)|∫-∞∞f(x)φk0(x)dx|≤Ak<∞,|∫-∞∞f(x)ψkn(x)dx|≤Bkn<∞,
and ℬ⊂L2(ℝ) the Paley-Wiener space, that is, the space of band limited functions, that is,(3.2)suppf̂⊂[-b,b],b<∞.According to the sampling theorem (see, e.g., [27] and references therein) we have the following.Theorem 3.1 (Shannon).
Iff(x)∈L2(ℝ) and suppf̂⊂[-π,π], the series
(3.3)f(x)=∑k=-∞∞αkφk0(x)
uniformly converges to f(x), and
(3.4)αk=f(k).Proof.
In order to compute the values of the coefficients we have to evaluate the series in correspondence of the integer:(3.5)f(h)=∑k=-∞∞αkφk0(h)=(2.5)∑k=-∞∞αkδkh=αh,
having taken into account (2.5).
The convergence follows from the hypotheses onf(x). In particular, the importance of the band limited frequency can be easily seen by applying the Fourier transform to (3.3):
(3.6)f̂(ω)=∑k=-∞∞f(k)φ̂k0(x)=(2.24)12π∑k=-∞∞f(k)e-iωkχ(ω+3π)=12πχ(ω+3π)∑k=-∞∞f(k)e-iωk
so that
(3.7)f̂(ω)={12π∑k=-∞∞f(k)e-iωk,ω∈[-π,π]0,ω∉[-π,π].
In other words, if the function is band limited (i.e., with compact support in the frequency domain), it can be completely reconstructed by a discrete Fourier series. The Fourier coefficients are the values of the function f(x) sampled at the integers.As a generalization of the Paley-Wiener space, and in order to generalize the Shannon theorem to unbounded intervals, we define the spaceℬψ⊇ℬ of functions f(x) such that the integrals(3.8)αk=def〈f(x),φk0(x)〉=(2.27)∫-∞∞f(x)φk0(x)̅dx,‍βkn=def〈f(x),ψkn(x)〉=(2.27)∫-∞∞f(x)ψkn(x)̅dx
exist and are finite. According to (2.26), (2.27), it is in the Fourier domain that(3.9)αk=def∫-∞∞f(x)φk0(x)dx=(14)2π〈f(x)̂,φk0(x)̂〉=2π∫-∞∞f̂(ω)φk0(ω)̅dω=(2.24)2π∫-∞∞f̂(ω)12πeiωkχ(ω+3π)dω=(2.23)∫-ππf̂(ω)eiωkdω,βkn=def∫-∞∞f(x)ψkn(x)dx=(2.27)2π〈f(x)̂,ψkn(x)̂〉=(2.24)-2π∫-∞∞f̂(ω)2-n/22πeiω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)]dω=(2.23)-2-n/2[∫2nπ2n+1πf̂(ω)eiω(k+1/2)/2ndω+∫-2n+1π-2nπf̂(ω)eiω(k+1/2)/2ndω],
so that(3.10)αk=∫-ππf̂(ω)eiωkdω‍βkn=-2-n/2[∫2nπ2n+1πf̂(ω)eiω(k+1/2)/2ndω+∫-2n+1π-2nπ‍f̂(ω)eiω(k+1/2)/2ndω].For the unbounded interval, let us prove the following.Theorem 3.2 (Shannon generalized theorem).
Iff(x)∈Bψ⊂L2(ℝ) and suppf̂⊆ℝ, the series
(3.11)f(x)=∑h=-∞∞αhφh0(x)+∑n=0∞∑k=-∞∞βknψkn(x)
converges to f(x), with αh and βkn given by (3.8) and (3.10). In particular, when suppf̂⊆[-2N+1π,2N+1π], it is
(3.12)f(x)=∑h=-∞∞αhφh0(x)+∑n=0N∑k=-∞∞βknψkn(x).Proof.
The representation (3.11) follows from the orthogonality of the scaling and Shannon wavelets (Theorems 2.1, 2.2, and 2.3). The coefficients, which exist and are finite, are given by (3.8). The convergence of the series is a consequence of the wavelet axioms.It should be noticed that(3.13)suppf̂=[-π,π]⋃n=0,…,∞[-2n+1π,-2nπ]∪[2nπ,2n+1π],
so that for a band limited frequency signal, that is, for a signal whose frequency belongs to the band [-π,π], this theorem reduces to the Shannon sampling theorem. More in general, the representation (3.11) takes into account more frequencies ranging in different bands. In this case we have some nontrivial contributions to the series coefficients from all bands, ranging from [-2Nπ,2Nπ]:(3.14)suppf̂=[-π,π]⋃n=0,…,N[-2n+1π,-2nπ]∪[2nπ,2n+1π].
In the frequency domain, (3.11) gives(3.15)f̂(ω)=∑h=-∞∞αhφ̂h0(ω)+∑n=0∞∑k=-∞∞‍βknψ̂kn(ω)f̂(ω)=(2.24)12π∑h=-∞∞αhe-iωhχ(ω+3π)‍-12π∑n=0∞∑k=-∞∞‍2-n/2βkne-iω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)].‍
That is,(3.16)f̂(ω)=12πχ(ω+3π)∑h=-∞∞αhe-iωh-12πχ(ω2n-1)∑n=0∞∑k=-∞∞2-n/2βkne-iω(k+1/2)/2n-12πχ(-ω2n-1)∑n=0∞∑k=-∞∞2-n/2βkne-iω(k+1/2)/2n.
Moreover, taking into account (2.5), (2.7), we can write (3.11) as(3.17)f(x)=∑h=-∞∞f(h)φh0(x)-∑n=0∞∑k=-∞∞2-n/2fn(2-n(k+12))ψkn(x)
with(3.18)fn(x)=def∑k=-∞∞〈f(x),ψkn(x)〉ψkn(x).


### 3.1. Error of the Shannon Wavelet Approximation

Let us fix an upper bound for the series of (3.11) in a such way that we can only have the approximation(3.19)f(x)≅∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x).
This approximation can be estimated by the followingTheorem 3.3 (Error of the Shannon wavelet approximation).
The error of the approximation (3.19) is given by
(3.20)|f(x)-∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x)|≤|f(-K-1)+f(K+1)-33π[f(2-N-1(-S-12))+f(2-N-1(S+32))]|.Proof.
The error of the approximation (3.19) is defined as
(3.21)f(x)-∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x)=∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)+∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)].
Concerning the first part of the r.h.s, it is
(3.22)∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)≤maxx∈ℝ[∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)]=∑h=-∞-K-1αhφh0(h)+∑h=K+1∞αhφh0(h)=(2.5)∑h=-∞-K-1αh+∑h=K+1∞αh=(3.3)∑h=-∞-K-1f(h)+∑h=K+1∞f(h),
and since f(x)∈L2(ℝ) is a decreasing function,
(3.23)∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)≤f(-K-1)+f(K+1).
Analogously, it is
(3.24)∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]≤maxx∈ℝ∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]=(2.18)∑n=N+1∞[∑k=-∞-S-1βknψkn(2-n-1(18k+7)3)+∑k=S+1∞βknψkn(2-n-1(18k+7)3)]=∑n=N+1∞[∑k=-∞-S-1βkn2n/233π+∑k=S+1∞βkn2n/233π]=33π∑n=N+1∞2n/2[∑k=-∞-S-1βkn+∑k=S+1∞βkn]=(3.17)-33π∑n=N+1∞2n/2[∑k=-∞-S-12-n/2f(2-n(k+12))+∑k=S+1∞2-n/2f(2-n(k+12))],
so that
(3.25)∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]≤-33π[f(2-N-1(-S-12))+f(2-N-1(S+32))]
from where (3.20) follows.


## 3.1. Error of the Shannon Wavelet Approximation

Let us fix an upper bound for the series of (3.11) in a such way that we can only have the approximation(3.19)f(x)≅∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x).
This approximation can be estimated by the followingTheorem 3.3 (Error of the Shannon wavelet approximation).
The error of the approximation (3.19) is given by
(3.20)|f(x)-∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x)|≤|f(-K-1)+f(K+1)-33π[f(2-N-1(-S-12))+f(2-N-1(S+32))]|.Proof.
The error of the approximation (3.19) is defined as
(3.21)f(x)-∑h=-KKαhφh0(x)+∑n=0N∑k=-SSβknψkn(x)=∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)+∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)].
Concerning the first part of the r.h.s, it is
(3.22)∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)≤maxx∈ℝ[∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)]=∑h=-∞-K-1αhφh0(h)+∑h=K+1∞αhφh0(h)=(2.5)∑h=-∞-K-1αh+∑h=K+1∞αh=(3.3)∑h=-∞-K-1f(h)+∑h=K+1∞f(h),
and since f(x)∈L2(ℝ) is a decreasing function,
(3.23)∑h=-∞-K-1αhφh0(x)+∑h=K+1∞αhφh0(x)≤f(-K-1)+f(K+1).
Analogously, it is
(3.24)∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]≤maxx∈ℝ∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]=(2.18)∑n=N+1∞[∑k=-∞-S-1βknψkn(2-n-1(18k+7)3)+∑k=S+1∞βknψkn(2-n-1(18k+7)3)]=∑n=N+1∞[∑k=-∞-S-1βkn2n/233π+∑k=S+1∞βkn2n/233π]=33π∑n=N+1∞2n/2[∑k=-∞-S-1βkn+∑k=S+1∞βkn]=(3.17)-33π∑n=N+1∞2n/2[∑k=-∞-S-12-n/2f(2-n(k+12))+∑k=S+1∞2-n/2f(2-n(k+12))],
so that
(3.25)∑n=N+1∞[∑k=-∞-S-1βknψkn(x)+∑k=S+1∞βknψkn(x)]≤-33π[f(2-N-1(-S-12))+f(2-N-1(S+32))]
from where (3.20) follows.


## 4. Reconstruction of the Derivatives

Letf(x)∈L2(ℝ) and let f(x) be a differentiable function f(x)∈Cp with p sufficiently high. The reconstruction of a function f(x) given by (3.11) enables us to compute also its derivatives in terms of the wavelet decomposition:(4.1)dℓdxℓf(x)=∑h=-∞∞αhdℓdxℓφh0(x)+∑n=0∞∑k=-∞∞βkndℓdxℓψkn(x),
so that, according to (3.11), the derivatives of f(x) are known when the derivatives(4.2)dℓdxℓφh0(x),dℓdxℓψkn(x)
are given.Indeed, in order to represent differential operators in wavelet bases, we have to compute the wavelet decomposition of the derivatives:(4.3)dℓdxℓφh0(x)=∑k=-∞∞λhk(ℓ)φk0(x),dℓdxℓψhm(x)=∑n=0∞∑k=-∞∞γ(ℓ)hkmnψkn(x),
being(4.4)λkh(ℓ)=def〈dℓdxℓφk0(x),φh0(x)〉,γ(ℓ)khnm=def〈dℓdxℓψkn(x),ψhm(x)〉
the connection coefficients [15–21, 26, 29] (or refinable integrals).Their computation can be easily performed in the Fourier domain, thanks to the equality (2.27). In fact, in the Fourier domain the ℓ-order derivative of the (scaling) wavelet functions is(4.5)dℓdxℓφkn(x)̂=(iω)ℓφ̂kn(ω),dℓdxℓψkn(x)̂=(iω)ℓψ̂kn(ω),
and according to (2.24),(4.6)dℓdxℓφkn(x)̂=(iω)ℓ2-n/22πe-iωk/2nχ(ω2n+3π),dℓdxℓψkn(x)̂=-(iω)ℓ2-n/22πe-iω(k+1/2)/2n[χ(ω2n-1)+χ(-ω2n-1)].Taking into account (2.27), we can easily compute the connection coefficients in the frequency domain(4.7)λkh(ℓ)=2π〈dℓdxℓφk0(x)̂,φh0(x)̂〉,γ(ℓ)khnm=2π〈dℓdxℓψkn(x)̂,ψhm(x)̂〉
with the derivatives given by (4.6).If we define(4.8)μ(m)=sign(m)={1,m>0,-1,m<0,0,m=0,
the following has been shown [16, 17].Theorem 4.1.
The any order connection coefficients (4.4)1 of the Shannon scaling functions φk0(x) are
(4.9)λkh(ℓ)={(-1)k-hiℓ2π∑s=1ℓℓ!πss![i(k-h)]ℓ-s+1[(-1)s-1],k≠h,iℓπℓ+12π(ℓ+1)[1+(-1)ℓ],k=h,or, shortly,(4.10)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].For the proof see [17].Analogously for the connection coefficients (4.4)2 we have the following.Theorem 4.2.
The any order connection coefficients (4.7)2 of the Shannon scaling wavelets ψkn(x) are
(4.11)γ(ℓ)khnm=δnm{iℓ(1-|μ(h-k)|)πℓ2nℓ-1ℓ+1(2ℓ+1-1)(1+(-1)ℓ)+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-1×{2ℓ+1[(-1)4h+s+(-1)4k+ℓ]-2s[(-1)3k+h+ℓ+(-1)3h+k+s]}πℓ2nℓ-1ℓ+1},
respectively, for ℓ≥1, and γ(0)khnm=δkhδnm.For the proof see [17].Theorem 4.3.
The connection coefficients are recursively given by the matrix at the lowest scale level:(4.12)γ(ℓ)khnn=2ℓ(n-1)γ(ℓ)kh11.Moreover it is(4.13)γ(2ℓ+1)khnn=-γ(2ℓ+1)hknn,γ(2ℓ)khnn=γ(2ℓ)hknn.If we consider a dyadic discretisation of thex-axis such that(4.14)xk=2-n(k+12),k∈ℤ
according to (2.8), the (4.3)2 at dyadic points xk=2-n(k+1/2) becomes(4.15)[ddxψkn(x)]x=xk=-2n/2∑h=-∞∞γ′khnn.For instance, inx1=2-1(1+1/2)(4.16)[ddxψ11(x)]x=x1=3/4=-21/2∑h=-∞∞γ1h11≅-21/2∑h=-22γ1h11=-21/2(16+14)=-5212.Analogously it is(4.17)φkn(2-n(k+12))=21+n/2π,k∈ℤ,
from where, in xk=(k+1/2), it is(4.18)[ddxφk0(x)]x=xk=2π∑h=-∞∞λkh.


## 5. Wavelet Solution of the Integrodifferential Equation

Let us consider the following linear integrodifferential equation:(5.1)Adudx=B∫-∞∞k(x,y)u(y)dy+u(x)+q(x)(A,B∈ℝ),
which includes as special cases the integral equation (A=0,B≠0) and the differential equation (A≠0,B=0). When A=B=0, there is the trivial solution u(x)=-q(x).It is assumed that the kernel is in the form:(5.2)k(x,y)=f(x)g(y),
and the given functions f(x)∈L2(ℝ), g(x)∈L2(ℝ), q(x)∈L2(ℝ), so that, according to (3.11)(5.3)f(x)=∑h=-∞∞fhφh0(x)+∑n=0∞∑k=-∞∞fknψkn(x),g(x)=∑h=-∞∞ghφh0(x)+∑n=0∞∑k=-∞∞gknψkn(x),q(x)=∑h=-∞∞qhφh0(x)+∑n=0∞∑k=-∞∞qknψkn(x),
with the wavelet coefficients fh,fkn,gh,gkn,qh,qkn given by (3.8).The analytical solution of (5.1) can be obtained as follows.Theorem 5.1.
The solution of (5.1), in the degenerate case (5.2), in the Fourier domain is
(5.4)û(ω)=2πB〈ĝ(ω),q̂(ω)/(Aiω-1)〉(1-2πB)〈ĝ(ω),f̂(ω)/(Aiω-1)〉f̂(ω)Aiω-1+q̂(ω)Aiω-1.Proof.
The Fourier transform of (5.1), with kernel as (5.2), is
(5.5)Adudx̂=Bf(x)̂∫-∞∞g(y)u(y)dy+u(x)̂+q(x)̂,Aiωû(ω)=2πBf̂(ω)〈ĝ(ω),û(ω)〉+û(ω)+q̂(ω),û(ω)=2πBf̂(ω)(Aiω-1)〈ĝ(ω),û(ω)〉+q̂(ω)(Aiω-1),
that is,
(5.6)û(ω)=2πBf̂(ω)(Aiω-1)〈ĝ(ω),û(ω)〉+q̂(ω)(Aiω-1).
By the inner product with ĝ(ω) there follows
(5.7)〈ĝ(ω),û(ω)〉=2πB〈ĝ(ω),f̂(ω)(Aiω-1)〉〈ĝ(ω),û(ω)〉+〈ĝ(ω),q̂(ω)(Aiω-1)〉,
so that
(5.8)〈ĝ(ω),û(ω)〉=〈ĝ(ω),q̂(ω)/(Aiω-1)〉(1-2πB)〈ĝ(ω),f̂(ω)/(Aiω-1)〉.
If we put this equation into (5.6), we get (5.4).Although the existence of solution is proven, the computation of the Fourier transform could not be easily performed. Therefore the numerical computation is searched in the wavelet approximation.The wavelet solution of (5.1) can be obtained as follows: it is assumed that the unknown function and its derivative can be written as(5.9)u(x)=∑h=-∞∞αhφh0(x)+∑n=0∞∑k=-∞∞βknψkn(x),dudx=∑h=-∞∞αhddxφh0(x)+∑n=0∞∑k=-∞∞ddxβknψkn(x)=(4.3)∑h=-∞∞αh∑s=-∞∞λhs′φs0(x)+∑n=0∞∑k=-∞∞βkn∑m=0∞∑s=-∞∞‍γ'sknmψsm(x),
and the integral can be written as(5.10)∫-∞∞g(y)u(y)dy=〈g,u〉=∑h=-∞∞αhgh+∑n=0∞∑k=-∞∞βkngkn.
There follows the system(5.11)∑h=-∞∞αh∑s=-∞∞λhs′φs0(x)+∑n=0∞∑k=-∞∞βkn∑m=0∞∑s=-∞∞γ'sknmψsm(x)=∑h=-∞∞αhφh0(x)+∑n=0∞∑k=-∞∞βknψkn(x)+[∑h=-∞∞αhgh+∑n=0∞∑k=-∞∞βkngkn][∑h=-∞∞fhφh0(x)+∑n=0∞∑k=-∞∞fknψkn(x)]+∑h=-∞∞qhφh0(x)+∑n=0∞∑k=-∞∞qknψkn(x),
and, according to the definition of the connection coefficients,(5.12)∑h=-∞∞αh∑s=-∞∞λhs′φs0(x)+∑n=0∞∑k=-∞∞∑s=-∞∞βknγ'sknnψsn(x)=∑h=-∞∞αhφh0(x)+∑n=0∞∑k=-∞∞βknψkn(x)+[∑h=-∞∞αhgh+∑n=0∞∑k=-∞∞βkngkn][∑h=-∞∞fhφh0(x)+∑n=0∞∑k=-∞∞fknψkn(x)]+∑h=-∞∞qhφh0(x)+∑n=0∞∑k=-∞∞qknψkn(x).
By the inner product and taking into account the orthogonality conditions (Theorems 2.1, 2.2, and 2.3) it is(5.13)∑h=-∞∞αhλhk′=αk+[∑h=-∞∞αhgh+∑n=0∞∑h=-∞∞βhnghn]fk+qk,
or(5.14)∑h=-∞∞(λhk′-δhk-ghfk)αh=[∑n=0∞∑h=-∞∞βhnghn]fk+qk,(k∈ℤ).Analogously, it is(5.15)∑n=0∞∑k=-∞∞βknγ'krnj=βrj+[∑h=-∞∞αhgh+∑n=0∞∑k=-∞∞βkngkn]frj+qrj
or, according to (4.11), and rearranging the indices(5.16)∑h=-∞∞βhn(γ′hknn-δhk)-fkn∑m=0∞∑h=-∞∞βhmghm=fkn∑h=-∞∞αhgh+qkn.
Thus the solution of (5.1) is (5.9)1 with the wavelet coefficients given by the algebraic system(5.17)∑h=-∞∞(λhk′-δhk-ghfk)αh=[∑n=0∞∑h=-∞∞βhnghn]fk+qk(k∈ℤ),∑h=-∞∞βhn(γ'hknn-δhk)-fkn∑m=0∞∑h=-∞∞βhmghm=fkn∑h=-∞∞αhgh+qkn(n∈ℕ,k∈ℤ)
and up to a fixed scale of approximation N,S:(5.18)∑h=-SS(λhk′-δhk-ghfk)αh=[∑n=0N∑h=-SSβhnghn]fk+qk(k∈ℤ),∑h=-SSβhn(γ′hknn-δhk)-fkn∑m=0N∑h=-SSβhmghm=fkn∑h=-NNαhgh+qkn(n∈ℕ,k∈ℤ).


## 6. Example

Let us consider the following equation:(6.1)dudx=∫-∞∞e-x2-|y|u(y)dy-x|x|u(x)-e-x2
with the condition(6.2)u(0)=1.
The analytical solution, as can be directly checked, is(6.3)u(x)=e-|x|.
Since(6.4)f(x)=e-x2,g(x)=e-|x|,q(x)=-e-x2
belong to L2(ℝ), let us find the wavelet approximation by assuming that also u(x) belongs to L2(ℝ), so that they can be represented according to (5.3), (5.9).At the level of approximationN=0,S=0, from (5.3) we have(6.5)f(x)=e-x2≅0.97φ00(x),g(x)=e-|x|≅0.80φ00(x)+0.04ψ00(x),q(x)=-e-x2≅-0.97φ00(x),
so that(6.6)f0=0.97,f00=0,g0=0.80,g00=0.04,q0=-0.97,q00=0.
System (5.18) becomes(6.7)(λ00′-δ00-g0f0)α0=β00g00f0+q0,β00(γ′0000-δ00)-f00β00g00=f00α0g0+q00,
and, since λ00′=0 and γ'0000=0, according to (6.6) we have(6.8)-1-0.80×0.97α0=-0.97,-β00=0,
whose solution is(6.9)α0=0.548,β00=0,
so that(6.10)u(x)≅0.548φ00(x).
As expected, the approximation is very row (Figure 1(a)); in fact in order to get a satisfactory approximation we have to solve system (5.18) at least at the levels N=0,S=5 as shown in Figure 1(b).Wavelet approximations (shaded) of the analytical solution (plain) of (6.1) obtained by solving (5.17).
(a)(b)


## 7. Conclusion

In this paper the theory of Shannon wavelets combined with the connection coefficients methods and the Petrov-Galerkin method has been used to find the wavelet approximation of integrodifferential equations. Among the main advantages there is the decorrelation of frequencies, in the sense that the differential operator is splitted into its different frequency bands.



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*Source: 408418-2010-04-11.xml*