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 of

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 [

Wavelets [

Among the many families of wavelets, Shannon wavelets [

are analytically defined;

are infinitely differentiable;

are sharply bounded in the frequency domain, thus allowing a decomposition of frequencies in narrow bands;

enjoy a generalization of the Shannon sampling theorem, which extend to all range of frequencies [

give rise to the connection coefficients which can be analytically defined [

The (Shannon wavelet) connection coefficients are obtained in [

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., [

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. Section

Shannon wavelets theory (see, e.g., [

From these functions a multiscale analysis [

By a direct computation it can be easily seen that

Thus, according to (

Since they belong to

The maximum and minimum values of these functions can be easily computed. The maximum value of the scaling function

The minimum of the wavelet

Let

The Fourier transform of (

For each

With respect to the inner product (

Shannon wavelets are orthonormal functions, in the sense that

For the proof see [

The translated instances of the Shannon scaling functions

See the proof in [

The scalar product of the (Shannon) scaling functions with respect to the corresponding wavelets is characterized by the following [

The translated instances of the Shannon scaling functions

Proof is in [

Let

According to the sampling theorem (see, e.g., [

If

In order to compute the values of the coefficients we have to evaluate the series in correspondence of the integer:

The convergence follows from the hypotheses on

As a generalization of the Paley-Wiener space, and in order to generalize the Shannon theorem to unbounded intervals, we define the space

For the unbounded interval, let us prove the following.

If

The representation (

It should be noticed that

Let us fix an upper bound for the series of (

The error of the approximation (

The error of the approximation (

Let

Indeed, in order to represent differential operators in wavelet bases, we have to compute the wavelet decomposition of the derivatives:

Their computation can be easily performed in the Fourier domain, thanks to the equality (

Taking into account (

If we define

The any order connection coefficients (_{1} of the Shannon scaling functions

or, shortly,

For the proof see [

Analogously for the connection coefficients (_{2} we have the following.

The any order connection coefficients (_{2} of the Shannon scaling wavelets

For the proof see [

The connection coefficients are recursively given by the matrix at the lowest scale level:

Moreover it is

If we consider a dyadic discretisation of the

_{2} at dyadic points

For instance, in

Analogously it is

Let us consider the following linear integrodifferential equation:

It is assumed that the kernel is in the form:

The analytical solution of (

The solution of (

The Fourier transform of (

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 (

Analogously, it is

_{1} with the wavelet coefficients given by the algebraic system

Let us consider the following equation:

At the level of approximation

Wavelet approximations (shaded) of the analytical solution (plain) of (

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.