Connection coefficients of Shannon wavelets. (English) Zbl 1117.65179

The paper presents an explicit computation of the matrix coefficients representing differential operators of arbitrary order in a Shannon wavelet basis. Shannon wavelets for which explicit expressions are available in physical and in Fourier space form an orthogonal basis of \(L^2(\mathbb R)\). They are compactly supported in Fourier space, exhibit a \(1/x\) decay in physical space and they can be derived from harmonic wavelets. The entries of the stiffness matrix required e.g. for Galerkin discretizations of differential equations fulfill recursion formulas.
In the present paper they are called connection coefficients, in the literature, however, this expression is used for the coefficients representing bilinear terms in a wavelet basis, i.e. the projection of the square of a wavelet onto a wavelet.


65T60 Numerical methods for wavelets
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
94A11 Application of orthogonal and other special functions
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems