On the computation of Battle-Lemarié’s wavelets

Abstract:

We propose a matrix approach to the computation of Battle-Lemarié’s wavelets. The Fourier transform of the scaling function is the product of the inverse $F({\mathbf {x}})$ of a square root of a positive trigonometric polynomial and the Fourier transform of a B-spline of order m. The polynomial is the symbol of a bi-infinite matrix B associated with a B-spline of order 2m. We approximate this bi-infinite matrix ${{\mathbf {B}}_{2m}}$ by its finite section ${A_N}$, a square matrix of finite order. We use ${A_N}$ to compute an approximation ${{\mathbf {x}}_N}$ of x whose discrete Fourier transform is $F({\mathbf {x}})$. We show that ${{\mathbf {x}}_N}$ converges pointwise to x exponentially fast. This gives a feasible method to compute the scaling function for any given tolerance. Similarly, this method can be used to compute the wavelets.