On the Computation of Battle-lemarie's Wavelets

We propose a matrix approach to the computation of BattleLemarié's wavelets. The Fourier transform of the scaling function is the product of the inverse F(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 B2m by its finite section As , a square matrix of finite order. We use As to compute an approximation \s of x whose discrete Fourier transform is F(x). We show that xs 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.