On the computation of Battle-Lemarié’s wavelets

Author:
Ming Jun Lai

Journal:
Math. Comp. **63** (1994), 689-699

MSC:
Primary 65T99; Secondary 41A15, 42C15, 65D07

DOI:
https://doi.org/10.1090/S0025-5718-1994-1248971-3

MathSciNet review:
1248971

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Abstract | References | Similar Articles | Additional Information

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 2*m*. 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.

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Additional Information

Keywords:
B-spline,
bi-infinite matrices,
exponential decay,
finite section,
positive operator,
Toeplitz matrix,
wavelet

Article copyright:
© Copyright 1994
American Mathematical Society