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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the computation of Battle-Lemarié’s wavelets
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by Ming Jun Lai PDF
Math. Comp. 63 (1994), 689-699 Request permission

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.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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