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: 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 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 by its finite section , a square matrix of finite order. We use to compute an approximation of **x** whose discrete Fourier transform is . We show that 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.

**[1]**Guy Battle,*A block spin construction of ondelettes. I. Lemarié functions*, Comm. Math. Phys.**110**(1987), no. 4, 601–615. MR**895218****[2]**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****[3]**Pierre Gilles Lemarié,*Ondelettes à localisation exponentielle*, J. Math. Pures Appl. (9)**67**(1988), no. 3, 227–236 (French, with English summary). MR**964171****[4]**Stephane G. Mallat,*Multiresolution approximations and wavelet orthonormal bases of 𝐿²(𝑅)*, Trans. Amer. Math. Soc.**315**(1989), no. 1, 69–87. MR**1008470**, https://doi.org/10.1090/S0002-9947-1989-1008470-5**[5]**-,*A theory for multiresolution signal decomposition*:*the wavelet representation*, IEEE Trans. Pattern Anal. and Machine Intelligence**11**(1989), 674-693.

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Additional Information

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

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

Article copyright:
© Copyright 1994
American Mathematical Society