On the computation of Battle-Lemarié’s wavelets
HTML articles powered by AMS MathViewer
- 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.References
- Guy Battle, A block spin construction of ondelettes. I. Lemarié functions, Comm. Math. Phys. 110 (1987), no. 4, 601–615. MR 895218
- 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, DOI 10.1137/1.9781611970104
- Pierre Gilles Lemarié, Ondelettes à localisation exponentielle, J. Math. Pures Appl. (9) 67 (1988), no. 3, 227–236 (French, with English summary). MR 964171
- Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(\textbf {R})$, Trans. Amer. Math. Soc. 315 (1989), no. 1, 69–87. MR 1008470, DOI 10.1090/S0002-9947-1989-1008470-5 —, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. and Machine Intelligence 11 (1989), 674-693.
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