On Wavelets and Prewavelets with Vanishing Moments in Higher Dimensions
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 443 KB
- Volume
- 90
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Using an approximation theory approach, we prove that a scaling function with suitable polynomial decay satisfies the Strang-Fix condition of order r # N if and only if the elements of any prewavelet set [ & ] & # E* with polynomial decay of the same order have vanishing integral moments up to order r&1. An analogous equivalence is established that does not involve any assumptions concerning decay; this yields a new characterization of the rate of L 2 -approximation of (stationary and nonstationary) multiresolution analyses in terms of a corresponding prewavelet set. Furthermore, we show that the existence of a scaling function with polynomial decay implies the existence of both an orthonormal scaling function and a wavelet set with polynomial decay of the same order. Several known constructions of wavelets and prewavelets are discussed in this respect.
1997 Academic Press
To ensure that the Fourier transform is (r&1) times continuously differentiable, we will need a suitable decay condition upon . A sufficient condition would be to require (1+& } &) r&1 # L 1 (R d ). In our setting it is article no. AT963063 46