@MISC{Ehrich_explicitinequalities, author = {Sven Ehrich}, title = {Explicit Inequalities for Wavelet Coefficients}, year = {} }

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Abstract

A fundamental principle for many applications of wavelets is that the size of the wavelet coefficients indicates the local smoothness of the represented function f . We show how explicit and best possible a priori bounds for wavelet coefficients can be obtained for any wavelet from the coefficients of its two scale relation. 1991 Mathematics Classification: 42C15,41A15 Key Words and Phrases: wavelet coefficients, bounds, two scale relation 1 Introduction One of the most important applications of wavelets is for compression, i.e., neglecting small coefficients in a wavelet decomposition of the type f X j2Z (' 0j ; f) ~ ' 0j + X i0 X j2Z (/ ij ; f) ~ / ij (1) of a function f 2 L 2 (R), where (f; g) = R R fg denotes the standard inner product and f 2 L 2 (I) () kfk L 2 (I) := aeZ I jf(x)j 2 dx oe 1 2 ! 1: For I = R, we write kfk 2 = kfk L 2 (R) . The representation (1) is an example of a biorthogonal wavelet decomposition, i.e., (/ ij ; ~ / i 0 j 0 ) =...