On the dimension function of orthonormal wavelets

Abstract

We announce the following result: Every orthonormal wavelet of L2 (R) is associated with a multiresolution analysis such that for the subspace Vo the integral translates of a countable at most family of functions is a tight frame. Orthonormal wavelets are square-integrable functions ' such that the set {2j/2'0(2jx k): j, k E Z} is an orthonormal basis of L2(R). Theorem 1. If a function b in L2(R) is an orthonormal wavelet, then we have (1) Z | b(?+2kr)I2=1 for a.e. (E R kEZ and (2) E (2 (+ 2kir))( + 2kir) = 0 for a.e. ( E R, j > 1. kEZ Let Wj be the closed linear span of the set {2j/2 (2jx k): k E Z}. An orthonormal wavelet b is associated with a multiresolution analysis (MRA) {Vj }j if the set {f4(x-k): k E Z} is an orthonormal basis of Vi nvl 1. For a complete, rigorous and comprehensive introduction to wavelet analysis, the appropriate definitions and a complete proof of the previous theorem the reader may refer to [5]. Journe gave the first example of an orthonormal wavelet not associated with an MRA. On the other hand Auscher proved (see [1]) that orthonormal wavelets (or simply wavelets) whose Fourier transforms satisfy a mild smoothness condition and a weak decay condition at oo are associated with an MRA. Let