Wavelets and pre-wavelets in low dimensions

Abstract

In Riemenschneider and Shen ( in “Approximation Theory and Functional Analysis” (C. K. Chui, Ed.), pp. 133–149, Academic Press, New York, 1991 ) an explicit orthonormal basis of wavelets for L 2( R s ), s=1,2,3, was constructed from a multiresolution approximation given by box splines. In other words, L 2( R s ) has the orthogonal decomposition ⊕ W ν . (∗) ν ϵ Z Orthonormal bases for the spaces W ν , are given by {2 νs 2 K μ(2 ν · −j)} , j∈ Z s , μ ϵ Z 2 s ⧹ 0, where Z s 2 and the “wavelets” K μ are 2 s − 1 cardinal splines with exponential decay. In this paper, we consider multiresolutions generated by suitable compactly supported and symmetric functions ϑ and explicitly construct 2 s − 1 compactly supported functions ϑ μ , μ ϵ Z 2 s ⧹ 0, such that the translates ϑ μ (· − j), j∈ Z s , are an unconditional basis for W 0. Thus, the functions ϑ μ (2 ν · − j), ν ϵ Z, j∈ Z s , μ ϵ Z 2 s ⧹ 0 comprise a basis for the orthogonal decomposition ( ∗) (the functions are orthogonal for different ν because the decomposition is orthogonal, but neither the translates nor the functions will be orthogonal for given ν). The functions are given as ϑ μ(·/2) 2 s = μ ∗′ β μ with the sequences β μ formed from a single sequence by translation and change in sign pattern. We also discuss various ways to regain some of the orthogonality lost by requiring compact support.