A general framework of compactly supported splines and wavelets

AbstractLet {Vk} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L2(R). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ϑ ϵ Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces Wk of Vk, ∈Z. In particular, the minimally supported ψ ϵ Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (ϑ, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L2 functions from the “spline” spaces Vk and “wavelet” spaces Wk, k∈Z. A very general duality principle, which yields the dual bases of both {ϑ(·−j):j∈Z and {η(·−j):j∈Z} for any η ϵ Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both ϑ and ψ have linear phases. Hence, “non-symmetric” ϑ and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φm and ψm have linear phases, but the odd-order B-wavelet only has generalized linear phases