On Compactly Supported Spline Wavelets and a Duality Principle

Abstract

Let <!-- MATH $\cdots \subset{V_{ - 1}} \subset{V_0} \subset{V_1} \subset \cdots$ --> be a multiresolution analysis of generated by the th order -spline . In this paper, we exhibit a compactly supported basic wavelet <!-- MATH ${\psi _m}(x)$ --> that generates the corresponding orthogonal complementary wavelet subspaces <!-- MATH $\cdots,{W_{ - 1}},{W_0},{W_1}, \ldots$ --> . Consequently, the two finite sequences that describe the two-scale relations of and <!-- MATH ${\psi _m}(x)$ --> in terms of <!-- MATH ${N_m}(2x - j),j \in \mathbb{Z}$ --> , yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases <!-- MATH $\{ {\tilde N_m}(x - j)\}$ --> and <!-- MATH $\{ {\tilde \psi _m}(x - j)\}$ --> , relative to <!-- MATH $\{ {N_m}(x - j)\}$ --> and <!-- MATH $\{ {\psi _m}(x - j)\}$ --> , respectively.