On Compactly Supported Spline Wavelets and a Duality Principle

Abstract

L2 generated by the mth order 5-spline Nm{x). In this paper, we exhibit a compactly supported basic wavelet i//m(x) that generates the corresponding orthogonal complementary wavelet subspaces ... , W_ \, Wo, Wx, ... . Consequently, the two finite sequences that describe the two-scale relations of Nm(x) and i//m(x) in terms of Nm(2x -j), ;6Z, 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 {Nm(x -j)} and {y/m(x -j)} , relative to {Nm(x -j)} and {y/m(x -j)}, respectively. Jo with Ni = X[o,\) Fr anY k, j Z, we set Nm,kJ(x) = Nm(2kx-j); and for each k , let Vk denote the L2 -closure of the algebraic linear span (Nm.kJ: jl).