On compactly supported spline wavelets and a duality principle

Abstract:

Let $\cdots \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset \cdots$ be a multiresolution analysis of ${L^2}$ generated by the $m$th order $B$-spline ${N_m}(x)$. In this paper, we exhibit a compactly supported basic wavelet ${\psi _m}(x)$ that generates the corresponding orthogonal complementary wavelet subspaces $\cdots ,{W_{ - 1}},{W_0},{W_1}, \ldots$. Consequently, the two finite sequences that describe the two-scale relations of ${N_m}(x)$ and ${\psi _m}(x)$ in terms of ${N_m}(2x - j),j \in \mathbb {Z}$, 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 $\{ {\tilde N_m}(x - j)\}$ and $\{ {\tilde \psi _m}(x - j)\}$, relative to $\{ {N_m}(x - j)\}$ and $\{ {\psi _m}(x - j)\}$, respectively.