On compactly supported spline wavelets and a duality principle

Abstract

Let ⋯ ⊂ V − 1 ⊂ V 0 ⊂ V 1 ⊂ ⋯ \cdots \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset \cdots be a multiresolution analysis of L 2 {L^2} generated by the m m th order B B -spline N m ( x ) {N_m}(x) . In this paper, we exhibit a compactly supported basic wavelet ψ m ( x ) {\psi _m}(x) that generates the corresponding orthogonal complementary wavelet subspaces ⋯ , W − 1 , W 0 , W 1 , … \cdots ,{W_{ - 1}},{W_0},{W_1}, \ldots . Consequently, the two finite sequences that describe the two-scale relations of N m ( x ) {N_m}(x) and ψ m ( x ) {\psi _m}(x) in terms of N m ( 2 x − j ) , j ∈ Z {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 { N ~ m ( x − j ) } \{ {\tilde N_m}(x - j)\} and { ψ ~ m ( x − j ) } \{ {\tilde \psi _m}(x - j)\} , relative to { N m ( x − j ) } \{ {N_m}(x - j)\} and { ψ m ( x − j ) } \{ {\psi _m}(x - j)\} , respectively.