On Compactly Supported Spline Wavelets and a Duality Principle

Let ⋯ ⊂ V- 1 ⊂ V₀ ⊂ V₁ ⊂ ⋯ be a multiresolution analysis of L² generated by the mth order B-spline Nₘ(x). In this paper, we exhibit a compactly supported basic wavelet ψ ₘ(x) that generates the corresponding orthogonal complementary wavelet subspaces ⋯ ,W- 1,W₀,W₁, …. Consequently, the two finite sequences that describe the two-scale relations of Nₘ(x) and ψ ₘ(x) in terms of Nₘ(2x - j),j ∈ 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 \ Nₘ(x - j)\ and \ ψ ₘ(x - j)\, relative to \ Nₘ(x - j)\ and \ ψ ₘ(x - j)\, respectively.