A cardinal spline approach to wavelets

While it is well known that the mth order B-spline Nₘ(x) with integer knots generates a multiresolution analysis, ⋯ ⊂ V- 1 ⊂ V₀ ⊂ ⋯, with the mth order of approximation, we prove that ψ (x): = L₂ₘ⁽ᵐ⁾(2x - 1), where L₂ₘ(x) denotes the $(2m)$th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces Wₖ. Note that for $m = 1$, when the B-spline N₁(x) is the characteristic function of the unit interval $[0,1)$, our basic wavelet L’₂(2x - 1) is simply the well-known Haar wavelet. In proving that Vk + 1 = Vₖ ⊕ Wₖ, we give the exact formulation of Nₘ(2x - j), j ∈ Z, in terms of integer translates of Nₘ(x) and ψ (x). This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.