A cardinal spline approach to wavelets

Abstract:

While it is well known that the $m$th order $B$-spline ${N_m}(x)$ with integer knots generates a multiresolution analysis, $\cdots \subset {V_{ - 1}} \subset {V_0} \subset \cdots$, with the $m$th order of approximation, we prove that $\psi (x): = L_{2m}^{(m)}(2x - 1)$, where ${L_{2m}}(x)$ denotes the $(2m)$th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces ${W_k}$. Note that for $m = 1$, when the $B$-spline ${N_1}(x)$ 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 ${V_{k + 1}} = {V_k} \oplus {W_k}$, we give the exact formulation of ${N_m}(2x - j), j \in \mathbb {Z}$, in terms of integer translates of ${N_m}(x)$ and $\psi (x)$. This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.