A Cardinal Spline Approach to Wavelets

While it is well known that the th order -spline with integer knots generates a multiresolution analysis, <!-- MATH $\cdots \subset {V_{ - 1}} \subset {V_0} \subset \cdots$ --> , with the th order of approximation, we prove that <!-- MATH $\psi (x): = L_{2m}^{(m)}(2x - 1)$ --> , where <!-- MATH ${L_{2m}}(x)$ --> denotes the th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces . Note that for , when the -spline is the characteristic function of the unit interval , our basic wavelet <!-- MATH ${L'_2}(2x - 1)$ --> is simply the well-known Haar wavelet. In proving that <!-- MATH ${V_{k + 1}} = {V_k} \oplus {W_k}$ --> , we give the exact formulation of <!-- MATH ${N_m}(2x - j), j \in \mathbb{Z}$ --> , in terms of integer translates of and . This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.