A Cardinal Spline Approach to Wavelets

While it is well known that the mth order 5-spline Nm(x) with integer knots generates a multiresolution analysis, • • • C V_x c V0 C • ■ • , with the with order of approximation, we prove that i//(x) := Ú1mJ¡{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 ß-spline Nx(x) is the characteristic function of the unit interval [0, 1), our basic wavelet L2(2x 1) is simply the well-known Haar wavelet. In proving that Vk+l = Vk ffi Wk , we give the exact formulation of Nm(2x j), j e Z , in terms of integer translates of Nm(x) and y/{x). This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.