Periodic Orthogonal Splines and Wavelets

Abstract

Periodic scaling functions and wavelets are constructed directly from non-stationary multiresolutions of L 2([0, 2π)), the space of square-integrable 2π-periodic functions. For a multiresolution { V k : k ≥ 0}, necessary and sufficient conditions for ∪ k≥0 V k to be dense in L 2([0, 2π)) and characterizations of a function φ k for which φ k (· - 2π j/2 k ), j = 0, 1, . , 2 k - 1, form a basis of V k are given. The construction of scaling functions and wavelets are done via orthogonal bases of functions, called orthogonal splines . Sufficient conditions are given for a sequence of scaling functions to generate a multiresolution. These conditions are also sufficient for the convergence of convolution operators with the scaling functions as kernels. Sufficient conditions are also given for the wavelets to generate a stable basis of L 2([0, 2π)). The orthogonal spline bases give rise to algorithms in which the equations are the finite Fourier transforms of the classical wavelet decomposition and reconstruction equations. Each equation in the orthogonal spline algorithms involves only two terms and its complexity does not depend on the length of the filter coefficients. The general construction given here includes periodic versions of known wavelets. Examples on periodic polynomial spline wavelets and an extension of Chui and Mhaskar′s trigonometric wavelets are given to illustrate the construction. These trigonometric wavelets, in particular Chui and Mhaskar′s wavelets, form a stable basis of L 2([0, 2π)).