Stability and linear independence associated with wavelet decompositions

Abstract

Wavelet decompositions are based on basis functions satisfying refinement equations.The stability, linear independence, and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets.In this paper we characterize these properties in terms of the mask sequence in the refinement equation that the basis function satisfies.Here and throughout this paper i denotes the imaginary unit yf^A.. Restricted to R, <f> becomes the Fourier transform of </>.As usual, for 1 < p < 00, we denote by LP(R) the Banach space of all complex-valued functions / for which \\f\\P--=(JR\f(x)\pdx) "<oo.