Spline Wavelet Bases of Weighted L p Spaces, 1 ≤p < ∞

Abstract

We study necessary conditions on the weight w for the spline wavelet systems to be bases in the weighted space LP (w) . In this article we study some questions arising in the investigation of the prob- lem of describing those classes of weight functions w for which the wavelet systems are bases or unconditional bases in the weighted spaces LP (w) = LP(R, w). A priori these conditions depend on the concrete system. But there are some features which are common to all wavelet systems. Here we examine only the case of spline wavelet systems defined on the real line R. For a given non-negative integer m, the spline wavelets are defined in the following way: Let V0 = {f E L2(R) n Cm-l(R) such that the restriction of f to each interval )n, n + 1 ( is a polynomial of degree < m}, where by C (R) we denote the class of functions on R whose derivatives of order r are continuous and by C-' (R) we denote the class of piecewise continuous functions on R. Defining Vj+j = {f(2x) : f(x) E Vj} , we get a multiscale analysis of L2(R) in the sense of Mallat and Meyer (see (M), (D)).