A substitute for summability in wavelet expansions

Abstract

Gibbs’phenomenon almost always appears in the expansions using classical orthogonal systems. Various summability methods are used to get rid of unwanted properties of these expansions. Similar problems arise in wavelet expansions, but cannot be solved by the same methods. In a previous work [10], two alternative procedures for wavelet expansions were introduced for dealing with this problem. In this article, we are concerned with the details of the implementation of one of the procedures, which works for the wavelets with compact support in the time domain. Estimates based on this method remove the excess oscillations. We show that the dilation equations which arise, though they contain an infinite number of terms, have coefficients which decrease exponnetially. In addition, an iteration relation for the positive estimation function is derived to reduce the amount of calculation in the approximation. Numerical experiments are given to illustrate the theoretical results.KeywordsCompact SupportScaling FunctionFilter CoefficientSummability FunctionSummability MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.