Singular wavelets on a finite interval

Abstract

Nonparametric methods are used in complex cases where model information is insufficient. A new method of nonparametric approximation, the singular wavelet method, is developed. The method includes a numerical algorithm based on the summation of a recurrent sequence of functions. The introduction explains the idea of the singular wavelet method to combine the theory of wavelets with kernel regression estimation of the Nadaraya - Watson type. This integration is realized by regularizing the wavelet transform. Usually kernel estimation is are considered as an example of nonparametric estimation. However, one parameter - the blur parameter - is still present in the traditional kernel regression algorithm. In the approximation by the method of singular value wavelet, the summation of kernel estimation of the type Nadaraya - Watson using the blur parameter takes place. In the main part of the work, the variant of wavelet transform regularization for the finite interval is considered. Theorems that formulate the properties of a wavelet transform with a singular wavelet are proved, an algorithm for approximating a function defined on a finite interval by a sequence of wavelet transforms is proposed.