Hyperbolic trigonometric functions as approximation kernels and their properties II: Wavelets

TL;DR

This paper introduces a novel approach to function approximation using hyperbolic radial basis functions to construct prewavelets, enabling localized time-frequency analysis and filtering.

Contribution

It extends previous work by developing new prewavelet constructions from hyperbolic radial basis functions applicable to various other basis functions.

Findings

Effective localized time-frequency decompositions

Applicable to multiquadrics and polynomial splines

Enhances function approximation and filtering techniques

Abstract

In a previous paper we have introduced a new class of radial basis functions that are powerful means to approximate functions by quasi-interpolation. In this article we extend the results to create new ways of approximating functions by prewavelets that are constructed from spaced spanned of the new hyperbolic radial basis functions. They consist of highly localised time-frequency decompositions that are suitable for analysis and filtering. The construction is sufficiently general to apply for large classes of other radial basis functions too - such as multiquadrics and their generalisations and thin-plate splines -, as well as, for example, polynomial splines.