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

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.