Hyperbolic trigonometric functions as approximation kernels and their properties I: generalised Fourier transforms

TL;DR

This paper introduces a new class of radial basis functions based on hyperbolic trigonometric functions, analyzes their generalized Fourier transforms, and demonstrates their effectiveness in constructing polynomially exact quasi-interpolants with numerical comparisons.

Contribution

It presents novel hyperbolic trigonometric-based radial basis functions and studies their Fourier transforms, providing explicit formulas and applications to quasi-interpolants in multiple dimensions.

Findings

New radial basis functions with hyperbolic trigonometric kernels

Explicit Fourier transform expansions and asymptotics

Numerical comparisons showing effectiveness of the new functions

Abstract

In this paper a new class of radial basis functions based on hyperbolic trigonometric functions will be introduced and studied. We focus on the properties of their generalised Fourier transforms with asymptotics. Therefore we will compute the expansions of these Fourier transforms with an application of the conditions of Strang and Fix in order to prove polynomial exactness of quasi-interpolants. These quasi-interpolants will be formed with special linear combinations of shifts of the new radial functions and we will provide explicit expressions for their coefficients. In establishing these new radial basis functions we will also use other, new classes of shifted thin-plate splines and multiquadrics of [11], [12]. There are numerical examples and comparisons of different constructions of quasi-interpolants, in several dimensions, varying the underlying radial basis functions.