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

TL;DR

This work develops a flexible class of hyperbolic-trigonometric radial basis kernels, notably $\phi(r)=r^{\beta}\tanh^{\alpha} r$ (and variants with $\log r$ factors), and analyzes their generalised Fourier transforms to derive near-origin and asymptotic behavior. By applying Strang–Fix conditions, the authors construct quasi-interpolants from shifted kernels, including non-classical designs that cancel kernel singularities with trigonometric factors to achieve polynomial reproduction up to a finite degree. They extend the kernel family to $g(x)=r^{\beta}(\tanh r)^{\alpha}$ (and variants with logarithms), enabling adjustable near-zero smoothness and global growth, and provide explicit coefficients for the associated quasi-Lagrange functions. Numerical experiments in 1D and 2D illustrate the method’s ability to reproduce polynomials with controlled convergence and demonstrate competitive accuracy across several kernel choices. Overall, the paper offers a tunable, spline-inspired RBF framework with rigorous Fourier-analytic foundations and practical quasi-interpolation schemes for high-dimensional approximation tasks.

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.

Figures (10)

• Figure 1: Quasi-Lagrange functions corresponding to $g_1$ (left) and $g_2$ (right).
• Figure 2: Quasi-interpolants corresponding to $g_1$ (left) and $g_2$ (right).
• Figure 3: Graphics of the absolute error between $f(x)$ and the quasi-interpolant based on $g_1(x)$ (left) and the one based on $g_2(x)$ (right).
• Figure 4: Quasi-Lagrange functions corresponding to $g_1$ (left) and $g_2$ (right).
• Figure 5: Quasi-interpolants corresponding to $g_1$ (left) and $g_2$ (right).

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Theorem 5
• Theorem
• Theorem
• Theorem