A shape preserving quasi-interpolation operator based on a new transcendental RBF

TL;DR

This paper introduces a new transcendental radial basis function based on hyperbolic tangent for quasi-interpolation, offering higher accuracy, better convergence, and shape preservation compared to traditional multiquadric methods.

Contribution

A novel transcendental RBF is proposed for quasi-interpolation, improving accuracy, convergence, and shape preservation over existing multiquadric-based methods.

Findings

Converges with a rate of O(h^2)

Preserves convexity and monotonicity

Demonstrates higher accuracy and efficiency in numerical experiments

Abstract

It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as a smooth approximant to f(r)=r with higher accuracy and better convergence properties than the multiquadric. Then Wu-Schaback's quasi-interpolation formula is rewritten using the proposed RBF. It preserves convexity and monotonicity. We prove that the proposed scheme converges with a rate of O(h^2). So it has a higher degree of smoothness. Some numerical experiments are given in order to demonstrate the efficiency and accuracy of the method.