Quasi-interpolation using generalized Gaussian kernels

TL;DR

This paper develops a high-accuracy quasi-interpolation framework using generalized Gaussian kernels, achieving optimal approximation order through kernel restriction and periodization, with extensions to nonperiodic and high-dimensional cases.

Contribution

It introduces a novel quasi-interpolation method with maximal approximation order based on generalized Gaussian kernels and extends it to nonperiodic and high-dimensional settings.

Findings

Achieves the highest approximation order s for periodic functions.

Extends the quasi-interpolant to nonperiodic functions with the same order.

Proposes a sparse grid approach to mitigate the curse of dimensionality.

Abstract

This paper focuses on developing a framework for constructing quasi-interpolation with the highest achievable approximation order from generalized Gaussian kernels with the help of kernel restriction trick and periodization technique. We first demonstrate that when we restrict generalized Gaussian kernels satisfying generalized Strang-Fix conditions of order s over a torus, the corresponding restricted kernels in tensor-product forms fulfill periodic Strang-Fix conditions of the same order s. Then, based on these restricted kernels, we construct a periodic quasi-interpolant in Schoenberg's form and derive its error estimates for periodic function approximation over a torus, which reveals that our quasiinterpolant attains the highest approximation order s. Finally, using the periodization technique, we extend the periodic quasi-interpolant to its nonperiodic counterpart with the highest approximation order s for approximating a general function defined over a cube via a torus-to-cube transformation. This result stands in stark contrast to classical quasi-interpolation counterparts, which often yield much lower approximation orders than those dictated by the generalized Strang-Fix conditions of generalized Gaussian kernels. Furthermore, we propose a sparse grid counterpart for high-dimensional function approximation to alleviate the curse of dimensionality. Numerical simulations confirm that our quasi-interpolation scheme is simple and computationally efficient.