On the $L_p$-error of approximation of bivariate functions by harmonic splines

TL;DR

This paper investigates the use of harmonic splines for bivariate function approximation, analyzing their $L_p$-error asymptotics and comparing advantages and drawbacks to polynomial splines.

Contribution

It introduces harmonic spline interpolation on grids, providing asymptotic error analysis and discussing its benefits and limitations compared to polynomial splines.

Findings

Asymptotic $L_p$-error formulas for harmonic spline approximation

Advantages of harmonic splines in certain approximation scenarios

Discussion of limitations and potential drawbacks

Abstract

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline "interpolation" (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the $L_p$-error for adaptive approximation by harmonic splines.