Optimal Approximation by $sk$-Splines on the Torus

TL;DR

This paper extends the concept of $sk$-splines to the $d$-dimensional torus, providing convergence rates for approximating functions using these splines, which match the optimal rates of trigonometric approximation in certain cases.

Contribution

It introduces a generalized $sk$-spline framework on the torus and derives convergence estimates for functions of the form $f=K*\varphi$, including Sobolev class functions.

Findings

Established convergence rates in $L^q$ norm for $sk$-spline interpolation.

Proved the approximation error is of the same order as best trigonometric approximation.

Provided optimal error estimates for functions in Sobolev spaces.

Abstract

Fixed a continuous kernel K on the $d$-dimensional torus, we consider a generalization of the univariate $sk$-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given function by its interpolating $sk$-splines, in the norm of $L^q$ for functions of the type $f=K*\varphi$ where $\varphi \in L^p$ and $1\leq p \leq 2 \leq q \leq \infty,\ 1/p - 1/q \geq 1/2$. The rate of convergence is obtained for functions f in Sobolev classes and this rate gives optimal error estimate of the same order as best trigonometric approximation, in a special case.