Lower bounds for the approximation with variation-diminishing splines

TL;DR

This paper establishes lower bounds for the approximation error of variation-diminishing splines, characterizes the spectrum of the Schoenberg operator, and confirms an open conjecture relating approximation error to smoothness measures.

Contribution

It provides new lower bounds for spline approximation errors and proves the equivalence between approximation error and the second order modulus of smoothness, advancing understanding of spline approximation.

Findings

Lower bounds for Schoenberg operator approximation error

Spectrum characterization of the Schoenberg operator

Equivalence between approximation error and second order modulus of smoothness

Abstract

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval $[0,1]$ in terms of classical moduli of smoothness depending on the degree of the spline basis using a functional analysis based framework. Thereby, we characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates. Finally, we prove the equivalence between the approximation error and the classical second order modulus of smoothness as an improved version of an open conjecture from 2002.