A lower bound for the uniform Schoenberg operator

TL;DR

This paper establishes a lower bound for the Schoenberg operator with equidistant knots, analyzes its iterates, and demonstrates the equivalence between approximation error and the second order modulus of smoothness.

Contribution

It provides a new lower bound estimate for the Schoenberg operator and explores the behavior of its iterates, linking approximation error to smoothness measures.

Findings

Established a lower bound for the Schoenberg operator.

Analyzed the behavior of iterates of the Schoenberg operator.

Proved the equivalence between approximation error and second order modulus of smoothness.

Abstract

We present an estimate for the lower bound for the Schoenberg operator with equidistant knots in terms of the second order modulus of smoothness. We investigate the behaviour of iterates of the Schoenberg operator and in addition, we show an upper bound of the second order derivative of these iterates. Finally, we prove the equivalence between the approximation error and the second order modulus of smoothness.