# Lower estimates for linear operators with smooth range

## TL;DR

This paper presents a new method for deriving lower bounds on approximation errors of linear operators with smooth ranges, connecting them to classical smoothness measures and applying the results to classical operators.

## Contribution

Introduces a novel approach to establish lower estimates for linear operators with smooth ranges, including explicit derivations for positive operators and applications to classical approximation methods.

## Key findings

- New method for lower approximation error estimates
- Explicit lower bounds for positive linear operators
- Insights into eigenvalues of Schoenberg's spline operator

## Abstract

We introduce a new method to prove lower estimates for the approximation error of general linear operators with smooth range in terms of classical moduli of smoothness and related $K$-functionals. In addition, we explicitly show how to derive lower estimates for positive linear operators with smooth range and apply this result to classical approximation operators. We finish with some remarks on the eigenvalues of Schoenberg's spline operator.