The Lebesgue constant for uniform approximation of differential forms

TL;DR

This paper investigates the Lebesgue constant in the uniform approximation of differential forms from weak data, analyzing its behavior under smooth transformations and establishing its relation to the projection operator's norm.

Contribution

It introduces a measure-theoretic framework linking the Lebesgue constant to the projection operator norm in differential form approximation, including its variation under smooth mappings.

Findings

The norm of the projection operator equals the Lebesgue constant under certain conditions.

The Lebesgue constant's variation under smooth mappings is estimated.

The framework applies to approximation schemes like weighted least squares and interpolation.

Abstract

In this work we address the problem of uniform approximation of differential forms starting from weak data defined by integration on rectifiable sets. We study approximation schemes defined by the projection operator L given by either generalized weighted least squares or interpolation. We show that, under a natural measure theoretic condition, the norm of such operator equals the Lebesgue constant of the problem. We finally estimate how the Lebesgue constant varies under the action of smooth mappings from the reference domain to a physical one, as is customarily done e.g. in finite element method.