Boundedness of Lebesgue constants and interpolating Faber bases

TL;DR

This paper explores conditions for bounded Lebesgue constants in polynomial interpolation on compact sets, examining their link to convergence and the existence of interpolating Faber bases.

Contribution

It establishes new relationships between bounded Lebesgue constants, convergence properties, and the existence of interpolating Faber bases in polynomial interpolation.

Findings

Bounded Lebesgue constants relate to uniform and pointwise convergence.

Existence of interpolating Faber bases is connected to boundedness conditions.

Conditions for boundedness depend on the structure of the compact set.

Abstract

We investigate some conditions under which the Lebesgue constants or Lebesgue functions are bounded for the classical Lagrange polynomial interpolation on a compact subset of $\mathbb R$. In particular, relationships of such boundedness with uniform and pointwise convergence of Lagrange polynomials and with the existence of interpolating Faber bases are discussed.