The order of Lebesgue constant of Lagrange interpolation on several intervals

TL;DR

This paper investigates the growth behavior of the Lebesgue constant in Lagrange interpolation over multiple intervals, providing bounds and explicit constructions, especially for two-interval cases, to understand interpolation stability.

Contribution

It offers new bounds for the Lebesgue constant on multiple intervals and explicit constructions for specific two-interval cases, advancing understanding of interpolation on complex sets.

Findings

Derived lower and upper bounds for Lebesgue constants on multiple intervals.

Provided explicit constructions for two-interval cases, including non-symmetric intervals.

Enhanced understanding of interpolation stability on complex sets.

Abstract

We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue constant. The case of two intervals of equal lengths is simpler, and an explicit construction for two non-symmetric intervals will be given only in a special case.