The Upper Bound for the Lebesgue Constant for Lagrange Interpolation in Equally Spaced Points of the Triangle

TL;DR

This paper establishes an explicit upper bound for the Lebesgue constant in Lagrange interpolation on equally spaced points within a triangle, refining previous general results specifically for two-dimensional cases.

Contribution

The paper provides a refined explicit upper bound for the Lebesgue constant in 2D triangle interpolation, improving understanding of interpolation stability.

Findings

Derived an explicit upper bound for the Lebesgue constant in 2D

Refined previous bounds for arbitrary simplices to the case of a triangle

Enhanced the theoretical understanding of interpolation stability in triangular domains

Abstract

An upper bound for the Lebesgue constant (the supremum norm) of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to n is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to n for an arbitrary d-dimensional simplex was established by the author. The explicit upper bound proved in this article refines this result for d=2.