Estimates of Lebesgue Constants for Lagrange Interpolation Processes by Rational Functions under Mild Restrictions to their Fixed Poles

TL;DR

This paper provides estimates for Lebesgue constants in Lagrange interpolation using rational functions with fixed poles, accommodating accumulation points, and introduces an inverse polynomial image method for analysis.

Contribution

It introduces a novel approach using an inverse polynomial image method to estimate Lebesgue constants for rational Lagrange interpolation with fixed poles.

Findings

Provides bounds for Lebesgue constants in rational interpolation

Handles cases with accumulation points of poles on intervals

Extends existing methods to rational functions with fixed poles

Abstract

We estimate the Lebesgue constants for Lagrange interpolation processes on one or several intervals by rational functions with fixed poles. We admit that the poles have accumulation points on the intervals. To prove it we use an analog of the inverse polynomial image method for rational functions with fixed poles.