Approximation by orthonormal polynomials associated with even exponential weights

TL;DR

This paper establishes quantitative approximation results using orthonormal polynomials linked to exponential weights e^(-Φ), with Φ being an even polynomial, supported by recursion relations, inequalities, and simulations on various functions.

Contribution

It introduces a new approximation framework for orthonormal polynomials with exponential weights based on recursion relations and Poincaré inequalities, including practical simulations.

Findings

Effective approximation for smooth and non-smooth functions

Validation through simulations in Gaussian and double well cases

Enhanced understanding of polynomial approximation with exponential weights

Abstract

In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$\Phi$ , where $\Phi$ is an even polynomial with positive leading coefficient. This result is a consequence of a recursion relation for the orthonormal polynomials and of the strong Poincar{\'e} inequality. Simulations are provided at the end of the article, on smooth, non-smooth functions as well as in the Gaussian and the double well case.