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Approximation by orthonormal polynomials associated with even exponential weights

Bastien Grosse

Abstract

In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e -$Φ$ , where $Φ$ 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{é} 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.

Approximation by orthonormal polynomials associated with even exponential weights

Abstract

In this paper, we prove a quantitative approximation result by orthonormal polynomials associated to an exponential weight of the form e - , where 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{é} 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.

Paper Structure

This paper contains 18 sections, 14 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Notations and main result
  4. Outline of the paper
  5. Preliminary results
  6. Properties of the orthonormal polynomials
  7. Functional inequality
  8. Convergence of the projection error
  9. Hermite case
  10. General case
  11. Numerical simulations
  12. Generating orthonormal polynomials
  13. The algorithms
  14. Comparison of the two methods
  15. Approximation of functions by orthonormal polynomials
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $r\geq 1$, $n\in\mathbb{N}$ and let $b_{n}$ be the unique positive solution of the equation There exists a constant $K>0$ such that for all functions $f : \mathbb{R} \mapsto \mathbb{R}$ having $r-1$ continuous derivatives, and such that $f^{(r)}$ is absolutely continuous and $f^{(r)}\in L^{2}(\rho)$:

Figures (5)

  • Figure 1: Asymptotic behavior of the recursion coefficients $\beta_{k}$ computed by the Chebychev algorithm, for the brute-foce and the recursion-based method.
  • Figure 2: Comparison of the brute-force and of the recurrence-based methods in the Hermite case.
  • Figure 3: Evolution of the projection error. The $y$-axis is in log scale. Orders of convergence are displayed in the legend.
  • Figure 4: Evolution of the projection error. The $y$-axis is in log scale. Orders of convergence are displayed in the legend.
  • Figure 5: Continuous line: $F_{n}$. Thick dotted line: $G_{n}$. The asymptote of $F_{n}$ is the straight dotted line.

Theorems & Definitions (28)

  • Definition 1.1: Freud weight
  • Theorem 1.1: LEVIN1987
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Theorem 2.1: MAGNUS198665
  • Lemma 2.1: YangChen
  • proof
  • Proposition 2.2
  • proof
  • ...and 18 more