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Integrals of Legendre polynomials and approximations

Abdelhamid Rehouma

TL;DR

This work studies integral Legendre polynomials $Q_n$ derived from Legendre polynomials and develops a framework of kernels, extremal problems, and Fourier-type expansions. It introduces the $n$-th kernel $K_n(x,y)$ with a Christoffel-Darboux form and shows reproducing properties and orthogonality with weight $w(x)=\frac{1}{1-x^2}$. An extremal problem is solved, yielding a minimal value $M$ and a minimizer proportional to $K_n(\cdot,0)$, while a Fourier expansion in the $Q_n$ basis is established with explicit coefficient formulas. The results provide a pathway to constructing new orthogonal systems via integral Legendre polynomials and enable refined approximations on $[-1,1]$ through $Q_n$-based transforms.

Abstract

We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.

Integrals of Legendre polynomials and approximations

TL;DR

This work studies integral Legendre polynomials derived from Legendre polynomials and develops a framework of kernels, extremal problems, and Fourier-type expansions. It introduces the -th kernel with a Christoffel-Darboux form and shows reproducing properties and orthogonality with weight . An extremal problem is solved, yielding a minimal value and a minimizer proportional to , while a Fourier expansion in the basis is established with explicit coefficient formulas. The results provide a pathway to constructing new orthogonal systems via integral Legendre polynomials and enable refined approximations on through -based transforms.

Abstract

We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.
Paper Structure (5 sections, 5 theorems, 104 equations)

This paper contains 5 sections, 5 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The n-th kernel for the integral Legendre polynomials
  3. Extremal problem for integral Legendre polynomials
  4. Main results.
  5. Fourier development and approximations

Key Result

Proposition 1

The functions $Q_{n}\left( x\right)$ and $Q_{n}\left( x\right)$ ($n\neq m$) are orthogonal with respect to the weight function $w\left( x\right) =\dfrac{1}{1-x^{2}}$.Then

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Proposition 4
  • Theorem 5