Orthogonality of polar Legendre polynomials and approximation

TL;DR

This paper investigates the properties and relationships of polar Legendre polynomials, deriving identities, relations, and extremal problems to enhance understanding of their approximation capabilities.

Contribution

It introduces new identities and extremal problems involving polar Legendre polynomials, expanding theoretical understanding of their approximation properties.

Findings

Derived new identities and relations for polar Legendre polynomials

Formulated extremal problems and minimization principles

Enhanced theoretical framework for approximation using polar Legendre polynomials

Abstract

Let $\{Q_{n}(x)\}$ be a system of integral Legendre polynomials of degree exactly n,and let $\{P_{n}(x)\}$ be polar polynomials primitives of integral Legendre polynomials. We derive some identities and relations and extremal problems and minimization involving of polar integral Legendre polynomials.