Lobatto and Radau positive quadrature formulas for linear combinations of Jacobi polynomials

TL;DR

This paper characterizes specific linear combinations of Jacobi polynomials that produce Lobatto and Radau positive quadrature formulas with high degree of exactness, useful for polynomial approximation problems.

Contribution

It identifies parameter conditions for Jacobi polynomial combinations that generate positive quadrature formulas with a designated node.

Findings

Determines parameter sets for positive quadratures with degree 2n+2 or 2n+1.

Establishes the existence of quadratures containing a specified node.

Provides tools for one-sided polynomial L1 approximation.

Abstract

For a given $\theta\in (-1,1)$, we find out all parameters $\alpha,\beta\in \{0,1\}$ such that, there exists a linear combination of Jacobi polynomials $J_{n+1}^{(\alpha,\beta)}(x)-C J_{n}^{(\alpha,\beta)}(x)$ which generates a Lobatto (Radau) positive quadrature formula of degree of exactness \textcolor{red}{$2n+2$ ($2n+1$)} and contains the point $\theta$ as a node. These positive quadratures are very useful in studying problems in one-sided polynomial $L_1$ approximation.