Spline Quadrature and semi-classical orthogonal Jacobi Polynomials

TL;DR

This paper develops a theory for spline quadrature rules on arbitrary intervals using semi-classical orthogonal Jacobi polynomials, applicable to various continuity classes, with some aspects based on a conjecture.

Contribution

It introduces a novel approach to spline quadrature rules leveraging semi-classical orthogonal Jacobi polynomials for arbitrary continuity and nonuniform subintervals.

Findings

Theory for spline quadrature rules with arbitrary continuity

Extension to nonuniform subintervals

Dependence on a conjecture for continuity class c ≥ 2

Abstract

A theory of spline quadrature rules for arbitrary continuity class in a closed interval $[a, b]$ with arbitrary nonuniform subintervals based on semi-classical orthogonal Jacobi polynomials is proposed. For continuity class $c \ge 2$ this theory depends on a conjecture.