Quadrature rules for $C^0$, $C^1$ splines, the real line, and the five (5) families

TL;DR

This paper introduces five families of simple-to-compute quadrature rules for $C^0$ and $C^1$ splines on the real line, applicable over one or two interval periods, extending classical Gauss-Legendre methods.

Contribution

It presents new families of quadrature rules for spline classes with simple formulas, expanding the tools for numerical integration of splines.

Findings

Five families of quadrature rules are formulated for $C^0$, $C^1$ splines.

Rules are designed for periods of one or two intervals on the real line.

Formulas for points and weights are straightforward, similar to Gauss-Legendre rules.

Abstract

The five (5) families of quadrature rules with periods of one or two intervals for the real line and spline classes $C^0$, $C^1$ are presented. The formulae allow one to calculate the points or weights of these quadrature rules in a very simple manner as for the classical Gauss-Legendre rules.