Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

TL;DR

This paper develops higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces, optimizing error estimates for integrating functions with derivatives in various function spaces, and provides exactness conditions for polynomial functions.

Contribution

It introduces a new class of quadrature formulas using monic polynomials to optimize error bounds in Lebesgue and related spaces, extending classical trapezoidal rules.

Findings

Derived sharp error estimates for the quadrature formulas.

Established conditions for exactness when integrating polynomials.

Showed that Legendre polynomial-based formulas are exact for polynomials up to degree 2n-1.

Abstract

If $f\!:\![a,b]\to\R$ such that $f^{(n)}$ is integrable then integration by parts gives the formula \begin{align*} &\intab f(x)\,dx = &\frac{(-1)^n}{n!}\sum_{k=0}^{n-1}(-1)^{n-k-1}\left[ \phi_n^{(n-k-1)}(a)f^{(k)}(a)- \phi_n^{(n-k-1)}(b)f^{(k)}(b)\right] +E_n(f), \end{align*} where $\phi_n$ is a monic polynomial of degree $n$ and the error is given by $E_n(f)=\frac{(-1)^n}{n!}\int_a^b f^{(n)}(x)\phi_n(x)\,dx$. This then gives a quadrature formula for $\int_a^bf(x)\,dx$. The polynomial $\phi_n$ is chosen to optimize the error estimate under the assumption that $f^{(n)}\in L^p([a,b])$ for some $1\leq p\leq\infty$ or if $f^{(n)}$ is integrable in the distributional or Henstock--Kurzweil sense. Sharp error estimates are obtained. It is shown that this formula is exact for all such $\phi_n$ if $f$ is a polynomial of degree at most $n-1$. If $\phi_n$ is a Legendre polynomial then the formula is exact for $f$ a polynomial of degree at most $2n-1$.