Euler-Maclaurin formulas for functions of bounded variation

TL;DR

This paper extends the Euler-Maclaurin formula to functions of bounded variation, broadening its applicability beyond smooth functions to include less regular functions.

Contribution

It formulates an analogue of the Euler-Maclaurin formula specifically for functions of bounded variation, a class not previously covered.

Findings

Established a new Euler-Maclaurin type formula for bounded variation functions

Extended the applicability of the Euler-Maclaurin formula to less regular functions

Provided theoretical foundations for numerical approximation of sums and integrals for bounded variation functions

Abstract

The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on $\mathbb R$. We formulate here the analogue for functions that are just of bounded variation.