An Euler-Maclaurin formula for polygonal sums
Classical Analysis and ODEs
2020-04-21 v1 Functional Analysis
Number Theory
Abstract
We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.
Cite
@article{arxiv.2004.09377,
title = {An Euler-Maclaurin formula for polygonal sums},
author = {Luca Brandolini and Leonardo Colzani and Sinai Robins and Giancarlo Travaglini},
journal= {arXiv preprint arXiv:2004.09377},
year = {2020}
}