An Euler-MacLaurin formula for polygonal sums

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 int...

Full description

Saved in:
Bibliographic Details
Published in:Transactions of the American Mathematical Society 2022-01, Vol.375 (1), p.151-172
Main Authors: Brandolini, Luca, Colzani, Leonardo, Robins, Sinai, Travaglini, Giancarlo
Format: Article
Language:English
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8462