An Euler-MacLaurin formula for polygonal sums

Brandolini, Luca; Colzani, Leonardo; Robins, Sinai; Travaglini, Giancarlo

doi:10.1090/tran/8462

An Euler-MacLaurin formula for polygonal sums
HTML articles powered by AMS MathViewer

by Luca Brandolini, Leonardo Colzani, Sinai Robins and Giancarlo Travaglini PDF

Trans. Amer. Math. Soc. 375 (2022), 151-172 Request permission

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.

References

Matthias Beck and Sinai Robins, Computing the continuous discretely, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York, 2015. Integer-point enumeration in polyhedra; With illustrations by David Austin. MR 3410115, DOI 10.1007/978-1-4939-2969-6
Nicole Berline and Michèle Vergne, Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron, Configuration spaces, Springer INdAM Ser., vol. 14, Springer, [Cham], 2016, pp. 67–105. MR 3615728
L. Brandolini, L. Colzani, S. Robins, and G. Travaglini, Pick’s theorem and convergence of multiple Fourier series, Amer. Math. Monthly 128 (2021), no. 1, 41–49. MR 4200451, DOI 10.1080/00029890.2021.1839241
R. Diaz, Q. N. Le, and S. Robins, Fourier transform of polytopes, solid angles, and discrete volume, submitted.
Alexey Garber and Igor Pak, Concrete polytopes may not tile the space, Mathematika 66 (2020), no. 4, 920–926. MR 4143157, DOI 10.1112/mtk.12052
Stavros Garoufalidis and James Pommersheim, Sum-integral interpolators and the Euler-Maclaurin formula for polytopes, Trans. Amer. Math. Soc. 364 (2012), no. 6, 2933–2958. MR 2888234, DOI 10.1090/S0002-9947-2012-05381-5
Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
Victor Guillemin and Shlomo Sternberg, Riemann sums over polytopes, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 7, 2183–2195 (English, with English and French summaries). Festival Yves Colin de Verdière. MR 2394539
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
Yael Karshon, Shlomo Sternberg, and Jonathan Weitsman, The Euler-Maclaurin formula for simple integral polytopes, Proc. Natl. Acad. Sci. USA 100 (2003), no. 2, 426–433. MR 1950644, DOI 10.1073/pnas.0237168100
Yael Karshon, Shlomo Sternberg, and Jonathan Weitsman, Euler-Maclaurin with remainder for a simple integral polytope, Duke Math. J. 130 (2005), no. 3, 401–434. MR 2184566, DOI 10.1215/S0012-7094-05-13031-9
Yael Karshon, Shlomo Sternberg, and Jonathan Weitsman, Exact Euler-Maclaurin formulas for simple lattice polytopes, Adv. in Appl. Math. 39 (2007), no. 1, 1–50. MR 2319562, DOI 10.1016/j.aam.2006.04.003
F. C. Machado and S. Robins, Coefficients of the solid angle and Ehrhart quasi-polynomials, arXiv:1912.08017.
G. H. Meisters, Polygons have ears, Amer. Math. Monthly 82 (1975), 648–651. MR 367792, DOI 10.2307/2319703
Ivan Niven, Irrational numbers, The Carus Mathematical Monographs, No. 11, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, N.Y., 1956. MR 0080123