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Hi,

I want to approximate a multiple sum of the form $$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$ where each $x_i$ is an integer between $0$ and $n$, by an integral $$\int_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)}dx_1dx_2\cdots dx_m\,.$$ I know that the Euler-Maclaurin formula can be used to derive the error term when $m=1$ but often see sums of this form with $m > 1$ approximated by integrals, though with little justification. I do not have much of a background in mathematical analysis so am not sure where to look for a reference for this.

Any help will be much appreciated.

flag	asked Apr 2 2010 at 17:21 bandini 221●7

1

In fact Euler-Maclaurin applies to your case as well. See mathoverflow.net/questions/10667 – Steve Huntsman Apr 2 2010 at 17:51

First of all, for a "generic" function $g$ your approximation by sum can be really bad. There are different techniques in asymptotic analysis (including Euler-Maclaurin) which can be used for your sums. – Wadim Zudilin Jun 9 2010 at 11:49

Steve Huntsman · Answer 1 · 2010-04-03 00:56:53Z UTC

2

See Chapter 10 of Computing the Continuous Discretely by Beck and Robins for an introduction, and the paper of Brion and Vergne cited in the link in my comment for the general case.

edited Apr 3 2010 at 0:56

answered Apr 2 2010 at 18:13

Steve Huntsman
8,139●1●22●56

Thanks Steve. Looks like a very interesting book. BTW, the google books view is limited but there's actually a full, non-printable, .pdf on the author's website math.sfsu.edu/beck/ccd.html – bandini Apr 3 2010 at 10:54

Approximating a multiple sum with an integral

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