Timeline for error estimates for multi-dimensional Riemann sums
Current License: CC BY-SA 3.0
15 events
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| Oct 15, 2011 at 2:54 | comment | added | Brendan McKay | There is a lot of theory of Euler-Maclaurin sums in multiple dimensions. A random recent example is Karshon, Sternberg, Weitsman, "Exact Euler–Maclaurin formulas for simple lattice polytopes", Advances in Applied Mathematics 39 (2007) 1–50, which has many references. Exactly how to extract an answer to your question from this theory is unclear at a quick glance. For years I have been looking for an exact bound on the error which remains useful when the number of dimensions goes to infinity, but two specialists I have asked about it did not know of any. | |
| Oct 14, 2011 at 13:24 | history | edited | J. M. isn't a mathematician |
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| Aug 19, 2011 at 2:27 | comment | added | Terry Tao | For asymptotics of oscillatory integrals such as the Fourier transform of the disk, you can try Chapter VIII of Stein's "Harmonic analysis". | |
| Aug 18, 2011 at 12:32 | comment | added | James Propp | @Terry: Thanks for the reference to the Iosevich article; it's very informative. | |
| Aug 18, 2011 at 5:14 | vote | accept | James Propp | ||
| Aug 16, 2011 at 6:40 | comment | added | James Propp | I only discussed $\sqrt{1-x^2-y^2}$ because I expect it to exhibit behavior typical for the case where $f$ is concave (note the divergence of the derivative of $\sqrt{1-x^2-y^2}$ at the boundary; in this respect the example is about as bad as it can be). I'm more interested right now in finding out about existing general-purpose off-the-shelf estimates than about any examples in particular. But let's focus on this example anyway. Can anyone suggest a (preferably online) derivation of the Fourier transform of the indicator function of the disc? I gather that Bessel functions play a role. | |
| Aug 15, 2011 at 19:34 | answer | added | Terry Tao | timeline score: 8 | |
| Aug 15, 2011 at 19:04 | comment | added | Terry Tao | If you want good estimates on the error for a specific f, rather than for all functions of bounded variation, and you are summing over the periodic lattice (and not just on the "worst" point in each cube, which is what a Riemann sum does) I would recommend using the Poisson summation formula and getting good bounds on the Fourier transform, possibly after first smoothing away any discontinuities (cf. the proof of the Landau (or Voronoi) error term in the Gauss circle problem, see e.g. ams.org/notices/200106/fea-iosevich.pdf ). | |
| Aug 15, 2011 at 11:34 | comment | added | James Propp | Also, I'd like an answer that is applicable to specific cases like $f(x,y)=\sqrt{1-x^2-y^2}$, for which the error of the Riemann sums appears to fall like $1/n^2$. | |
| Aug 15, 2011 at 4:13 | comment | added | James Propp | Sorry my question wasn't clearer. What I'm after are multi-dimensional analogues of Euler-Maclaurin summation, and things like that. | |
| Aug 15, 2011 at 1:59 | history | edited | James Propp | CC BY-SA 3.0 |
I forgot to include the hypothesis of bounded variation
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| Aug 15, 2011 at 0:31 | answer | added | Gerry Myerson | timeline score: 4 | |
| Aug 14, 2011 at 23:36 | comment | added | Gerhard Paseman | I can imagine some cropped, two dimensional version of abs(sin(1/x)/x) having very bad estimates, especially if the local maxima were at points in the domain with irrational coordinates. I don't have the error estimates, but my gut suggests that worse than O(1/n) is acheivable, and that the situation does not get better as the dimension grows. Gerhard "Ask Me About System Design" Paseman, 2011.08.12 | |
| Aug 14, 2011 at 22:39 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Aug 14, 2011 at 22:25 | history | asked | James Propp | CC BY-SA 3.0 |