Timeline for Approximate sum by an integral: valid or not?
Current License: CC BY-SA 3.0
16 events
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| Jul 14, 2017 at 15:08 | answer | added | fedja | timeline score: 3 | |
| Jul 14, 2017 at 12:44 | comment | added | valle | @fedja Thank you. I look forward to your answer. | |
| Jul 14, 2017 at 12:28 | comment | added | fedja | Only analyticity near the points of global maximum is really required (plus the fact that the global maximum does not occur at the endpoints). OK, I'll post the details today when I have time :-) | |
| Jul 14, 2017 at 11:43 | comment | added | valle | @fedja Oh, note that my $f$ and $g$ are analytic over a restricted domain. They are not entire. So the contour manipulations should not be too extreme. | |
| Jul 14, 2017 at 8:24 | comment | added | valle | @fedja Thanks. My $f$ and $g$ are analytic. If you offer a proof for this case that would be enough for me. Strange that this is isn't in textbooks (I looked around and couldn't find it). | |
| Jul 14, 2017 at 0:33 | comment | added | fedja | Yes, I can. But will it be enough for you? If it will, I'll post a detailed explanation today or tomorrow. The trickery consists of the Poisson summation formula and shifting the contour to bound the Fourier transform. It is the second part that requires analyticity. I haven't attempted a proof that analyticity is necessary for the exponential decay in all cases, but it is possible to show that no weaker "natural" assumption will suffice. However, if your functions are, indeed, analytic, then why should you care, and if they are not, the chance that there is some non-generic effect is slim.. | |
| Jul 14, 2017 at 0:02 | comment | added | valle | @fedja Can you offer a proof of these last statements? Why is analycity required? | |
| Jul 13, 2017 at 23:50 | comment | added | fedja | There is one general thing I can say though: if $f,g$ are analytic in a neighborhood of $[0,1]$ and $f$ attains its global maximum only at interior points, then you are fine. This condition can be relaxed a bit, but, alas, not by much. | |
| Jul 13, 2017 at 23:07 | comment | added | fedja | Unfortunately, the exponential decay is a very rare case. I'd rather prefer to see the functions you have in mind and try to figure out what rate I can guarantee for them than to try to find the weakest assumptions under which the decay is exponential. | |
| Jul 13, 2017 at 22:54 | comment | added | valle | @fedja I have some very specific functions in mind, but I do not want to meddle this post in too much details. What kind of techniques are available so that I prove these things? Are there some general conditions under which $R_n$ decays exponentially? Oh, you can assume that the max of $f(x)$ occurs in the interior. | |
| Jul 13, 2017 at 21:03 | comment | added | fedja | Only small neighborhoods near the points where $f$ attains its maximal value really matter. If the maximum is attained at some inner point, you are fine with $c=1$. If it is attained only at endpoints, then the computation of $c$ is a bit more complicated. The question about the speed of convergence is more delicate. It depends on many factors but is rarely exponential (though it may be; say, if $g(x)=1$, $f(x)=x-x^2$). With finite smoothness you'll normally get just a power decay. | |
| Jul 13, 2017 at 15:27 | history | edited | valle | CC BY-SA 3.0 |
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| Jul 13, 2017 at 15:06 | answer | added | M. Dus | timeline score: 2 | |
| Jul 13, 2017 at 15:04 | review | Close votes | |||
| Jul 13, 2017 at 17:38 | |||||
| Jul 13, 2017 at 15:01 | history | edited | valle | CC BY-SA 3.0 |
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| Jul 13, 2017 at 14:42 | history | asked | valle | CC BY-SA 3.0 |