Timeline for "sinc-ing" integral
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2017 at 7:12 | answer | added | Nemo | timeline score: 11 | |
| Mar 2, 2017 at 7:00 | comment | added | Nemo | This result and a more general one is due to Stoermer, see Whittaker and Watson, Modern Analysis, chapter 6, example 6 at page 122. | |
| Mar 2, 2017 at 0:00 | vote | accept | T. Amdeberhan | ||
| Mar 1, 2017 at 23:50 | comment | added | T. Amdeberhan | Paul and Keaton:Thank you both for the added discussion. The Fourier transform of $sinc(x)$ gives a square signal. If you take increasing powers $sinc^n(x)$ and compute its Fourier, the resulting signal gets smoother and smoother and eventual yielding a normal distribution curve. | |
| Mar 1, 2017 at 23:29 | comment | added | paul garrett | @KeatonHamm, ah, doesn't surprise me that signal-processing people would know this, but/and I didn't know how lively and considered-mathematics such a subject was/is! My comment is based on the palpable fact that such useful and poignant facts haven't made it into many standard "real analysis" texts, so the vast majority of grad students, hence, mathematicians, have no idea that such a thing is/could-be true. I used to live in that land myself! :) | |
| Mar 1, 2017 at 23:26 | answer | added | Noah Stein | timeline score: 10 | |
| Mar 1, 2017 at 23:17 | comment | added | Keaton Hamm | This fact about convolutions of sincs is, I would say, widely known. Certainly it is at least to harmonic analysts and anyone doing signal processing. This is also how you define B-splines. | |
| Mar 1, 2017 at 23:11 | comment | added | paul garrett | I cannot immediately see the independence of the "b", but I am aware of an apparently not-so-widely-known fact, that "sinc"'s convolved with each other behave remarkably. Namely, up to constants and normalizations, ${\sin x\over x}$ is the Fourier transform of the char fcn of an interval symmetrical about the origin. Thus, the convolution of several sinc's (with constants inserted) is the Fourier transform of the product of several char fcns of symmetrical intervals, which is just the FT of the char fcn of the smallest among them. If this is in the right direction for you, I could elaborate. | |
| Mar 1, 2017 at 23:02 | history | edited | T. Amdeberhan | CC BY-SA 3.0 |
added 2 characters in body
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| Mar 1, 2017 at 22:23 | history | asked | T. Amdeberhan | CC BY-SA 3.0 |