Timeline for Radial limit does not exist almost everywhere
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 28, 2015 at 21:48 | comment | added | GH from MO | @MattYoung: Thank you! I am glad you scanned the paper and posted to your webpage. I read it quickly (skipping some details), and found it very nice. | |
| May 28, 2015 at 17:43 | comment | added | Matt Young | @GHfromMO I scanned the paper and put it on my website at link. My only electronic copy was on a 3.5" floppy that is long gone, so this is the best I can do. I wouldn't be surprised if the proof could be greatly simplified, but it's what I came up with at that time. | |
| May 26, 2015 at 21:33 | comment | added | GH from MO | @MattYoung: Interesting story. I think it would be worthwhile to preserve this document in more durable form (although paper proved to be durable, but more copies would be better). Can you scan it and post it somewhere? If you have the word file we could try to make a pdf out of it, there are lots of converters around. | |
| May 26, 2015 at 21:25 | comment | added | Matt Young | I think the final result worked for coefficients whose $l^2$ norm is infinite. I originally looked at the case where all the coefficients are $1$ but Hejhal showed me how to do the more general case. My only copy of the thesis is a printout. I typed it up in Microsoft Word before anyone told me about tex. | |
| May 26, 2015 at 21:09 | comment | added | Lucia | @MattYoung: Slightly more general version -- lacunary Fourier series with coefficients whose $\ell^2$ norm is infinite. Is that what you looked at, or with coefficients being $1$? | |
| May 26, 2015 at 21:02 | comment | added | Matt Young | In my undergrad senior thesis I showed something along these lines. That is, the distribution function of the partial sums of a lacunary Fourier series have Gaussian moments. I assume this is what @GHfromMO might be referring to. | |
| May 26, 2015 at 20:32 | history | edited | Lucia | CC BY-SA 3.0 |
Corrected a small error
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| May 26, 2015 at 20:10 | comment | added | GH from MO | Perhaps your idea even works for Zygmund's general theorem (i.e. for all Hadamard lacunary series)! | |
| May 26, 2015 at 18:59 | comment | added | Lucia | @GHfromMO: Thanks for pointing that out -- I'll correct it. For the second part, my idea would be to compute moments on the circle with radius $r$. Since the powers of $2$ don't have too many linear relations, one would get an approximately (complex) Gaussian with variance tending to infinity (this is because of the assumption on coefficients). Then it follows that on each circle of radius sufficiently close to $1$, most points (measure close to $1$) will have a large value of the function. From that the result follows. | |
| May 26, 2015 at 18:00 | comment | added | GH from MO | Very nice. I think in the first term of (1), $e^{-2^n/2^{N+50}}$ should be $e^{-2^{n}/2^{N+50}} - e^{-2^n/2^N}$, and the last display should be adjusted accordingly. The second part of the problem states a special case of Zygmund's theorem: if $\sum_n |a_n|^2=\infty$, then $\sum_n a_n z^{2^n}$ has no radial limit at almost every point of the unit circle. Do you see a simple treatment for this more general statement? | |
| May 26, 2015 at 5:03 | history | answered | Lucia | CC BY-SA 3.0 |