Timeline for A question about pointwise convergence of Fourier transform in $N$-dimensions
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 19, 2014 at 13:09 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 19, 2014 at 13:03 | vote | accept | Rajesh D | ||
| Nov 19, 2014 at 12:59 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| S Nov 15, 2014 at 13:20 | history | bounty ended | CommunityBot | ||
| S Nov 15, 2014 at 13:20 | history | notice removed | CommunityBot | ||
| Nov 15, 2014 at 7:06 | comment | added | Rajesh D | Related : mathoverflow.net/q/187215/14414 | |
| Nov 15, 2014 at 7:02 | comment | added | Rajesh D | Please explain the downvotes! | |
| Nov 11, 2014 at 9:29 | history | edited | Willie Wong | CC BY-SA 3.0 |
fixed typo with MathJax.
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| Nov 10, 2014 at 15:36 | comment | added | Rajesh D | My first goal now is to establish localization principle for rectangular partial sums, for this class of functions. I need some solid references and papers. I dont have access to any math journals, so any help will be appreciated. Especially the Cesari-Tonelli result I've heard established localization principle for a wider class of functions. I don't have access to it. | |
| Nov 9, 2014 at 11:48 | comment | added | Rajesh D | H. Hardy 1 . On double Fourier series, and especially those which represent the double zeta function with real and incommensurable parameters. I don't have access to it. Wonder what was Hardy doing with double zeta function and its Fourier series? | |
| Nov 9, 2014 at 10:31 | comment | added | Rajesh D | @WillieWong : There are a couple of places you have typos/missing dollar in equations in "Higher dimension revisited". Request if you could correct them. 1. exp^x, 2. a limit not under $$." | |
| Nov 7, 2014 at 14:22 | comment | added | paul garrett | A too-short answer, but it seems that any comment would likely get lost in the others... Also, not responding directly to the literal question, but to the context: the notion of "wave-front set" would seem to me to be one of the concepts the questioner might find useful in refining the formulation of the issue (e.g., refining to the point that the assertions are not easily shown faulty in various ways, e.g., coordinate-(in)dependence as @WillieWong comments). | |
| S Nov 7, 2014 at 11:51 | history | bounty started | Rajesh D | ||
| S Nov 7, 2014 at 11:51 | history | notice added | Rajesh D | Draw attention | |
| Nov 7, 2014 at 11:45 | comment | added | Rajesh D | Od there any good source/book which has all the machinery to prove/disprove such a result for N=2. I still think result hold except at theta = npi/2. | |
| Nov 6, 2014 at 10:24 | answer | added | Willie Wong | timeline score: 4 | |
| Nov 6, 2014 at 10:20 | history | edited | Willie Wong | CC BY-SA 3.0 |
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| Nov 6, 2014 at 9:01 | comment | added | Rajesh D | @Willie : I can replace $[0,2\pi)^{N-1}$ with the open set $(0,2\pi)^{N-1}$ without any loss as the function is $2\pi$ periodic, so i hope that would rectify the recursive definition. But i still agree there are flaws. | |
| Nov 6, 2014 at 8:52 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 6, 2014 at 8:48 | comment | added | Rajesh D | @Willie : I made up these definitions on my own and they dont have any sanctity except to explain the question in a easy manner. My main motivation is generalizing mathoverflow.net/q/165038/14414 to images and $N>2$ ( but only for jumps of function rather than any of its derivatives). I have moved away from jumps of derivatives of the function altogether. | |
| Nov 6, 2014 at 8:47 | comment | added | Rajesh D | Hi @WillieWong : Thanks for the valid points. I just realize they hamper but hope they are not fatal. So I have edited the question to give a version for $N=2$, with which I am at comfort and clear with my intentions and leave it to mathematicians for $N>2$, if at all its possible in a meaningful way. I hope my motivations for definitions are clear with the $N=2$ case. | |
| Nov 6, 2014 at 8:40 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 5, 2014 at 9:27 | comment | added | Willie Wong | Maybe I should clarify that my point (1) above is meant to say that it makes no sense to define $\mathcal{V}(\Omega)$ recursively because $[0,2\pi)^{N-1}$ is not open. | |
| Nov 5, 2014 at 9:27 | comment | added | Willie Wong | A few random comments: (1) the domain of definition of $J_P^\theta$ is not an open subset of $\mathbb{R}^{N-1}$. (2) What is $\theta$-coordinate? You are identifying $\mathbb{S}^{N-1}$ with essentially $\mathbb{T}^{N-1}$ and that is problematic in my opinion. (3) What's up with $\phi$? By your definition when $N = 3$ you have $\phi = \frac{4\pi}{8} - \sum_{j = 1}^2 \theta_j$ so when $\theta_1 + \theta_2 > \pi/2$, which happens for a large chunk of $[0,2\pi)^2$ your $\phi$ is negative. // Did you come up with all these definitions yourself? If so please include motivations on why such defn. | |
| Nov 5, 2014 at 6:26 | comment | added | Rajesh D | @TerryTao and other Harmonic analysis experts, what do you think about this problem. Also let me know the defects in it, especially the formula for $\phi$, I am not sure I got it what I intended it to. Thanks | |
| Nov 4, 2014 at 14:12 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 4, 2014 at 10:10 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 4, 2014 at 9:38 | history | edited | Rajesh D | CC BY-SA 3.0 |
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| Nov 4, 2014 at 9:02 | history | asked | Rajesh D | CC BY-SA 3.0 |