Timeline for Support of functions on compact groups, and their Fourier transforms
Current License: CC BY-SA 3.0
12 events
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| Jan 17, 2013 at 18:04 | comment | added | Yemon Choi | @Jafar: thanks for clarifying. If we replace your group $G$ with the circle group, then what is the analogous result that you hope for? (Usually in harmonic analysis, if you can't do something for $T$ then one has very little chance for non-abelian compact groups.) | |
| Jan 17, 2013 at 16:02 | comment | added | Marc Palm | Why did you change the question? My answer remains valid. Use induction and the representation theory of compact Lie groups. The function can be recovered here as well. same formula. | |
| Jan 17, 2013 at 11:32 | history | edited | Jafar | CC BY-SA 3.0 |
added 549 characters in body
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| Jan 16, 2013 at 7:49 | comment | added | Jochen Wengenroth | Perhaps, Jafar has something like the Paley-Wiener-Schwartz theorem in mind which describes the convex hull of the support (of a test function on $\mathbb R^n$ or a distribution with compact support) by growth conditions of the Fourier-Laplace transform. | |
| Jan 16, 2013 at 1:39 | comment | added | Yemon Choi | Paul, my problems with the question go further: the words "recover" and "function" are open to different interpretations. (After all, why stop at functions, why not go for suitable tempered distributions?) | |
| Jan 16, 2013 at 1:14 | comment | added | paul garrett | ... expansions has a straightforward answer for compact Lie groups: "even worse" than for "ordinary" Fourier series, pointwise convergence typically fails, for Baire-category-corollary reasons. Nevertheless, the $L^2$ convergence determines an $L^2$ function, which, if it is a.e. a continuous function, unambiguously determines its support. | |
| Jan 16, 2013 at 1:12 | comment | added | paul garrett | As reflected in the other comments and answers, the question itself is ambiguous. E.g., if "Fourier transform" means expansion in "characters" (=traces of irreds), then only conjugation-invariant functions are "represented" by "spectral expansions". If at the other end, "Fourier expansion" means expression as sum of functions from irreducibles, then there is a standard Sobolev-lemma that would assert that sufficiently smooth functions are represented (in various senses) by their spectral expansions. The subtler question of whether continuous functions are represented by their spectral... | |
| Jan 15, 2013 at 19:14 | history | edited | Yemon Choi | CC BY-SA 3.0 |
replaced the distinctly uninformative title
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| Jan 15, 2013 at 18:56 | comment | added | Yemon Choi | Also, I assume that you are referring to the matrix-valued Fourier transform? | |
| Jan 15, 2013 at 18:50 | comment | added | Yemon Choi | Could you be a bit more precise about what you mean by "recover"? are you looking for an algorithm? | |
| Jan 15, 2013 at 15:24 | answer | added | Marc Palm | timeline score: 1 | |
| Jan 15, 2013 at 15:05 | history | asked | Jafar | CC BY-SA 3.0 |