Timeline for Does there exist a smooth version of Cohen's factorization theorem?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 2, 2015 at 19:24 | comment | added | Marc Palm | Yes, I forgot the span in the sentence, but the theorem does mention it. Sorry. | |
| Jun 21, 2015 at 1:51 | history | edited | EPS | CC BY-SA 3.0 |
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| Jun 20, 2015 at 2:27 | history | edited | EPS | CC BY-SA 3.0 |
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| Jun 19, 2015 at 21:57 | comment | added | paul garrett | Not clear to me, either. The argument is not at all trivial, even after Casselman's discussion... Entire functions and such... | |
| Jun 19, 2015 at 21:45 | comment | added | EPS | @paulgarrett I was looking at the same essay:) but Casselman doesn't mention any "symmetric" factorization $g*g^*$ theorem there. Perhaps it is not to hard to see, but not very clear to me. | |
| Jun 19, 2015 at 21:40 | comment | added | paul garrett | I don't really know about the latest on this. Bill Casselman has a nice essay about D-M and its proof on his web-site ... which probably you can find as quickly as I can. :) | |
| Jun 19, 2015 at 21:32 | comment | added | EPS | @paulgarrett D-M should be a strong positive result for many practical purposes. I just wanted to know what work has been done in this direction. As a separate question, does this follow from D-M that any function $f\in C_c^\infty(G)$ is a linear combination of the convolutions of the form $g*g^*$ in $C_c^\infty(G)*C_c^\infty(G)$? I think this is true but I don't have a reference for it. I really appreciate your help and clarification. | |
| Jun 19, 2015 at 21:08 | comment | added | paul garrett | It's not only approximated, but equal. (The weaker and much easier "approximation" assertion is essentially Garding's from 1947 or so, which would also cope directly with approximation of smooth by convolutions). But then, in any case, you can certainly approximate arbitrary smooth by test functions, if that's enough for you. To get exact equality with $C^\infty*C^\infty_c$ seems tricky... Is approximation good enough for you? | |
| Jun 19, 2015 at 21:02 | comment | added | EPS | @paulgarrett You are absolutely right, D-M result is completely relevant to my question. My interpretation of D-M in this context is that any function in $C_c^\infty(G)$ can be approximated by a linear combination of the functions in $C_c^\infty(G)*C_c^\infty(G)$. Now I have two questions. (1) is this the right conclusion from D-M? (2) if the answer to the previous question is yes, then isn't the same conclusion clear from the fact that any function in $f\in C_c^\infty(G)$ can be approximated by $g_t*f$, where $g_t$ is an approximate unit? Sorry if these are too naive questions. | |
| Jun 19, 2015 at 18:26 | comment | added | paul garrett | So the D-M result is not addressing some aspect of things relevant to your purposes? Yes, the D-M more directly addresses test functions, i.e., compactly-supported, not all smooth, but that's not a fatal problem. A technical point is that finite sums are necessary in that context, so not every test function is exactly a single convolution of two. Can you clarify your issue? | |
| Jun 19, 2015 at 18:02 | comment | added | EPS | @paulgarrett The version of Dixmier-Malliavin that I have in mind is (roughly) the following: "Every smooth vector in a Frechet representation $(\pi, V)$ belongs to the Garding space." | |
| Jun 19, 2015 at 17:59 | comment | added | paul garrett | Isn't this exactly the Dixmier-Malliavin result? | |
| Jun 19, 2015 at 17:58 | review | First posts | |||
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| Jun 19, 2015 at 17:56 | history | asked | EPS | CC BY-SA 3.0 |