Timeline for A Paley–Wiener theorem for a Volterra equation on compact operators
Current License: CC BY-SA 4.0
13 events
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| S Apr 3, 2023 at 12:02 | history | bounty ended | CommunityBot | ||
| S Apr 3, 2023 at 12:02 | history | notice removed | CommunityBot | ||
| Apr 1, 2023 at 2:37 | comment | added | fedja | "I don't see how to mimic the proof of Newman here." Just follow it literally until you hit the non-commutativity issue and show us where you get. There is a small trick in the end, which is to freeze the non-smooth entries before looking at the FT of the convolution, but let's go over Newman's scheme first: it is very elegant and powerful, so I'd rather let you try it yourself. BTW, when responding to me, you'd better add @fedja somewhere, so that I would be alerted that something is going on :-) | |
| S Mar 26, 2023 at 9:54 | history | bounty started | Quentin | ||
| S Mar 26, 2023 at 9:54 | history | notice added | Quentin | Authoritative reference needed | |
| Mar 26, 2023 at 9:53 | history | edited | Quentin | CC BY-SA 4.0 |
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| Mar 22, 2023 at 13:42 | comment | added | Quentin | Thank you. I tried to do something in this spirit. If $k_n(t) = P_n k(t) P_n$, I believe that $I - \hat{k}_n(z)$ is indeed invertible for all $\Re(z) \geq 0$, provided n is large enough. So by the standard Paley-Wiener's theorem, there exists $r_n \in L^1(\mathbb{R}_+)$ such that $r_n(t)= k_n(t) + \int_0^t{k_n(t-s) r_n(s) ds}$. Now my main problem is how to check that $sup_{n \in \mathbb{N}} \int_0^\infty{ \lVert r_n(t) \rVert dt} < \infty$ ? Did I miss something? I don't see how to mimic the proof of Newman here. | |
| Mar 21, 2023 at 5:10 | comment | added | fedja | If we are in a separable Hilbert space $H$, why don't we just use a family of finite-dimensional spaces $H_n$ expanding to $H$ and the corresponding family of orthogonal projections $P_n$? We have $\|P_nk(t)P_n-k(t)\|\to 0$ point-wise (due to compactness) and $\|k(t)\|$ is the integrable majorant, so $P_nkP_n$ tends to $k$ in $L^1$, after which we can approximate $P_nkP_n$ (and, thus, $k$) by a $C_0^\infty$ operator-valued function $k_1(t)$ in $L^1$ and mimic the Newman proof of Wiener's theorem (ams.org/proc/1975-048-01/S0002-9939-1975-0365002-8/… )? | |
| Mar 20, 2023 at 15:25 | history | edited | Quentin | CC BY-SA 4.0 |
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| Mar 20, 2023 at 14:48 | comment | added | LSpice | Spelling note: almost certainly you meant Paley–Wiener (not ‘Palay’). I edited accordingly. | |
| Mar 20, 2023 at 14:45 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading; link to book; deleted "thanks"
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| S Mar 20, 2023 at 14:41 | review | First questions | |||
| Mar 20, 2023 at 14:46 | |||||
| S Mar 20, 2023 at 14:41 | history | asked | Quentin | CC BY-SA 4.0 |