Timeline for Special Schwartz function on the positive interval
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 10 at 20:17 | vote | accept | SnowRabbit | ||
| Jan 5 at 1:58 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Jan 4 at 18:33 | comment | added | fedja | @GiorgioMetafune For you I can tell it in a couple of lines. $PQP$ is a self-adjoint compact (HS) positive definite operator of norm at most $1$. The eigenvalue $1$ is impossible because the eigenvector would be a function with bounded support and spectrum simultaneously. Hence, the norm is strictly less than $1$. If you are interested in quantitative bounds, look at the book by Havin and Joricke "The Uncertainty Principle in Harmonic Analysis" for proofs and references. | |
| Jan 4 at 18:23 | comment | added | Giorgio Metafune | Everything is very nice. Where is a proof of the contractivity of $PQ$ for long intervals? | |
| Jan 4 at 17:51 | history | edited | fedja | CC BY-SA 4.0 |
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| Jan 4 at 17:08 | comment | added | Christian Remling | This is too funny!! As I was writing my comment on your answer to the other question, I was thinking to myself, this question here is probably of that type too... (and I hadn't seen your answer here yet) | |
| Jan 4 at 17:06 | comment | added | fedja | @AlekseiKulikov OK. Yeah, for short intervals it is a trivial HS-norm bound. I'll add it a bit later :-) | |
| Jan 4 at 16:36 | comment | added | Aleksei Kulikov | Nice argument! I had something quite different and way more technical in mind (cooking a function in the unit disk and then translating it to the upper half-plane). But I think you should give some explanation/reference to the fact that $PQ$ is contractive, while it is quite old I don't think it is that well-known (I assume you have a Hilbert--Schmidt norm argument in mind?) | |
| Jan 4 at 16:19 | history | answered | fedja | CC BY-SA 4.0 |