Timeline for Is there a nonpolynomial $C^\infty$ function $f$ such that $\sup_{x \in \mathbb{R}} \lvert f^{(q)}(x) \rvert \leq (\ln q)^{-q}$ for every $q >1$?
Current License: CC BY-SA 4.0
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| Jan 19, 2023 at 18:26 | comment | added | Terry Tao | The usual Fourier uncertainty principle tells us that trig polynomials of degree $n$ behave roughly like constants at spatial scales of length $1/n$. After a trig substitution, this implies that polynomials of degree n that are controlled on an interval of length $I$ should be have roughly like constants at scales $|I|/n$, at least in the bulk of the interval (and with errors measured relative to the size of the polynomial on $I$). So if $|I|/n$ can be made to go to infinity, one can hope to force the polynomial to be constant, as is done in my argument here. | |
| Jan 19, 2023 at 1:41 | comment | added | Salini Mendisi | "we have bounded control on a polynomial on an interval of length much wider than its degree (a situation in which one expects favorable estimates thanks to the uncertainty principle)" - Do you have a reference for this version of the uncertainty principle? Or maybe how this fits into some general uncertainty principle heuristics? | |
| Jan 18, 2023 at 2:16 | comment | added | Isaac | I see. Thank you for point that out. I reformulated my question to be a bit more "realistic". Could you please help me? : mathoverflow.net/questions/438766/… | |
| Jan 18, 2023 at 0:33 | comment | added | fedja | @Isaac Terry's argument does not change and remains equally valid if you mechanically replace every instance of $\ln q$ by $A(q)$ in it. | |
| Jan 18, 2023 at 0:10 | vote | accept | Isaac | ||
| Jan 18, 2023 at 0:10 | comment | added | Isaac | Thank you for your wonderful answer. What I am looking for is a smooth function $f(x)$ whose $q^{th}$ order derivative $f^{(q)}$ is uniformly bounded by some $A(q)^{-q}$ where $A(q) \to \infty$ as $q \to \infty$. I just chose $A(q)=\log q$ but it does not seem to work. Is there such $A(q)$ and nonpolynomial $f(x)$? Could you please help me once more? | |
| Jan 17, 2023 at 23:59 | history | answered | Terry Tao | CC BY-SA 4.0 |