Timeline for Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces
Current License: CC BY-SA 3.0
6 events
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| Aug 14, 2014 at 13:52 | comment | added | user14166 | Thanks a lot for your answer! I was beginning to think this would be left forgotten. I agree with Dirk though that this doesn't give a heuristic interpretation. I like @Goulifet 's stochastic result, but this doesn't distinguish $B_{p,q}$ spaces for $q \neq \infty$. What really happens when we increase '$q$-ness', but not all the way to infinity? | |
| Aug 14, 2014 at 11:46 | comment | added | Dunham | I have not verified the details, but I suspect that we could distinguish Besov spaces of different $q$ parameters using functions whose Fourier transforms behave like \begin{equation*} \widehat{f}(\omega) \sim \frac{1}{|w|^{\beta} (\log(|\omega|))^{\alpha}} \end{equation*} for $|\omega|$ sufficiently large. | |
| Aug 14, 2014 at 11:15 | comment | added | Goulifet | To answer your last question, this situation happens for stochastic processes. It's typically the case for the Gaussian white noise on the torus, that is in $B^{-1/2}_{p,\infty}$ almost surely but almost surely not in $B^{-1/2}_{p,q}$ for $q<\infty$. (Source: arxiv.org/abs/1010.6219) | |
| Aug 14, 2014 at 9:23 | comment | added | Dirk | That's all well but I don't get a heuristic interpretation of $q$ out of it. Sure, the embeddings are clear but in what sense is $q$ some "fine tuning"? Probably the case $q=\infty$ is interesting: Although the spaces $B^s_{p,\infty}$ are not Sobolev spaces one somehow sees a link to Sobolev regularity. Do you know an example of a function in $B^s_{p,\infty}$ that does not lie in $B^s_{p,p}$? Maybe for $p=2$? | |
| Aug 14, 2014 at 9:19 | history | edited | Dirk | CC BY-SA 3.0 |
only fixed umlauts
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| Aug 14, 2014 at 8:54 | history | answered | Dunham | CC BY-SA 3.0 |