Timeline for Approximation of Hölder functions by Fourier series
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Feb 27, 2023 at 18:39 | comment | added | Willie Wong | @GiorgioMetafune thanks, that's good enough. The first one you listed is exactly what I meant by "$C^l$ embeds into $W^{l,2}$ on compacts" in my previous comment. I also suspected there's another proof using finer details of how spherical harmonics behave; it is good to see that confirmed in your second answer. | |
| Feb 27, 2023 at 18:27 | comment | added | Giorgio Metafune | @WillieWong The other one I know uses more on spherical harmonics, zonals, sup-norm estimates. If you need, I can write down it. | |
| Feb 27, 2023 at 18:25 | comment | added | Giorgio Metafune | @WillieWong One argument is short but uses elliptic regularity. For $\psi \in C^{2\ell}$ write $\psi=\sum_k (\psi, \psi_k)\psi_k$ in $L^2$ and $$\Delta^\ell \psi=\sum_k (\Delta^\ell \psi, \psi_k)\psi_k=\sum_k (\psi, \Delta^\ell \psi_k)\psi_k=\sum_k \lambda^\ell_k ( \psi, \psi_k)\psi_k=\Delta^\ell (\sum_k (\psi, \psi_k)\psi_k).$$ This gives that the series yielding $\psi$ converges in $H^{2\ell}$, by elliptic regularity, and by Sobolev embedding uniformly, for $\ell$ large. | |
| Feb 27, 2023 at 13:44 | comment | added | Giorgio Metafune | @WillieWong I will answer later in the late afternoon....(UTC+1) | |
| Feb 26, 2023 at 23:50 | comment | added | Willie Wong | @GiorgioMetafune: on the sphere, how does one argue that if $f\in C^l$ for large $l$ then the projections $\langle f, \psi_k\rangle$ is small for large $k$? Are you just using that $C^l$ embeds into $W^{l,2}$ on compacts? | |
| Feb 26, 2023 at 18:10 | comment | added | user500030 | @Giorgio-Metafune, I thought about Sobolev theorems, but settled on $\Vert grad \psi_k\Vert^2= \lambda_k$, althogh it was naturally to continue with $\Vert\Delta \psi_k\Vert^2=\lambda_k^2$ and so on.... | |
| Feb 26, 2023 at 15:31 | comment | added | Giorgio Metafune | Please, quote my name if you reply to me, to let me receive a notification. If $-\Delta \psi_k=\lambda_k \psi_k$ then $(-\Delta)^\ell \psi_k=\lambda_k^\ell \psi_k$ and by elliptic estimates the norm of $\psi_k$ in the Sobolev space $H^{2\ell}$ is boounded by $\lambda_k^\ell$. Since $\lambda_k \approx k^{2/N}$ choosing $2\ell >N/2$ by Sobolev embedding you get a rough estimate in the sup norm. I am counting eigenvalues and eigenfunctions with multiplicity (in the comment before all spherical harmonics of degrre $k$ belong to the same $\lambda_k$). | |
| Feb 26, 2023 at 12:19 | comment | added | user500030 | That's the most interesting thing to me: for what $p$ one have $\sup\limits_Q |\psi_k(x)|\approx k^p$ ? if it’s already been found by someone | |
| Feb 26, 2023 at 8:36 | comment | added | Giorgio Metafune | This is not true in higher dimension, basicly because the sequence of eigenfunctions need not be uniformly bounded. An example of this situation is the Laplace Beltrami on the unit sphere $S^{N-1}$ of $\mathbb R^N$. The eigenfunctions are the spherical harmonics, if $k$ is the degree, the supnorm is like $k^{N-2}$ and the expansion converges in the supnorm if $f \in C^l$ with $l>(N-1)/2$. | |
| Feb 26, 2023 at 5:22 | comment | added | user500030 | I cerrected. Thank you. | |
| Feb 26, 2023 at 4:46 | history | edited | user500030 | CC BY-SA 4.0 |
added 7 characters in body
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| Feb 25, 2023 at 23:26 | history | edited | Willie Wong |
edited tags
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| Feb 25, 2023 at 21:39 | history | edited | LSpice | CC BY-SA 4.0 |
Holder -> Hölder
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| Feb 25, 2023 at 21:37 | answer | added | Bazin | timeline score: 1 | |
| Feb 25, 2023 at 14:19 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
typo in the title
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| Feb 25, 2023 at 14:15 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing and formatting
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| S Feb 25, 2023 at 11:39 | review | First questions | |||
| Feb 25, 2023 at 12:19 | |||||
| S Feb 25, 2023 at 11:39 | history | asked | user500030 | CC BY-SA 4.0 |