Timeline for Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Jan 5, 2020 at 17:49 | comment | added | Giorgio Metafune | I have only a comment concerning $f(x)=\frac{1}{|x|}$ in the ball $B$. As pointed out, the eigenfunctions consist of Bessel functions $u_k$ multiplied by spherical harmonics $P_n$ but since $f$ is radial all coefficients vanish except those arising from $u_k P_0$. Then the expansion is a Fourier Bessel one (1d) and there are classical results saying that a Fourier Bessel expansion converges pointwise to $f$ if and only if the Fourier series of $f$ converges to $f$. These can be found in Chapter 4 of the book Eigenfunction Expansions by Titchmarsh. However, I did not check the details. | |
| Nov 21, 2019 at 17:35 | comment | added | paul garrett | About (especially uniform) pointwise convergence of eigenfunction expansions: for example, on $[0,2\pi]^n$, the $L^2$ Sobolev spaces $H^s$ are inside $C^o$ for $s\ge {n\over 2}+\epsilon$ for $\epsilon>0$. But this may not be exactly what you're asking... | |
| Nov 21, 2019 at 13:41 | comment | added | user39481 | @paulgarret: Could you explain your intuition about 'the effect gets worse as dimensions go up"? | |
| Nov 20, 2019 at 1:34 | history | edited | user39481 | CC BY-SA 4.0 |
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| Nov 20, 2019 at 1:27 | comment | added | user39481 | @Christian Remling: Sorry, I should replace 'pointwise' with 'a.e. pointwise' in my questions, indeed the Carleson theorem holds only for a.e. pointwise convergence. I am interested if something analogous can be said in our case. Thank you to point my mistake out. | |
| Nov 19, 2019 at 23:12 | comment | added | paul garrett | Not-so-smooth points do affect the convergence of eigenfunction expansions at other points, and the effect gets worse as dimensions go up. But this is not "an answer", because I do not have counter-examples to hand. Asking for a generalization of Carleson's theorem is probably toooo optimistic, anyway, though, again, I don't know for sure. | |
| Nov 19, 2019 at 21:45 | history | edited | user39481 | CC BY-SA 4.0 |
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| Nov 19, 2019 at 21:26 | history | edited | user39481 | CC BY-SA 4.0 |
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| Nov 19, 2019 at 20:42 | history | edited | YCor |
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| Nov 19, 2019 at 20:40 | review | First posts | |||
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| Nov 19, 2019 at 20:38 | history | asked | user39481 | CC BY-SA 4.0 |