Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^2(\Omega)$-orthonormal basis made of eigenfunctions ordered by the corresponding eigenvalues. Actually, using rigged Hilbert spaces, $\{w_k\}_{k=1}^{+\infty}$ (up to renormalization) is an orthonormal basis in $H^1(\Omega)$. Hence for any $f\in H^1(\Omega)$ we have $$\sum_{k=1}^{N}a_k w_k(x)\to f(x) \quad \text{in}\quad H^1(\Omega),\quad \text{where}\quad a_k:=\int_{\Omega}w_k(x) f(x) \,dx.$$

Furthermore it is well-known that $w_k(x)\in C^\infty(\bar{\Omega})$ and is real analytical in $\Omega$.

Suppose that $\Omega \subset \mathbb{R}^3$ and $\Omega$ contains the origin.

Main questions.

  1. Can we hope that the convergence of $\sum_{k=1}^N a_k w_k(x)$ to $f(x)=\frac{1}{|x|}$ is pointwise a.e. pointwise in any compact set $K\subset \Omega\setminus \{0\}$? In such case, what can we say about the asymptotic behaviour of the series $S(x)=\sum_{k=1}^{+\infty}a_kw_k(x)$ near $x=0$ (suspecting that $S(x) \sim \frac{1}{|x|} )$?
  2. If $\Omega=B_1(0)$, using that the eigenfunctions of the Laplacian are the Bessel functions multiplied by spherical armonics, can we assert something more in this special case?

A general problem for $f(x)$ in $\Omega \subset \mathbb{R}^n$: Can we hope that the converge inside $\Omega$ is pointwise a.e. pointwise where $f(x)$ is continuous? This sounds like a Carleson theorem for a general (not only rectangular) domain with a non-trigonometric basis. I suppose that this is an open problem. Related question

Any related idea, counterexample or reference is welcome!

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    $\begingroup$ 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. $\endgroup$ Nov 19 '19 at 23:12
  • $\begingroup$ @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. $\endgroup$
    – user39481
    Nov 20 '19 at 1:27
  • $\begingroup$ @paulgarret: Could you explain your intuition about 'the effect gets worse as dimensions go up"? $\endgroup$
    – user39481
    Nov 21 '19 at 13:41
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    $\begingroup$ 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... $\endgroup$ Nov 21 '19 at 17:35
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    $\begingroup$ 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. $\endgroup$ Jan 5 '20 at 17:49

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