# Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

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!

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