I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, generating an orthonormal basis of eigenfunctions $\varphi_k$ in $L^2(\Omega)$. Given a function $f\in C^{0}(\overline{\Omega})$ which is realanalytic inside $\Omega$, what can be said in general about the decay of the Fourier coefficients $f_k=(f,\varphi_k)_{L^2}$ of $f$ with respect the basis $\varphi_k$? Thanks a lot!
If the domain $\Omega$ were a closed analytic manifold and $\mathcal{L}$ were elliptic of order $m$ with analytic coefficients, the relevant result would be due to
Seeley, R. T., Eigenfunction expansions of analytic functions, Proc. Am. Math. Soc. 21, 734738 (1969). ZBL0183.10102.
Theorem. Denoting by $\lambda_k$ the eigenvalue of $\varphi_k$, $f$ is analytic iff $$\sup_k s^{\sqrt[m]{\lambda_k}} f_k < \infty$$ for some $s>1$.
I suppose that a similar result should apply for a domain $\Omega \subset \mathbb{R}^n$, once the boundary conditions and the selfadjointness of $\mathcal{L}$ are taken care of. But you should check that Seeley's argument continues to apply in your case.

$\begingroup$ Thank you very much! An exponential decay is indeed something which I expected. I will check Seeley's argument but as far as I see from the last page the criterion is appliable on any (self adjoint) elliptic operator $\mathcal{L}$ of any order. $\endgroup$ May 18 '20 at 11:24