Decay of Fourier coefficients of real analytic functions

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 real-analytic 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
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 self-adjointness of $$\mathcal{L}$$ are taken care of. But you should check that Seeley's argument continues to apply in your case.
• 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. May 18 '20 at 11:24