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The question starts with the well known facts that: if $f$ is a smooth function on $S^1$, then its Fourier series converges to it in smooth topology.

This must be true in more general setting. I have the following generalizations in mind.

  1. let $(M,g)$ be a closed Riemannian manifold, let $\phi_n$ be the eigenfunctions of the Laplace operator, counting multiplicity and normalized. If $c_n=\int_M f \phi_n dV_g$, does the series $\sum_n c_n \phi_n$ converges to $f$ in smooth topology?

  2. Let $E$ be some vector bundle over $(M,g)$ with a reasonable metric and connection. There are some elliptic operators whose eigensections form an orthonormal basis of $L^2$. Say, $\phi_n$ is a basis, normalized. Repeat the above question for a smooth section $f$ of $E$. In particular, what about differential forms, i.e. $E=\Lambda^k(M)$, and the Hodge Lapalce operator?

My guess is that these are true, but can not find any reference. A proof would require good estimates on variaous derivatives of the eigenfunctions (and sections in the second case), which I do not know how to prove.

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    $\begingroup$ Does the answer to this question answer your first question? (The convergence considered there is in $L^\infty$, but I think the argument ought to work in $C^k$ also.) $\endgroup$
    – Leo Moos
    Commented Mar 7, 2023 at 11:09
  • $\begingroup$ Thank you. I think, following your advice, I can give a proof to both the above claims. $\endgroup$
    – Hao Yin
    Commented Mar 7, 2023 at 11:48
  • $\begingroup$ The statement of 1, holds for the Laplace operator on a compact Riemannian manfold. The crucial fact is that the eigenvalues go to infinity like a power ($>1$) of $n$ and so it is valid in many situations (Schrödinger operators, Laplacian operator on a bounded manifold with boundary, given suitable boundary conditions--Dirichlet or Neumann). The growth conditions on the eigenvalues can be obtained by classical results in the interesting special cases or by variants of the Weyl inequality in the more abstract situations. $\endgroup$
    – terceira
    Commented Mar 7, 2023 at 12:43
  • $\begingroup$ See Thm. 10.4.20 of this book www3.nd.edu/~lnicolae/Lectures.pdf $\endgroup$
    – Liviu Nicolaescu
    Commented Mar 7, 2023 at 15:02

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