Let $u \in L^2(M)$ on some closed Riemannian manifold. We can write $$u = \sum_{k \geq 0}(u,\varphi_k)\varphi_k$$ if $\varphi_k$ is some o.n basis of $L^2$ with is orthogonal in $H^1$ (eg. eigenfunctions of appropriate operator). So (I believe) for almost all $x$, $$u(x) = \sum_{k \geq 0}(u,\varphi_k)\varphi_k(x).$$ How does one know if this sum is uniformly convergent on $x \in M$? If say $u \in H^1(M)$, it would be nice to say $$\nabla u(x) = \sum_{k \geq 0}(u,\varphi_k)\nabla \varphi_k(x)$$ and this commutation of operators need uniform continuity of the RHS too. So this is why I want to know.

(this question was posted by someone some time ago in MSE and was unanswered for months but sadly I cannot find a link anymore)

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