Let $M$ be a closed (i.e. compact without boundary) manifold with positive sectional curvature. Let us denote the $i$-th eigenfunctions of the Laplacian operator by $\phi_i$.
During some computations I encountered the expression $\sum_{i=0}^{\infty}(\phi_i(x))^2$. I believe this sum should converge at least for almost every $x\in M$, but I have no way to prove it or disprove it. Is there any general estimates on this type of expression for the eigenfunctions?
I include the restriction on the sectional curvature because this is what happens in my example but I don't know if it is crucial or not.
On a circle, with inner product $\int_{0}^{2\pi}fg dx$, the normalized eigenfunctions are $$ \frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt{\pi}}\cos(nx),\frac{1}{\sqrt{\pi}}\sin(nx),\;\; n=1,2,3,\cdots. $$ In this case, the sum in question does not converge: $$ \frac{1}{2\pi}1^2+\sum_{n=1}^{\infty}\frac{1}{\pi}(\cos^2(nx)+\sin^2(nx)) $$