On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds

Abstract

On a closed analytic manifold $(M,g)$, let ${\varphi }_i$ be the eigenfunctions of ${\Delta }_g$ with eigenvalues ${\lambda }_i^2$ and let $f:=\prod {\varphi }_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that ${⟨f,{\varphi }_i⟩}_{L^2(M)}$ decays exponentially as soon as ${\lambda }_i>C\sum {\lambda }_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\Vert f{\Vert }_{L^2(M)}$, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients.