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

On a closed analytic manifold (M,g), let phi i be the eigenfunctions of Delta g with eigenvalues lambda(2)(i) and let f := Pi phi(kj) be a finite product of Laplace-Beltrami eigenfunctions. We show that < f,phi(i)>(2)(L)(M) decays exponentially as soon as lambda(i) > C Sigma lambda(kj) for some constant C depending only on M. Moreover, by using a lower bound on parallel to f parallel to(2)(L)(M), we show that 99% of the L-2-mass of f can be recovered using only finitely many Fourier coefficients. (c) 2024 Published by Elsevier Inc.