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 ᵢ be the eigenfunctions of g with eigenvalues ᵢ² and let f: = ₊䲛 be a finite product of Laplace-Beltrami eigenfunctions. We show that f, ᵢ ₋ℂ (₌) decays exponentially as soon as ᵢ > C ₊䲛 for some constant C depending only on M. Moreover, by using a lower bound on \| f \|₋ℂ (₌), we show that 99\% of the L²-mass of f can be recovered using only finitely many Fourier coefficients.