Fourier coefficients of restrictions of eigenfunctions

Let{e_(j)}be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g).Let H■M be a submanifold and{ψ_(k)}be an orthonormal basis of Laplace eigenfunctions of H with the induced metric.We obtain joint asymptotics for the Fourier coefficients<γHe_(j),ψ_(k)>L^(2)(H)=∫He_(j),ψ_(k)dV_(H)of restrictionsγHe_(j)of e_(j)to H.In particular,we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum{(μ_(k),λ_(j))}^(∞)_(j,k-0)of the(square roots of the)Laplacian△_(M)on M and the Laplacian△_(H)on H in a family of suitably‘thick'regions in R^(2).Thick regions include(1)the truncated coneμ_(k)/λ_(j)∈[a,b]■(0,1)andλ_(j)≤λ,and(2)the slowly thickening strip|μ_(k)-cλ_(j)|≤w(λ)andλ_(j)≤λ,where w(λ)is monotonic and 1■w(λ)≤λ^(1/2).Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.