  Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact
Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let
$\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the
induced metric. We obtain joint asymptotics for the Fourier coefficients \[
  \langle \gamma_H e_j, \psi_k \rangle_{L^2(H)} = \int_H e_j \overline \psi_k
\, dV_H, \] of restrictions $\gamma_H e_j$ 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 $\{(\mu_k, \lambda_j)\}_{j,k - 0}^{\infty}$ of the
(square roots of the) Laplacian $\Delta_M$ on $M$ and the Laplacian $\Delta_H$
on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions
include (1) the truncated cone $\mu_k/\lambda_j \in [a,b] \subset (0,1)$ and
$\lambda_j \leq \lambda$, and (2) the slowly thickening strip $|\mu_k -
c\lambda_j| \leq w(\lambda)$ and $\lambda_j \leq \lambda$, where $w(\lambda)$
is monotonic and $1 \ll w(\lambda) \lesssim \lambda^{1 - 1/n}$. Key tools for
obtaining these asymptotics include the composition calculus of Fourier
integral operators and a new multidimensional Tauberian theorem.
