  Let $(M,g)$ be a smooth, compact, Riemannian manifold and $\{\phi_h\}$ a
sequence of $L^2$-normalized Laplace eigenfunctions on $M$. For a smooth
submanifold $H\subset M$, we consider the growth of the restricted
eigenfunctions $\phi_h|_H$ by testing them against a sequence of functions
$\{\psi_h\}$ on $H$ whose wavefront set avoids $S^*H$. That is, we study what
we call the generalized Fourier coefficients: $\langle
\phi_h,\psi_h\rangle_{L^2(H)}$. We give an explicit bound on these coefficients
depending on how the defect measures for the two collections of functions
$\phi_h$ and $\psi_h$ relate. This allows us to get a little$-o$ improvement
whenever the collection of recurrent directions over the wavefront set of
$\psi_h$ is small. To obtain our estimates, we utilize geodesic beam
techniques.

Analysis of PDEs