We obtain approximation bounds for products of quasimodes for the Laplace-Beltrami operator, on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes $uv$ by a low-degree vector space $B_{ n}$, and we prove that the size of the space $\dim(B_{ n})$ is small. In our paper, we first study bilinear quasimode estimates of all dimensions $d = 2, 3$, $d = 4,5$ and $d \ge 6$, respectively, to make the highest frequency disappear from the right hand. Furthermore, the result of the case $\lambda=\mu$ of bilinear quasimode estimates improves $L^{ 4}$ quasimodes estimates of Sogge-Zelditch in \cite{ sogge6} when $d \ge 8$. And on this basis, we give approximation bounds in $H^{ -1}$ norm. We also prove approximation bounds for the products of quasimodes in $L^{ 2}$ norm using the results of $L^{ p}$-estimates for quasimodes in \cite{ sogge3}. We extend the results of Lu-Steinerberger in \cite{ lu} to quasimodes.
Categories: math.AP