# Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature

@article{Xi2017ImprovedGP, title={Improved Generalized Periods estimates on Riemannian Surfaces with Nonpositive Curvature}, author={Yakun Xi}, journal={arXiv: Analysis of PDEs}, year={2017} }

We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficient of eigenfunctions $e_\lambda$ over a period geodesic $\gamma$ goes to 0 at the rate of $O((\log\lambda)^{-1/2})$, if $0<\nu<c_0\lambda$, given any $0<c_0<1$. No such result is possible for the sphere $S^2$ or the flat torus $\mathbb T^2$. Our proof consists of a further refinement of a recent paper by Sogge, Xi and Zhang on the geodesic period integrals ($\nu=0… Expand

#### 3 Citations

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

- Mathematics
- Forum Mathematicum
- 2021

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions eλe_{\lambda} over a closed smooth… Expand

Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

- Physics, Mathematics
- The Journal of Geometric Analysis
- 2018

We show that for a smooth closed curve $$\gamma $$γ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $$e_\lambda $$eλ and $$e_\mu $$eμ restricted to $$\gamma… Expand

Fourier coefficients of restrictions of eigenfunctions

- Physics, Mathematics
- 2020

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… Expand

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