On the approximate normality of eigenfunctions of the Laplacian
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Abstract:
The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If $X$ is a random point on a manifold $M$ and $f$ is an eigenfunction of the Laplacian with $L^2$-norm one and eigenvalue $-\mu$, then \[ d_{TV}(f(X),Z)\le \frac {2}{\mu }\mathbb {E}\big |\|\nabla f(X)\|^2-\mathbb {E}\|\nabla f(X) \|^2\big |.\] This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.References
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Additional Information
- Elizabeth Meckes
- Affiliation: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44122
- MR Author ID: 754850
- Email: elizabeth.meckes@case.edu
- Received by editor(s): May 17, 2007
- Received by editor(s) in revised form: October 9, 2007
- Published electronically: May 4, 2009
- Additional Notes: This research was supported by fellowships from the ARCS Foundation and the American Institute of Mathematics.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 5377-5399
- MSC (2000): Primary 58J50; Secondary 60F05, 58J65
- DOI: https://doi.org/10.1090/S0002-9947-09-04661-3
- MathSciNet review: 2515815