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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the approximate normality of eigenfunctions of the Laplacian


Author: Elizabeth Meckes
Journal: Trans. Amer. Math. Soc. 361 (2009), 5377-5399
MSC (2000): Primary 58J50; Secondary 60F05, 58J65
Published electronically: May 4, 2009
MathSciNet review: 2515815
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Abstract | References | Similar Articles | Additional Information

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

$\displaystyle d_{TV}(f(X),Z)\le\frac{2}{\mu}\mathbb{E}\big\vert\Vert\nabla f(X)\Vert^2-\mathbb{E}\Vert\nabla f(X) \Vert^2\big\vert.$

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.


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Additional Information

Elizabeth Meckes
Affiliation: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44122
Email: elizabeth.meckes@case.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04661-3
PII: S 0002-9947(09)04661-3
Keywords: Eigenfunctions, Laplacian, value distributions, spherical harmonics, random waves, Stein's method, normal approximation
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.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.