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Shannon Sampling and Weak Weyl’s Law on Compact Riemannian Manifolds

  • Isaac Z. PesensonEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 275)

Abstract

The well known Weyl’s asymptotic formula gives an approximation to the number \(\mathcal {N}_{\omega }\) of eigenvalues (counted with multiplicities) on an interval \([0,{\,}\omega ]\) of an elliptic second-order differential self-adjoint non-negative operator on a compact Riemannian manifold \(\mathbf{M}\). In this paper we approach this question from the point of view of Shannon-type sampling on compact Riemannian manifolds. Namely, we give a direct proof that \(\mathcal {N}_{\omega }\) is comparable to cardinality of certain sampling sets for the subspace of \(\omega \)-bandlimited functions on \(\mathbf{M}\).

Keywords

Compact riemannian manifolds Spectral geometry Weyl’s law 

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

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