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Published in:
2019 | OriginalPaper | Chapter
Shannon Sampling and Weak Weyl’s Law on Compact Riemannian Manifolds
Author : Isaac Z. Pesenson
Published in: Analysis and Partial Differential Equations: Perspectives from Developing Countries
Publisher: Springer International Publishing
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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 \(\mathbf{M}\).