Applications of M.G. Krein's Theory of Regular Symmetric Operators to Sampling Theory

Details Applications of M.G. Krein's Theory of Regular Symmetric Operators to Sampling Theory Applications of M.G. Krein's Theory of Regular Symmetric Operators to Sampling Theory

2007-01-25

arxiv.org/abs/math/0701756v2

Mathematics

Spectral Theory

15 pages; v2: minor changes in abstract, addition of PACS numbers, changes in some keywords, some few changes in the introduct

Scientific paper

10.1088/1751-8113/40/31/017

The classical Kramer sampling theorem establishes general conditions that allow the reconstruction of functions by mean of orthogonal sampling formulae. One major task in sampling theory is to find concrete, non trivial realizations of this theorem. In this paper we provide a new approach to this subject on the basis of the M. G. Krein's theory of representation of simple regular symmetric operators having deficiency indices (1,1). We show that the resulting sampling formulae have the form of Lagrange interpolation series. We also characterize the space of functions reconstructible by our sampling formulae. Our construction allows a rigorous treatment of certain ideas proposed recently in quantum gravity.

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