Applications of Krein's theory of regular symmetric operators to sampling theory

Abstract

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. * Mathematics Subject Classification(2000): 41A05, 46E22, 47B25, 47N50, 47N99, 94A20. †

Symmetric Operators and Reproducing Kernel Hilbert Spaces

“…Krein calls the gauge u quasi-regular if the set of all φ ∈ H for whichφ is analytic in a region containing R is dense in H. In particular, if N u ∩ R = ∅ then u is a quasi-regular gauge. Krein asserts that one can always choose u so that N u ∩ R = ∅ [8,11]. Since the original paper in which this is proven is in Russian, and the author is not aware of a translation, here is an original and simple proof of this fact:…”

“…Let δ z := ϕ z ϕ z ,u . It is not difficult to see that this is a meromorphic function on C with simple poles at points of the set N u (observe that the poles of ϕ z at the points of σ (B ) coincide with those of ϕ z , u ) [8]. Furthermore, we have the following: It is not hard to see, with the aid of the above lemma, that δ z ∈ Ker(B * − z) for every z ∈ C\N u .…”

“…In [8], the authors exploit Krein's theory to show that if u is chosen so that N u ∩R = ∅, then the reproducing kernel Hilbert spaces u H ⊂ L 2 (R, dσ Q ) have the sampling property. Their result can be seen as a simple consequence of the following original and simple theorem.…”

“…The fact that any symmetric operator B with the sampling property is unitarily equivalent to the operator of multiplication by the independent variable in a reproducing kernel Hilbert space with the sampling property was first proven by Kempf in [3], prior to [8], and without the use of Krein's theory of spectral representations of such symmetric operators [7]. Kempf's approach to proving this result stems from Theorem 2.…”

“…It is known that if B is a symmetric operator which is simple, regular, has deficiency indices (1,1), and is densely defined on some domain Dom(B) ⊂ H of a separable Hilbert space H, that there exists a unitary transformation U which maps H onto a Hilbert space H ⊂ L 2 (R, dμ), which is a reproducing kernel Hilbert space of meromorphic functions with the sampling property [3,7,8]. Furthermore, under this unitary transformation, M := U BU * , the image of B, acts as multiplication by the independent variable on the dense domain U Dom(B) ⊂ H .…”