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On sampling associated with singular Sturm-Liouville eigenvalue problems: The limit-circle case. (English) Zbl 1041.34078

The aim of this paper is to derive a general sampling theorem for solutions of second order singular eigenvalue problems with limit-circle endpoint(s). More precisely, the authors consider the Sturm-Liouville problem involving the equation \[ -y''+q(x)y=\lambda y\tag{\(*\)} \] and the condition \(y(a)\cos\alpha+y'(a)\sin\alpha=0\), \(\alpha\in[0,\pi)\). Assuming that the limit-circle case holds at \(\infty\), another boundary condition at \(\infty\) is derived using Fulton’s approach, see C. T. Fulton [Trans. Am. Math. Soc. 229, 51–63 (1977; Zbl 0358.34021)]. Then, for any \(\lambda\in\mathbb{C}\), sampling expansions of the solutions to (\(*\)) satisfying the boundary conditions are given. As applications, sampling expansions for the Legendre and Bessel functions as well as for the associated integral transforms are presented.
Reviewer: Pavel Rehak (Brno)

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B24 Sturm-Liouville theory
94A20 Sampling theory in information and communication theory

Citations:

Zbl 0358.34021
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References:

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