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Annaby, M. H.; Bustoz, J.; Ismail, M. E. H.

On sampling theory and basic Sturm-Liouville systems. (English) Zbl 1128.39014

J. Comput. Appl. Math. 206, No. 1, 73-85 (2007).
The authors consider the basic Sturm-Liouville problem involving the second order \(q\)-difference equation with \(q\)-Jackson difference operator. The main purpose of this paper is to derive some \(q\)-analogs of the results for the “classical” Sturm-Liouville problem, i.e., involving the differential equation. They establish two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson’s type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expresssed in terms of basic Green’s function of the basic Sturm-Liouville systems. Examples involving basic sine and cosine transforms are given.
Reviewer: Pavel Rehak (Brno)

Cited in 14 Documents

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
34B24 Sturm-Liouville theory
47B39 Linear difference operators
39A12 Discrete version of topics in analysis
39A70 Difference operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
62D05 Sampling theory, sample surveys

Keywords:

sampling theory; Green function; basic Sturm-Liouville problem; second order \(q\)-difference equation; \(q\)-Jackson difference operator; integral transforms; basic sine and cosine transforms
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References:

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