A sampling theorem for transforms with discontinuous kernels

Authors: Mahmoud Annaby¹; Gerhard Freiling²

Source: Applicable Analysis, Volume 83, Number 10, October 2004 , pp. 1053-1075(23)

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Abstract:

We present a sampling representation for the transform where the kernel is a solution of an nth order differential equation ell

(y) =

y. Here the differential expression ell

is defined to be ell

₁ on [-1,0) and ell

₂ on (0,1], ell

₁ and

₂ are in general two different nth order differential expressions. This includes the case when ell

₁ and

₂ are identical with discontinuous coefficients at x = 0. So, in addition to the boundary conditions, the eigenvalue problems in question contain n compatibility conditions. The problem is not necessarily self adjoint, but conditions are imposed on it in which the eigenfunctions are a Riesz basis.

Keywords: Sampling theory; Expansion in eigenfunctions; Riesz bases; Spectral problems with discontinuous coefficients; AMS Subject Classifications: 41A05; 34L10; 34B05

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00036810410001657224

Affiliations: 1: Department of Mathematics Faculty ofScience Cairo University Giza Egypt 2: Universität Duisburg-Essen Fachbereich 11 Mathematik Lotharstrasse. 65 47057 Duisburg Germany

Publication date: 2004-10-01

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