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Entropy 2012, 14(11), 2192-2226; doi:10.3390/e14112192
Article

Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals

1, 2 and 1,*
Received: 29 August 2012 / Revised: 6 October 2012 / Accepted: 8 October 2012 / Published: 5 November 2012
(This article belongs to the Special Issue Information Theory Applied to Communications and Networking)
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Abstract

The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.
Keywords: sampling formulae; differentiation formulae; non-bandlimited functions; aliasing error; Hilbert transforms; formulae with remainders; derivative-free error estimates; Bernstein’s inequality sampling formulae; differentiation formulae; non-bandlimited functions; aliasing error; Hilbert transforms; formulae with remainders; derivative-free error estimates;  Bernstein’s inequality
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Butzer, P.L.; Schmeisser, G.; Stens, R.L. Shannon’s Sampling Theorem for Bandlimited Signals and Their Hilbert Transform, Boas-Type Formulae for Higher Order Derivatives—The Aliasing Error Involved by Their Extensions from Bandlimited to Non-Bandlimited Signals. Entropy 2012, 14, 2192-2226.

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