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Axioms 2015, 4(1), 71-83; doi:10.3390/axioms4010071

Boas’ Formula and Sampling Theorem

1
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
2
College of Engineering, Nihon University, Koriyama 963-8642, Japan
*
Author to whom correspondence should be addressed.
Academic Editor: Hari M. Srivastava
Received: 14 November 2014 / Revised: 30 December 2014 / Accepted: 19 January 2015 / Published: 26 January 2015
View Full-Text   |   Download PDF [241 KB, uploaded 26 January 2015]

Abstract

In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation of the Shannon sampling theorem for a function which is the Fourier transform of a distribution involving the Dirac delta function. It is also shows that, by the argument giving Boas’ formula for the derivative f'(x) of a function f(x), we can derive the corresponding formula for f'''(x), by which we can obtain an upperbound of |f'''(x)+3R2f'(x)|. Discussions are given also on an extension of the Szegö theorem for trigonometric series, which Boas mentioned in the same paper. View Full-Text
Keywords: Shannon sampling theorem; Boas’ formula; generalized Parseval formula; Bernstein theorem; Szegö theorem Shannon sampling theorem; Boas’ formula; generalized Parseval formula; Bernstein theorem; Szegö theorem
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. (CC BY 4.0).

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Morita, T.; Sato, K.-I. Boas’ Formula and Sampling Theorem. Axioms 2015, 4, 71-83.

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