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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
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Author to whom correspondence should be addressed.
Academic Editor: Hari M. Srivastava
Axioms 2015, 4(1), 71-83; https://doi.org/10.3390/axioms4010071
Received: 14 November 2014 / Revised: 30 December 2014 / Accepted: 19 January 2015 / Published: 26 January 2015
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
MDPI and ACS Style

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

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