Boas’ Formula and Sampling Theorem
AbstractIn 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
Share & Cite This Article
Morita, T.; Sato, K.-I. Boas’ Formula and Sampling Theorem. Axioms 2015, 4, 71-83.
Morita T, Sato K-I. Boas’ Formula and Sampling Theorem. Axioms. 2015; 4(1):71-83.Chicago/Turabian Style
Morita, Tohru; Sato, Ken-ichi. 2015. "Boas’ Formula and Sampling Theorem." Axioms 4, no. 1: 71-83.