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)
|. Discussions are given also on an extension of the Szegö theorem for trigonometric series, which Boas mentioned in the same paper.
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