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.
Highlights
Pólya and Szegö has taken up the Bernstein theorem for trigonometric series in their famous book [1]
We assume that g(t) is continuous in [−R, R], and h(t) is integrable in (−R, R) and has the Fourier series, so that h(t) is expressed as follows: h(t) = lim cn einπt/R
) for n ∈ Z are the Fourier coefficients of F (t), and
Summary
Pólya and Szegö has taken up the Bernstein theorem for trigonometric series in their famous book [1]. 11]), the theorem is given as follows. The generalized theorems are concerned with a function f (x) which can be expressed as follows:. When we see the formula (3), we expect that there must exist a sampling theorem which is applicable to the function f (x), and (3) must be obtained by its differentiation. This is the motivation of the present paper. If we denote the total variation of α(t) by T , (2) shows |f (x)| 5 T for all x ∈ R
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
