{'fraction_non_alphanumeric': 0.04610318331503842, 'fraction_numerical': 0.006586169045005488, 'mean_word_length': 4.772151898734177, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Boas' Formula and Sampling Theorem

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 Szego theorem for trigonometric series, which Boas mentioned in the same paper.