Abstract

Some bibliographic comments are made concerning the genesis of sampling expansions for periodic signals that have only a finite number of terms (harmonic-limited) in their Fourier series developments; such expansions have been used recently by Stark [ J. Opt. Soc. Am. 69, 1519 ( 1979)] to simplify image restoration in computerized tomography applications. The results normally attributed to Cahn [in H. S. Black, Modulation Theory ( Van Nostrand, New York, 1953)] and Maeda [ J. Inst. Electron. Commun. Eng. Jpn. 50, 1472 ( 1967)] for uniformly and nonuniformly spaced samples, respectively, taken over one period of the signal are shown to have been derived by Cauchy [ C. R. Acad. Sci. 12, 283 ( 1841)] in an 1841 paper. A simple proof is then given for a generalized class of harmonic-limited signals for which the finite number of samples is nonuniformly spaced; this more general result contains the results of Cahn, of Maeda, and of Stark as special cases.

© 1984 Optical Society of America

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