INFORMATION AND CONTROL 34, 55-65 (1977) A Sampling Theorem for Duration-Limited Functions with Error Estimates P. L. BUTZER AND W'. SPLETTSTOSSER Lehrstuhl A fiir Mathematik, Rheinisch-Westfdlische Teehnische Hochschule Aachen, Aachen, West Germany In contrast to the classical Shannon sampling theorem, signal functions are considered which are not band-limited but duration-limited. It is shown that these functions can be approximately represented by a discrete set of samples. The error is estimated that arises when only a finite number of samples is selected. 1. INTRODUCTION The well-known sampling theorem set up by Whittaker (1915), Kotel'nikov (1933), and Shannon (1940/1949), plays a basic role in communication, control, and data processing. It states that every real-valued deterministic signal f(t) that is band-limited to [--zrW, ~W] can be reconstituted by ~.f(~_)sinTr(Wt--k) ~=_~ .(wt - k) (t ~ ~). (1.1) Since by Fourier transform theory band-limited signals must necessarily extend for an infinite time and, conversely, a signal which only exists for a certain time must have a spectrum (Fourier transform) which contains fre- quencies up to infinity, Wunsch (1963) and Kioustelidis (1969) have considered another model for a sampling theorem, one which is just as practical, namely to reconstruct duration- (or time-) limited functions from samples. Whereas in the Wunsch paper no proof of the corresponding result is supplied, arguments in the proof of the Kioustelidis paper are not complete; see also the review of the latter paper. 1 Furthermore, neither author mentioned hypotheses upon the signal function under which their sampling theorem would hold. Nevertheless, the Kioustelidis paper inspired the present authors to establish a sampling theorem for continuous duration-limited functions whose spectrum f^(v) = (1/(27r) 1/2) f(t) e-i~* at (v c R) (1.2) 1 Doetsch, G. (1970), Math. Reviews 40, 4716. 55 Copyright © 1977 by Academic Press, Inc. All rights of reproduction in any form reserved. ISSN 0019-9958