For f(t) a real-valued signal band-limited to [Formula Omitted] 1) and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error when f(t) is approximated at a generic time t by an appropriate selection of N1 + N2 + 1 terms from its Shannon sampling series expansion, the latter expansion being associated with the full band [-π, π] and thus involving samples of f taken at the integer points. Results are presented for two cases: 1) the Fourier transform F(ω) is such that |F(ω)|2is integrable on [-πr, πr] (finite energy case), and 2) |F(ω)| is integrable on [- πr, πr]. In case 1) it is shown that the truncation error magnitude is bounded above by [Formula Omitted] where E denotes the signal energy and 9 is independent of Nr, N, and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above by [Formula Omitted] where M is the maximum signal amplitude and h is independent of N1, N2 and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values of r close to unity. © 1969 IEEE
