Recently, it was shown with the help of Fourier analysis that by incorporating a Gaussian multiplier into the truncated classical sampling series, one can approximate bandlimited signals of finite energy with an error that decays exponentially as a function of the number of involved samples. Based on complex analysis, we show for a slightly modified operator that this approximation method applies not only to bandlimited signals of finite energy, but also to bandlimited signals of infinite energy, to classes of non-bandlimited signals, to all entire functions of exponential type (including those whose samples increase exponentially), and to functions analytic in a strip and not necessarily bounded. Moreover, the method extends to non-real argument. In each of these cases, the use of 2N + 1 samples results in an error bound of the form Me−αN, where M and α are positive numbers that do not depend on N. The power of the method is illustrated by several examples.