a single band, so that the passband technique does not apply. In other words, the frequency spectrum is sparse. Traditionally, the necessary sampling rate is thus 2B. Using compressed sensing techniques, the signal could be perfectly reconstructed if it is sampled at a rate slightly lower than 2EB. With this approach, reconstruction is no longer given by a formula, but instead by the solution to a linear optimization program. Another example where sub-Nyquist sampling is optimal arises under the additional constraint that the samples are quantized in an optimal manner, as in a combined system of sampling and optimal lossy compression. This setting is relevant in cases where the joint effect of sampling and quantization is to be considered, and can provide a lower bound for the minimal reconstruction error that can be attained in sampling and quantizing a random signal. For stationary Gaussian random signals, this lower bound is usually attained at a sub-Nyquist sampling rate, indicating that sub-Nyquist sampling is optimal for this signal model under optimal quantization. Historical background The sampling theorem was implied by the work of Harry Nyquist in 1928, in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result and discussed the sinc-function impulse response of a band-limiting filter, via its integral, the step-response sine integral; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English). The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon. The mathematician E. T. Whittaker published similar results in 1915, J. M. Whittaker in 1935, and Gabor in 1946 ("Theory of communication"). In 1948 and 1949, Claude E. Shannon published the two revolutionary articles in which he founded the information theory. In Shannon 1948 the sampling theorem is formulated as “Theorem 13”: Let f(t) contain no frequencies over W. Then where . It was not until these articles were published that the theorem known as “Shannon’s sampling theorem” became common property among communication engineers, although Shannon himself writes that this is a fact which is common knowledge in the communication art. A few lines further on, however, he adds: "but in spite