Whittaker-Shannon sampling theorem

One procedure for recovering the continuous (band-limited) function exactly is provided by the Whittaker-Shannon sampling theorem, which is expressed by the equation... 

A function that is compact in momentum space is equivalent to the band-limited Fourier transform of the function. Confinement of such a function to a finite volume in phase space is equivalent to a band-limited function with finite support. (The support of a function is the set for which the function is nonzero.) The accuracy of a representation of this function is assured by the Whittaker-Kotel nikov-Shannon sampling theorem (29-31). It states that a band-limited function with finite support is fully specified, if the functional values are given by a discrete, sufficiently dense set of equally spaced sampling points. The number of points is determined by Eq. (26). This implies that a value of the function at an intermediate point can be interpolated with any desired accuracy. This theorem also implies a faithful representation of the nth derivative of the function inside the interval of support. In other words, a finite set of well-chosen points yields arbitrary accuracy. 

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