quoting Shannon's original paper: Let be the spectrum of Then because is assumed to be zero outside the band If we let where is any positive or negative integer, we obtain: On the left are values of at the sampling points. The integral on the right will be recognized as essentially the nth coefficient in a Fourier-series expansion of the function taking the interval to as a fundamental period. This means that the values of the samples determine the Fourier coefficients in the series expansion of Thus they determine since is zero for frequencies greater than B, and for lower frequencies is determined if its Fourier coefficients are determined. But determines the original function completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function completely. Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via sinc functions, what we now call the Whittaker–Shannon interpolation formula as discussed above. He does not derive or prove the properties of the sinc function, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between rect (the rectangular function) and sinc was well known. Let be the nth sample. Then the function is represented by: As in the other proof, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to bandlimited stationary random processes. Notes Application to multivariable signals and images The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to time-dependent signals and is normally formulated in that context. However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables. Grayscale images, for example, are often represented as two-dimensional arrays (or matrices) of real numbers representing the relative intensities of pixels (picture elements) located at the intersections of row and column sample locations. As a result, images require two independent variables, or indices, to specify each pixel uniquely—one for the row, and one for the column. Color images typically consist of a composite of three separate grayscale images, one to represent each of the three primary colors—red, green, and blue, or RGB for short. Other colorspaces using 3-vectors for colors include HSV, CIELAB, XYZ, etc. Some colorspaces such as