Exact Interpolation of Band-Limited Functions

A general formula is derived for the spectrum of a multiply-periodic, amplitude modulated sequence of pulses. The result is used to show that a function which lies in a frequency band (W0, W0+W) is completely determined by its values at a properly chosen set of points of density 2W. This verifies a supposition commonly accepted in communication theory. The well-known, exact interpolation formula for a function f(t) in a band (O, W) is f(t)=Σnf(n/2W)sinπ(2Wt−n)π(2Wt−n). The function is thus determined by its values at a set of evenly spaced points ½W apart. For a function f(t) in a band (W0, W0+W) it is shown that an exact interpolation formula is f(t)=Σnf(n/W)s(t−n/W)+f(n/W+k)s(n/W+k−t)in which k is subject to weak restrictions and s(t)=cos2π(W0+W)t−(r+1)πWk−cos2π(rW−W0)t−(r+1)πWK2πWtsin(r+1)πWK+cos2π(rW−W0)t−rπWk−cos2πW0t−rπWk2πWtsinrπWk. Thus, the function is determined by its values at a set of points of density 2W, but the points consist of two similar groups with spacing 1/W, shifted with respect to each other.