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 $${\mathrm{f(t)=\Sigma }}_n\text{f(n/2W)}\frac{\sin \text{π(2Wt−n)}}{\text{π(2Wt−n)}}$$ . 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 $${\mathrm{f(t)=\Sigma }}_n\mathrm{[f(n/W)s(t-n/W)+f(n/W+k)s(n/W+k-t)]}$$ in which k is subject to weak restrictions and $$\mathrm{s(t)=}\frac{\cos {\text{[2π(W}}_0\text{+W)t−(r+1)πWk]−}\cos {\text{[2π(rW−W}}_0\text{)t−(r+1)πWK]}}{\text{2πWt}\sin \text{(r+1)πWK}}+\frac{\cos {\text{[2π(rW−W}}_0\mathrm{)t-r\pi Wk]-}\cos {\text{[2πW}}_0\mathrm{t-r\pi Wk]}}{\text{2πWt}\sin \mathrm{r\pi Wk}}$$ . 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.