In the previous post, we looked at an elegant equation involving integrals of the sinc function and computed the corresponding integrals for the jinc function.
It turns out the analogous equation holds for sums as well:
As before, we’d like to compute these two sums and see whether we can compute the corresponding sums for the jinc function.
The Poisson summation formula says that a function and its Fourier transform produce the same sums over the integers:
Recall from the previous post that the Fourier transform of sinc is the function π box(π x) where the box function is 1 on [-1/2, 1/2] and zero elsewhere. The only integer n with πn inside [-1/2, 1/2] is 0, so the sum of sinc(n) over the integers equals π. A very similar argument shows that the sum of jinc(n) over the integers equals its Fourier transform at 0, which equals 2.
Let tri(x) be the triangle function, defined to be 1 – |x| for -1 < x < 1 and 0 otherwise. Then the Fourier transform of tri(x) is sinc2(π ω) and so π tri(π x) and sinc2 are Fourier transform pairs. The Poisson summation formula says the sum of sinc2 over the integers is the sum of π tri(π x) over the integers, which is π.
I don’t know the Fourier transform of jinc2 and doubt it’s easy to compute. Maybe the sum could be computed more easily without Fourier transforms, e.g. using contour integration.