# What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function:

$$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$

We use the Poisson summation formula. Define $$f(x) \equiv \sin(\pi x) / (\pi x)$$. Then the sum we are trying to solve is $$g(y) = \sum_{n=-\infty}^\infty f(n-y) \, .$$ The Poisson summation formula converts the sum over values of $$f$$ to a sum over values of the Fourier transform of $$f$$.

# Poisson summation

Note that $$g(y)$$ is periodic with period $$1$$. The Fourier series coefficients of $$g$$ are by definition \begin{align} g_\nu &= \int_0^1 dy \, g(y)e^{-i 2 \pi \nu y} \\ &= \int_0^1 dy \, \sum_{n=-\infty}^\infty f(n-y) e^{-i 2 \pi \nu y} \\ (\text{Let }x\equiv n-y) \qquad &= \sum_{n=-\infty}^\infty \int_{n-1}^n dx \, f(x) e^{-i 2 \pi \nu (n-x)} \\ &= \int_{-\infty}^\infty dx \, f(x) e ^{i 2 \pi \nu x} \\ &= \tilde{f}(-\nu) \, . \end{align} where $$\tilde{f}$$ is the Fourier transform of $$f$$.

By definition of the Fourier series, \begin{align} g(y) &= \sum_{\nu = -\infty}^\infty e^{i 2 \pi \nu y} g_\nu \\ \text{so} \qquad \sum_{n=-\infty}^\infty f(n-y) &= \sum_{\nu=-\infty}^\infty e^{-i 2 \pi \nu y} \tilde{f}(\nu) \end{align} which is the Poisson summation formula

# Solution to the problem

Using the Poisson summation formula, we can write $$g(y) = \sum_{n=-\infty}^\infty f(n-y) = \sum_{\nu=-\infty}^\infty \tilde{f}(\nu) e^{-i 2 \pi \nu y} \, .$$ What is $$\tilde{f}$$? We can easily compute that the Fourier transform of the tophat function $$T(x) = \left\{ \begin{array}{l} 1, \qquad -1/2 < x <1/2 \\ 0, \qquad \text{otherwise} \end{array} \right.$$ is $$\tilde{T}(\nu)=\sin(\pi \nu) / (\pi \nu)$$. By duality of the Fourier transform, that means that $$\tilde{f}$$ is the tophat function $$T$$. Therefore we have $$g(y) = \sum_{\nu=-\infty}^\infty T(\nu) e^{-i 2 \pi \nu y} = 1 \, .$$ This is a remarkable result: no matter how much you shift your sample points on a sinc function, the sum of those samples is constant.


Note that

$\ds{\bracks{{\sin\pars{\pi\bracks{z - y}} \over \pi\pars{z - y}} + {\sin\pars{\pi\bracks{z + y}} \over \pi\pars{z + y}}} \expo{-2\pi\,\verts{\Im\pars{z}}} \stackrel{\mrm{as}\ \Im\pars{z}\ \to\ \pm\infty}{\large\sim} \pm{\expo{-\pi\verts{\Im\pars{z}}} \over \pi\,\Im\pars{z}} \stackrel{\mrm{as}\ \Im\pars{z}\ \to\ \pm\infty}{\large\to}{\large 0}}$

such that the sum can be evaluated by means of the Abel-Plana Formula: \begin{align} \mrm{g}\pars{y} & = -\,{\sin\pars{\pi y} \over \pi y} + \int_{0}^{\infty}\bracks{% {\sin\pars{\pi\bracks{x - y}} \over \pi\pars{x - y}} + {\sin\pars{\pi\bracks{x + y}} \over \pi\pars{x + y}}}\dd x \\[2mm] & + {1 \over 2} \bracks{{\sin\pars{\pi\bracks{x - y}} \over \pi\pars{x - y}} + {\sin\pars{\pi\bracks{x + y}} \over \pi\pars{x + y}}}_{\ x\ =\ 0} \\[5mm] & = -\,{\sin\pars{\pi y} \over \pi y} + {1 \over 2}\int_{-\infty}^{\infty}\bracks{% {\sin\pars{\pi\bracks{x - y}} \over \pi\pars{x - y}} + {\sin\pars{\pi\bracks{x + y}} \over \pi\pars{x + y}}}\dd x \\[2mm] & + {\sin\pars{\pi y} \over \pi y} = {1 \over 2}\int_{-\infty}^{\infty}{\sin\pars{\pi x} \over \pi x}\,\dd x + {1 \over 2}\int_{-\infty}^{\infty}{\sin\pars{\pi x} \over \pi x}\,\dd x = \bbx{\Large 1} \end{align}