Poisson Summation at Half Integers

Question

In the proof of Poisson Summation, for a Schwarz function $f$, you define $$ F(x)=\sum_{n\in\mathbb{Z}}f(x+n) $$ and you show that $$ F(x)=\sum_{n\in\mathbb{Z}}\hat{f}(n)e^{2\pi inx} $$ Then plugging in $x=0$ gives the classical Poisson summation formula. However, if you plug in $x=\frac{1}{2}$, I find it interesting as you get what I call the Poisson summation for half integers, and it will be given as $$ \sum_{n\in\mathbb{Z}}f(n+\frac{1}{2})=\sum_{n\in\mathbb{Z}}(-1)^n\hat{f}(n) $$ I found that this was interesting as the left-hand side is the function at half integers while the right is still just summing over the Fourier transform of the integers (but now with changing sign). I was curious if this formula is useful in anyway to get any sort of information.

Answer 1

Shifting to half integers doesn't bring any new information. It's equivalent to multiplying the Fourier transform by $e^{i\pi\xi}$. So in particular, that formula is not more useful than the original one.

In fact shifting by any amount doesn't bring any new information. All those formulas are strictly equivalent and can be obtained from one another.