Compute $\sum_{n \in Z} \exp \left(ia*(n^2-2fn)\right) $ with $a>0\ $

Question

I want to Compute $$\sum_{n \in Z} \exp \left(ia*(n^2-2fn)\right)$$ with $a>0\ $ being a real number. My final result should be in the form of $$ \sum_{n=-\infty}^{\infty} \boldsymbol{c}(\boldsymbol{n}) \boldsymbol{\delta}(\boldsymbol{f}-\boldsymbol{\beta} \boldsymbol{n}) $$ where I have to express c and $\beta$ coefficients. I tried to break the sum into even and off and make use of the DTFT table but with no result. Any suggestions??

Answer 1

$$g(x)=\sum_{n \in Z} \exp \left(ia(n^2-2xn)\right), \qquad h(x)=\sum_{n=-\infty}^{\infty}c(n)\delta(x-bn)$$ both series converge in the sense of distributions.

$g$ is $\pi/a$ periodic. If $g=h$ then $h$ is $\pi/a$ periodic ie. $b=\frac{\pi}{ak}$ and $c$ is $k$-periodic.

Whence its Fourier coefficients $e^{ian^2}$ are $k$ periodic.

• If $a \not \in 2\pi\Bbb{Q}$ then this is impossible.

• If $a=2\pi l/k $ then $b=\frac{\pi}{ak}$ and $c(n)/b$ is the inverse FFT of $e^{ian^2},n\in 0,\ldots k-1$.