Discrete-time Fourier transform

Remarks

The series converges absolutely and uniformly in $\omega$, so $X$ is continuous and $2\pi$-periodic, with $\abs{X(\omega)} \le \sum_n \abs{x_n}$. The sequence is recovered by $$x_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega)\, e^{i \omega n}\, d\omega,$$ so $x_{-n}$ is the $n$-th coefficient of the Fourier series of $X$.

Example: for $\abs{a} < 1$, let $x_n = a^n$ for $n \ge 0$ and $x_n = 0$ for $n < 0$. Then $\sum_n \abs{x_n} = 1/(1 - \abs{a}) < \infty$, and the geometric series gives $X(\omega) = \sum_{n \ge 0} (a e^{-i\omega})^n = \frac{1}{1 - a e^{-i\omega}}$.

Non-example: the constant sequence $x_n = 1$ is not absolutely summable, and $\sum_n e^{-i\omega n}$ diverges for every $\omega$ because its terms do not tend to $0$. As a distribution its transform is a Dirac comb of period $2\pi$, scaled by $2\pi$.

By the Riesz–Fischer and Parseval theorems the transform extends to an isometry, up to the factor $2\pi$, from $\ell^2(\Z)$ onto $L^2$ of a period, with the series converging in $L^2$. Some authors use the frequency $f = \omega / (2\pi)$, which gives period $1$.