Periodic extension

A function's shape copied every $T$ to build a periodic signal out of a non-periodic slice.

The periodic extension of $f$ on an interval $[-T/2,T/2]$ is the periodic function built by copying that one window's shape infinitely many times, shifted by every integer multiple of $T$ . Letting $T\to\infty$ pulls the copies infinitely far apart, which is exactly the limiting argument that turns a Fourier series into a Fourier transform.

Examples

Extending $f(x)=x$ on $(-\pi,\pi)$ periodically produces a sawtooth wave repeating every $2\pi$ .