Everywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then backtrack the thing using windows. Is inifinity necessary? Or can we do processing without this do-undo process?
Fourier analysis requires a function to be defined on a (locally compact, abelian) group. The set of real numbers is a group under addition; this is the setting of Fourier transform. So is the circle $\mathbb R/(a\mathbb Z)$ for some $a>0$; this is the setting of Fourier series. So is the cyclic group $\mathbb Z/(n\mathbb Z)$, which is the setting of the discrete Fourier transform.
An interval $[a,b]$ is not a group. So, even though we may only have a function defined on such an interval, from the mathematical point of view the Fourier transform is taken on the real line. The function can be set to $0$ outside of $[a,b]$.