# Why a periodic function can be expressed by a set of finite numbers — question about Fourier transform?

For a periodic function $f(x)$ with period $p$, Fourier transform says that it can be "expressed" by a set of infinite many number.

$$f(x)=\sum_{k=-\infty}^\infty F[k]e^{2\pi i kx/p}$$

where

$$F[k]=\frac{1}{p}\int_{x=0}^pf(x)e^{-2\pi ikx/p}dx$$

Why the information of $f(x)$ in the an interval $[0,p]$ can be expressed by a set of infinity many numbers $F[k]$? It seems that there are something in common between the interval and all the integers, what is that common thing in mathematic language? Does that mean there are as many real numbers as in the interval $[0,1]$ as there are of integers?

• cardinal(set of interesting functions in a interval) < cardinal(set of arbitrary functions in a interval). – Martín-Blas Pérez Pinilla Feb 7 '14 at 19:01