Fourier series

Definition

For $f\in L^2(0,1)$, its Fourier series is

$$f(t)=\sum _{k=-\infty }^{\infty }\hat{f}(k)e^{2\pi ikt}$$

where the equality is understood in the $L^2$ sense, and

$$\hat{f}(k)={\int }_0^{1}f(t)e^{-2\pi ikt}dt,k\in \mathbb{Z}.$$

Here $L^2(0,1)$ is the space of functions with finite squared integral:

$${\int }_0^1|f(t){|}^{2}dt<\infty .$$

Thus the partial sums converge to $f$ in mean square, i.e. their squared error has integral tending to $0$.