Fourier Series

analysis functional-analysis

Definition

Fourier Series

A Fourier series is an expansion of a periodic function in terms of complex exponentials or trigonometric functions.

On $[-\pi ,\pi ]$, a function $f\in L^2[-\pi ,\pi ]$ can be expanded as

$$f(x)=\frac{1}{\sqrt{2\pi }}\sum _{n=-\infty }^{\infty }{\lambda }_n{e}^{inx},$$

where the coefficients are given by

$${\lambda }_n=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }{e}^{-inx}f(x)dx.$$

The functions ${\left\{\frac{1}{\sqrt{2\pi }}{e}^{inx}\right\}}_{n\in \mathbb{Z}}$ form an orthonormal basis of $L^2[-\pi ,\pi ]$.

The coefficients ${\lambda }_n$ are called the Fourier coefficients.