Fourier series

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

Fourier series express a piecewise continuous, bounded, periodic function as a linear combination of orthogonal sine and cosine functions. The seeds of the modern theory were developed by Joseph Fourier.

The Fourier series of a function f(t) is of the form:

$f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(\omega _{n}t)+b_{n}\sin(\omega _{n}t)]$

where, n is an integer and

$\omega _{n}=n{\frac {2\pi }{T}}$

$a_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\cos(\omega _{n}t)\,dt$

$b_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\sin(\omega _{n}t)\,dt$

Fourier series can be generalized to Fourier transformations for other classes of functions, for instance the space $L^{2}(\mathbb {R} ^{n})$square integrable functions. Even more generally, it is possible to carry out Fourier analysis in the setting of locally compact abelian topological groups, where the fundamental observation of Pontryagin duality provides the necessary theoretical underpinnings.