Fourier series

In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

History

The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768 - 1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.

Definition

Consider a real-valued function, $s(x)$, that is integrable on a interval of length $P$, which will be the period of the Fourier series. Common examples of analysis intervals are: $x\in [0,1],$ and $P = 1.$$x\in [-\pi ,\pi ],$and $P=2\pi .$The analysis process determines the weights, indexed by integer $n$, which is also the number of cycles of the $n^{\text{th}}$harmonic in the analysis interval.