Editing Operator (mathematics) (section)

==== Fourier series and Fourier transform ====
{{Main|Fourier series|Fourier transform}}
The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively [[invertible function|invertible]]. No information is lost, as there is an inverse transform operator. In the simple case of [[periodic function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:<math display="block">f(t)=\frac{\ a_0\ }{2}+\sum_{n=1}^{\infty}\ a_n\cos(\omega\ n\ t) + b_n\sin(\omega\ n\ t) </math> The tuple {{math|( ''a''{{sub|0}}, ''a''{{sub|1}}, ''b''{{sub|1}}, ''a''{{sub|2}}, ''b''{{sub|2}}, ... )}} is in fact an element of an infinite-dimensional vector space [[Sequence space|{{math|''ℓ''{{i sup|2}} }}]], and thus Fourier series is a linear operator.

When dealing with general function <math>\mathbb{R} \to \mathbb{C}</math>, the transform takes on an [[integral]] form:
:<math display="block">f(t) = {1\over\sqrt{2\pi}}\int_{-\infty}^{+\infty}{g(\omega)\ e^{i\ \omega\ t}\ \mathrm{d}\ \omega} </math>