Editing Operator (mathematics) (section)

=== Probability theory ===
{{Main|Probability theory}}

Operators are also involved in probability theory, such as [[expected value|expectation]], [[variance]], and [[covariance]], which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a [[dot product]]: Every variance is a dot product of a vector with itself, and thus is a [[quadratic norm]]; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the [[Pearson correlation coefficient]]; expected value is basically an integral operator (used to measure weighted shapes in the space).

==== 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>

==== Laplace transform ====
{{Main|Laplace transform}}
The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations.

Given {{nobr|{{math|''f'' {{=}} ''f''(''s'')}}}}, it is defined by:<math display="block"> F(s)=\operatorname\mathcal{L}\{f\}(s)=\int_0^\infty e^{-s\ t}\ f(t)\ \mathrm{d}\ t </math>