In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point.
An example of a nonlocal operator is the Fourier transform.
be a topological space,
a set,
a function space containing functions with domain
a function space containing functions with domain
are called equivalent at
if there exists a neighbourhood
An operator
A nonlocal operator is an operator which is not local.
For a local operator it is possible (in principle) to compute the value
using only knowledge of the values of
in an arbitrarily small neighbourhood of a point
For a nonlocal operator this is not possible.
Differential operators are examples of local operators[citation needed].
A large class of (linear) nonlocal operators is given by the integral transforms, such as the Fourier transform and the Laplace transform.
For an integral transform of the form where
is some kernel function, it is necessary to know the values of
almost everywhere on the support of
in order to compute the value of
An example of a singular integral operator is the fractional Laplacian The prefactor
π
involves the Gamma function and serves as a normalizing factor.
The fractional Laplacian plays a role in, for example, the study of nonlocal minimal surfaces.
[1] Some examples of applications of nonlocal operators are: This mathematical analysis–related article is a stub.
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