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Nonlocal operator

Class of operator mapping

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Source updated April 25, 2026

In mathematics, a nonlocal operator is a mapping that maps a space of functions on a topological space to another space of functions on some domain in such a way that the value of the function output at a given point cannot be determined solely from the values of the input function in an arbitrary neighbourhood of any point. An example of a nonlocal operator is the Fourier transform.

01Formal definition

Let $X$ be a topological space, $Y$ a set, $F(X)$ a function space containing functions with domain $X$ , and $G(Y)$ a function space containing functions with domain ⁠ $Y$ ⁠. Two functions $u$ and $v$ in $F(X)$ are called equivalent at $x\in X$ if there exists a neighbourhood $N$ of $x$ such that $u(x')=v(x')$ for all ⁠ $x'\in N$ ⁠. An operator $A:F(X)\to G(Y)$ is said to be local if for every $y\in Y$ there exists an $x\in X$ such that $Au(y)=Av(y)$ for all functions $u$ and $v$ in $F(X)$ which are equivalent at ⁠ $x$ ⁠. A nonlocal operator is an operator which is not local.

For a local operator it is possible (in principle) to compute the value $Au(y)$ using only knowledge of the values of $u$ in an arbitrarily small neighbourhood of a point ⁠ $x$ ⁠. For a nonlocal operator this is not possible.

02Examples

Differential operators are examples of local operators. 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

$(Au)(y)=\int \limits _{X}u(x)\,K(x,y)\,dx,$

where $K$ is some kernel function, it is necessary to know the values of $u$ almost everywhere on the support of $K(\cdot ,y)$ in order to compute the value of $Au$ at ⁠ $y$ ⁠.

An example of a singular integral operator is the fractional Laplacian

$(-\Delta )^{s}f(x)=c_{d,s}\int \limits _{\mathbb {R} ^{d}}{\frac {f(x)-f(y)}{|x-y|^{d+2s}}}\,dy.$

The prefactor $c_{d,s}:={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\Gamma (-s)|}}$ 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.

03Applications

Some examples of applications of nonlocal operators are:

Time series analysis using Fourier transformations
Analysis of dynamical systems using Laplace transformations
Image denoising using non-local means
Modelling Gaussian blur or motion blur in images using convolution with a blurring kernel or point spread function

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Sources and credits

This article is adapted from the Wikipedia article “Nonlocal operator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Nonlocal operator

01Formal definition

02Examples

03Applications

Sources and credits

Related topics

Fractional calculus

Linear map

Nonlocal Lagrangian