Nonlocal operator

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.

Formal 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.

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

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.

Applications
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