Nonlocal operator

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

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

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

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

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

An example of a singular integral operator is the fractional Laplacian

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

The prefactor ${\displaystyle c_{d,s}:={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\Gamma (-s)|}}}$ 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.[1]

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[2]
• Modelling Gaussian blur or motion blur in images using convolution with a blurring kernel or point spread function

• Fractional calculus
• Linear map
• Nonlocal Lagrangian
• Action at a distance

• Nonlocal equations wiki