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.
Formal definition Let X {\displaystyle X} be a topological space, Y {\displaystyle Y} a set, F ( X ) {\displaystyle F(X)} a function space containing functions with domain X {\displaystyle X} , and G ( Y ) {\displaystyle G(Y)} a function space containing functions with domain Y {\displaystyle Y} . Two functions u {\displaystyle u} and v {\displaystyle v} in F ( X ) {\displaystyle F(X)} are called equivalent at x ∈ X {\displaystyle x\in X} if there exists a neighbourhood N {\displaystyle N} of x {\displaystyle x} such that u ( x ′ ) = v ( x ′ ) {\displaystyle u(x')=v(x')} for all x ′ ∈ N {\displaystyle x'\in N} . An operator A : F ( X ) → G ( Y ) {\displaystyle A:F(X)\to G(Y)} is said to be local if for every y ∈ Y {\displaystyle y\in Y} there exists an x ∈ X {\displaystyle x\in X} such that A u ( y ) = A v ( y ) {\displaystyle Au(y)=Av(y)} for all functions u {\displaystyle u} and v {\displaystyle v} in F ( X ) {\displaystyle F(X)} which are equivalent at x {\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 A u ( y ) {\displaystyle Au(y)} using only knowledge of the values of u {\displaystyle u} in an arbitrarily small neighbourhood of a point x {\displaystyle 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
( A u ) ( y ) = ∫ X u ( x ) K ( x , y ) d x , {\displaystyle (Au)(y)=\int \limits _{X}u(x)\,K(x,y)\,dx,}
where K {\displaystyle K} is some kernel function, it is necessary to know the values of u {\displaystyle u} almost everywhere on the support of K ( ⋅ , y ) {\displaystyle K(\cdot ,y)} in order to compute the value of A u {\displaystyle Au} at y {\displaystyle y} . An example of a singular integral operator is the fractional Laplacian
( − Δ ) s f ( x ) = c d , s ∫ R d f ( x ) − f ( y ) | x − y | d + 2 s d y . {\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 c d , s := 4 s Γ ( d / 2 + s ) π d / 2 | Γ ( − s ) | {\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.
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
See also Fractional calculus Linear map Nonlocal Lagrangian Action at a distance
References
External links Nonlocal equations wiki