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^{[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

(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.^[1]

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

References

^ Caffarelli, L.; Roquejoffre, J.-M.; Savin, O. (2010). "Nonlocal minimal surfaces". Communications on Pure and Applied Mathematics. 63 (9): 1111–1144. arXiv:0905.1183. doi:10.1002/cpa.20331. S2CID 10480423.
^ Buades, A.; Coll, B.; Morel, J.-M. (2005). "A Non-Local Algorithm for Image Denoising". 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05). Vol. 2. San Diego, CA, USA: IEEE. pp. 60–65. doi:10.1109/CVPR.2005.38. ISBN 9780769523729. S2CID 11206708.