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 $u(x^{\prime })=v(x^{\prime })$ for all $x^{\prime }\in N$. 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

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

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

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

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

External links

• Nonlocal equations wiki