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
:$(-\Delta)^sf(x) = c_ \int\limits_ \frac\,dy.$
The prefactor $c_ := \frac$ 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

Mathematical analysis
Functions and mappings

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