Nonlocal operator

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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[edit]

Let be a topological space, a set, a function space containing functions with domain , and a function space containing functions with domain . Two functions and in are called equivalent at if there exists a neighbourhood of such that for all . An operator is said to be local if for every there exists an such that for all functions and in which are equivalent at . A nonlocal operator is an operator which is not local.

For a local operator it is possible (in principle) to compute the value using only knowledge of the values of in an arbitrarily small neighbourhood of a point . For a nonlocal operator this is not possible.

Examples[edit]

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

where is some kernel function, it is necessary to know the values of almost everywhere on the support of in order to compute the value of at .

An example of a singular integral operator is the fractional Laplacian

The prefactor 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[edit]

Some examples of applications of nonlocal operators are:

See also[edit]

References[edit]

  1. ^ Caffarelli, L.; Roquejoffre, J.-M.; Savin, O. (2010). "Nonlocal minimal surfaces". Communications on Pure and Applied Mathematics: n/a–n/a. doi:10.1002/cpa.20331.
  2. ^ 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). 2. San Diego, CA, USA: IEEE. pp. 60–65. doi:10.1109/CVPR.2005.38. ISBN 9780769523729.

External links[edit]