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 [math]\displaystyle{ X }[/math] be a topological space, [math]\displaystyle{ Y }[/math] a set, [math]\displaystyle{ F(X) }[/math] a function space containing functions with domain [math]\displaystyle{ X }[/math], and [math]\displaystyle{ G(Y) }[/math] a function space containing functions with domain [math]\displaystyle{ Y }[/math]. Two functions [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] in [math]\displaystyle{ F(X) }[/math] are called equivalent at [math]\displaystyle{ x\in X }[/math] if there exists a neighbourhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ x }[/math] such that [math]\displaystyle{ u(x')=v(x') }[/math] for all [math]\displaystyle{ x'\in N }[/math]. An operator [math]\displaystyle{ A: F(X) \to G(Y) }[/math] is said to be local if for every [math]\displaystyle{ y\in Y }[/math] there exists an [math]\displaystyle{ x\in X }[/math] such that [math]\displaystyle{ Au(y) = Av(y) }[/math] for all functions [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] in [math]\displaystyle{ F(X) }[/math] which are equivalent at [math]\displaystyle{ x }[/math]. A nonlocal operator is an operator which is not local.
For a local operator it is possible (in principle) to compute the value [math]\displaystyle{ Au(y) }[/math] using only knowledge of the values of [math]\displaystyle{ u }[/math] in an arbitrarily small neighbourhood of a point [math]\displaystyle{ x }[/math]. 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
- [math]\displaystyle{ (Au)(y) = \int \limits_X u(x)\, K(x, y)\, dx, }[/math]
where [math]\displaystyle{ K }[/math] is some kernel function, it is necessary to know the values of [math]\displaystyle{ u }[/math] almost everywhere on the support of [math]\displaystyle{ K(\cdot, y) }[/math] in order to compute the value of [math]\displaystyle{ Au }[/math] at [math]\displaystyle{ y }[/math].
An example of a singular integral operator is the fractional Laplacian
- [math]\displaystyle{ (-\Delta)^sf(x) = c_{d,s} \int\limits_{\mathbb{R}^d} \frac{f(x)-f(y)}{|x-y|^{d+2s}}\,dy. }[/math]
The prefactor [math]\displaystyle{ c_{d,s} := \frac{4^s\Gamma(d/2+s)}{\pi^{d/2}|\Gamma(-s)|} }[/math] 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
See also
References
- ↑ Caffarelli, L.; Roquejoffre, J.-M.; Savin, O. (2010). "Nonlocal minimal surfaces" (in en). Communications on Pure and Applied Mathematics 63 (9): 1111–1144. doi:10.1002/cpa.20331.
- ↑ Buades, A.; Coll, B.; Morel, J.-M. (2005). A Non-Local Algorithm for Image Denoising. 2. San Diego, CA, USA: IEEE. pp. 60–65. doi:10.1109/CVPR.2005.38. ISBN 9780769523729.
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