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 formula_1 be a topological space, formula_2 a set, formula_3 a function space containing functions with domain formula_1, and formula_5 a function space containing functions with domain formula_2. Two functions formula_7 and formula_8 in formula_3 are  called equivalent at formula_10 if there exists a neighbourhood formula_11 of formula_12 such that formula_13 for all formula_14. An operator formula_15 is said to be local if for every formula_16 there exists an formula_10 such that formula_18 for all functions formula_7 and formula_8 in formula_3 which are equivalent at formula_12. A nonlocal operator is an operator which is not local. For a local operator it is possible (in principle) to compute the value formula_23 using only knowledge of the values of formula_7 in an arbitrarily small neighbourhood of a point formula_12. 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 where formula_27 is some kernel function, it is necessary to know the values of formula_7 almost everywhere on the support of formula_29 in order to compute the value of formula_30 at formula_31. An example of a singular integral operator is the fractional Laplacian Applications. Some examples of applications of nonlocal operators are: