• Source: 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


{\displaystyle X}

be a topological space,



Y


{\displaystyle Y}

a set,



F
(
X
)


{\displaystyle F(X)}

a function space containing functions with domain



X


{\displaystyle X}

, and



G
(
Y
)


{\displaystyle G(Y)}

a function space containing functions with domain



Y


{\displaystyle Y}

. Two functions



u


{\displaystyle u}

and



v


{\displaystyle v}

in



F
(
X
)


{\displaystyle F(X)}

are called equivalent at



x
∈
X


{\displaystyle x\in X}

if there exists a neighbourhood



N


{\displaystyle N}

of



x


{\displaystyle x}

such that



u
(

x
′

)
=
v
(

x
′

)


{\displaystyle u(x')=v(x')}

for all




x
′

∈
N


{\displaystyle x'\in N}

. An operator



A
:
F
(
X
)
→
G
(
Y
)


{\displaystyle A:F(X)\to G(Y)}

is said to be local if for every



y
∈
Y


{\displaystyle y\in Y}

there exists an



x
∈
X


{\displaystyle x\in X}

such that



A
u
(
y
)
=
A
v
(
y
)


{\displaystyle Au(y)=Av(y)}

for all functions



u


{\displaystyle u}

and



v


{\displaystyle v}

in



F
(
X
)


{\displaystyle F(X)}

which are equivalent at



x


{\displaystyle x}

. A nonlocal operator is an operator which is not local.
For a local operator it is possible (in principle) to compute the value



A
u
(
y
)


{\displaystyle Au(y)}

using only knowledge of the values of



u


{\displaystyle u}

in an arbitrarily small neighbourhood of a point



x


{\displaystyle 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




(
A
u
)
(
y
)
=

∫

X


u
(
x
)

K
(
x
,
y
)

d
x
,


{\displaystyle (Au)(y)=\int \limits _{X}u(x)\,K(x,y)\,dx,}


where



K


{\displaystyle K}

is some kernel function, it is necessary to know the values of



u


{\displaystyle u}

almost everywhere on the support of



K
(
⋅
,
y
)


{\displaystyle K(\cdot ,y)}

in order to compute the value of



A
u


{\displaystyle Au}

at



y


{\displaystyle y}

.
An example of a singular integral operator is the fractional Laplacian




(
−
Δ

)

s


f
(
x
)
=

c

d
,
s



∫



R


d







f
(
x
)
−
f
(
y
)



|

x
−
y


|


d
+
2
s






d
y
.


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


The prefactor




c

d
,
s


:=




4

s


Γ
(
d

/

2
+
s
)



π

d

/

2



|

Γ
(
−
s
)

|






{\displaystyle c_{d,s}:={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\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




References






External links


Nonlocal equations wiki

Kata Kunci Pencarian:

• Operator nonlokal
• Kucing Schrödinger
• Nonlocal operator
• Nonlocality
• Nonlocal
• Integral transform
• Quantum nonlocality
• Fractional Laplacian
• Calculus on finite weighted graphs
• Laplace transform
• Elitzur's theorem
• Non-local means