Integral operator

A linear integral operator is a mathematical object from functional analysis . This object is a linear operator that can be represented with a certain integral notation with an integral kernel.

definition

Be and open subsets and be a measurable function . A linear operator between the function spaces is called an integral operator if it is divided by ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ ${\displaystyle D\subset \mathbb {R} ^{n}}$ ${\displaystyle K\colon \Omega \times D\to \mathbb {C} }$ ${\displaystyle T\colon A\to B}$ ${\displaystyle A,\,B}$

${\displaystyle (Tf)(x)=\int _{\Omega }K(t,x)f(t)\mathrm {d} t}$

can be represented. The function is called integral kernel or kernel of for short . An obviously need some Regularitätsanforderungen be made so that the integral exists at all. These requirements depend on the domain of the integral operator. The integral kernels are often from the space of continuous functions or from the space of square-integrable functions . If it applies to an integral kernel and to all , then the integral kernel is called symmetrical. ${\displaystyle K:\Omega \times D\to \mathbb {C} }$ ${\displaystyle T}$ ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle D=\Omega }$ ${\displaystyle K(u,v)=K(v,u)}$ ${\displaystyle u,v\in D}$

Examples

Tensor product integral kernel

Let be two square integrable functions . The tensor product of these functions is defined as ${\displaystyle g,h\in L^{2}(\mathbb {R} ,\mathbb {C} )}$

${\displaystyle (g\otimes h)(x,y):=g(x){\overline {h(y)}},}$

where is the complex conjugation . The tensor product can be used as the integral kernel of the operator with ${\displaystyle {\bar {.}}}$ ${\displaystyle g\otimes h}$ ${\displaystyle T\colon L^{2}(\mathbb {R} ,\mathbb {C} )\to L^{2}(\mathbb {R} ,\mathbb {C} )}$

${\displaystyle (Tf)(x)=\int _{\mathbb {R} }(g\otimes h)(x,y)f(y)\mathrm {d} y:=\int _{\mathbb {R} }g(x){\overline {h(y)}}f(y)\mathrm {d} y}$

be used. This integral operator is well defined on. ${\displaystyle L^{2}(\mathbb {R} ,\mathbb {C} )}$

Volterra operator

The integral operator

${\displaystyle Tf(x)=\int _{0}^{x}f(t)\mathrm {d} t}$

is defined for all functions, for example . It is called the Volterra operator and can be used to determine an antiderivative of . Its integral core is given by ${\displaystyle f\in L^{2}([0,1])}$ ${\displaystyle f}$ ${\displaystyle K}$

${\displaystyle K(x,t)={\begin{cases}1,&t\leq x\\0,&t>x\end{cases}}.}$

There is a Hilbert-Schmidt operator . ${\displaystyle K\in L^{2}([0,1]\times [0,1])}$ ${\displaystyle T}$

Fredholm integral operator

Be a continuous function. Then an integral operator is through ${\displaystyle K\colon [0,1]\times [0,1]\to \mathbb {C} }$

${\displaystyle (Tf)(x)\colon =\int _{0}^{1}K(t,x)f(t)\mathrm {d} t}$

for everyone and defined. This operator is continuous and maps between the function spaces. This integral operator is an example of a Fredholm integral operator and is its kernel, which is also called the Fredholm kernel. A general Fredholm integral operator is characterized by the fact that, in contrast to the Volterra operator, the integral limits are fixed and the integral operator is a linear compact operator . ${\displaystyle f\in C([0,1])}$ ${\displaystyle C([0,1])\to C([0,1])}$ ${\displaystyle K}$

Cauchy's integral formula

→ Main article : Cauchy's integral formula

Cauchy's integral formula is defined as

${\displaystyle (Tf)(x)={\frac {1}{2\pi \mathrm {i} }}\oint _{\Gamma }{\frac {f(t)}{t-x}}\mathrm {d} t,}$

where is a closed curve in around the point . If then a holomorphic function is the extension of the function to a larger area. But this integral operator is also used in the theory of partial differential equations to investigate non-holomorphic functions. The integral kernel of Cauchy's integral formula is . ${\displaystyle \Gamma }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle t\colon =x}$ ${\displaystyle f}$ ${\displaystyle Tf}$ ${\displaystyle f|_{\Gamma }}$ ${\displaystyle {\tfrac {1}{2\pi \mathrm {i} (t-x)}}}$

Integral transformations

Traditionally, some integral operators are called integral transformations. For example, they play an important role in signal processing and serve to improve the handling and analysis of the information content of a signal . The integral kernel , which is a function of the target variable and the time variable, is essential for integral transformations . By multiplying the signal with the integral kernel and then integrating it over the base space in the time domain, the so-called image function is formed in the image area : ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle u}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle \Omega }$ ${\displaystyle X}$ ${\displaystyle \Omega '}$

${\displaystyle X(u)=\int _{\Omega }x(t)K(u,t)\mathrm {d} t,\qquad u\in \Omega '}$

If the integral kernel fulfills the reciprocity condition, that is, there is an “inverse kernel” , the signal can be reconstructed from the image function . The group of self-reciprocal nuclei plays an essential role in practical application in the field of signal processing. A kernel is then self-reciprocal if: ${\displaystyle K^{-1}}$ ${\displaystyle X}$ ${\displaystyle x}$

${\displaystyle K^{-1}(t,u)=K^{*}(u,t)}$

with the complex conjugation of the integration core . An example of an integral transform with a self-reciprocal kernel is the Fourier transform . ${\displaystyle K^{*}}$ ${\displaystyle K}$

Another important form in signal processing are the convolution kernels, which only depend on the difference or on. The transformation or inverse transformation can then be expressed with the convolution as: ${\displaystyle t-u}$ ${\displaystyle u-t}$

${\displaystyle X(u)=\int _{\Omega }x(t)K(u-t)\mathrm {d} t=x(t)*K(t),\qquad u\in \Omega '}$

${\displaystyle x(t)=\int _{\Omega '}X(u)K^{-1}(t-u)\mathrm {d} u=X(u)*K^{-1}(u),\qquad t\in \Omega }$

An example of an integral transformation with a convolution kernel is the Hilbert transformation .

In the following table some known, invertible integral transformations with the corresponding integral kernel , integration range and "inverse integral kernel" are listed. ${\displaystyle K}$ ${\displaystyle \Omega }$ ${\displaystyle K^{-1}}$

transformation	symbol	${\displaystyle K}$	${\displaystyle \Omega }$	${\displaystyle K^{-1}}$	${\displaystyle \Omega '}$
Fourier transform	${\displaystyle {\mathcal {F}}}$	${\displaystyle {\frac {e^{-\mathrm {i} u\cdot t}}{(2\pi )^{n/2}}}}$	${\displaystyle \mathbb {R} ^{n}\,}$	${\displaystyle {\frac {e^{+\mathrm {i} u\cdot t}}{(2\pi )^{n/2}}}}$	${\displaystyle \mathbb {R} ^{n}\,}$
Hartley transformation	${\displaystyle {\mathcal {H}}}$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$	${\displaystyle \mathbb {R} \,}$	${\displaystyle {\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}}$	${\displaystyle \mathbb {R} \,}$
Mellin transformation	${\displaystyle {\mathcal {M}}}$	${\displaystyle t^{u-1}\,}$	${\displaystyle ]0,\infty [}$ ${\displaystyle \ }$	${\displaystyle {\frac {t^{-u}}{2\pi \mathrm {i} }}\,}$	${\displaystyle c+\mathrm {i} \mathbb {R} }$ ${\displaystyle \ }$
Two-sided Laplace transform	${\displaystyle {\mathcal {B}}}$	${\displaystyle e^{-ut}\,}$	${\displaystyle \mathbb {R} \,}$	${\displaystyle {\frac {e^{+ut}}{2\pi \mathrm {i} }}}$	${\displaystyle c+\mathrm {i} \mathbb {R} }$
Laplace transform	${\displaystyle {\mathcal {L}}}$	${\displaystyle e^{-ut}\,}$	${\displaystyle ]0,\infty [}$ ${\displaystyle \ }$	${\displaystyle {\frac {e^{+ut}}{2\pi \mathrm {i} }}}$	${\displaystyle c+\mathrm {i} \mathbb {R} }$
Weierstrasse transformation	${\displaystyle {\mathcal {W}}}$	${\displaystyle {\frac {e^{-(u-t)^{2}/4}}{\sqrt {4\pi }}}\,}$	${\displaystyle \mathbb {R} \,}$	${\displaystyle {\frac {e^{+(u-t)^{2}/4}}{\mathrm {i} {\sqrt {4\pi }}}}}$	${\displaystyle c+\mathrm {i} \mathbb {R} }$
Abel transformation		${\displaystyle {\frac {2t}{\sqrt {t^{2}-u^{2}}}}\chi _{(u,\infty )}(t)}$	${\displaystyle \mathbb {R} \,}$	${\displaystyle {\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}\chi _{(t,\infty )}(u){\frac {\rm {d}}{{\rm {d}}u}}}$	${\displaystyle \mathbb {R} \,}$
Hilbert transformation	${\displaystyle {\mathcal {H}}il}$, ${\displaystyle {\mathcal {H}}}$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$	${\displaystyle \mathbb {R} \,}$	${\displaystyle {\frac {1}{\pi }}{\frac {1}{u-t}}}$	${\displaystyle \mathbb {R} \,}$
Hankel transformation with Bessel function of the first kind and ν th order ${\displaystyle \operatorname {J} _{\nu }(ut)}$	${\displaystyle {\mathcal {H}}_{\nu }}$	${\displaystyle t\operatorname {J} _{\nu }(ut)}$	${\displaystyle ]0,\infty [}$ ${\displaystyle \ }$	${\displaystyle u\operatorname {J} _{\nu }(ut)}$	${\displaystyle ]0,\infty [}$ ${\displaystyle \ }$
Stieltjes transformation	${\displaystyle {\mathcal {S}}}$	${\displaystyle {\frac {1}{u+t}}}$	${\displaystyle ]0,\infty [}$ ${\displaystyle \ }$

Integral transformations can be extended to higher dimensions , for example two-dimensional integral transformations play an essential role in image processing . When expanding to two dimensions, the functions of one variable are set to functions of two variables; the integral kernels are then functions with four variables. In the case of independent variables, the kernels can be factored and then put together as a product of two simple kernels.

Singular integral

Singular integrals are integral operators that have an integral kernel with singularity. That is, the integral kernel cannot be Lebesgue integrable on the diagonal. Therefore, the term integral has to be adapted for the integral kernels defined below.

Standard integral core

Be the diagonal in . Then a continuous function is called the standard kernel ${\displaystyle D:=\{(x,y)\in \mathbb {R} ^{n}\times \mathbb {R} ^{n}\mid x=y\}}$ ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$

${\displaystyle K:\mathbb {R} ^{n}\setminus D\to \mathbb {C} }$

with the following two properties:

1. ${\displaystyle K(x,y)\leq {\frac {C}{|x-y|^{n}}}}$
2. ${\displaystyle |\nabla _{x}K(x,y)|+|\nabla _{y}K(x,y)|\leq {\frac {C}{|x-y|^{n+1}}}.}$

The gradients are to be understood in the distributional sense .

Singular integral operator

Be a standard integral kernel. Then the operator is called ${\displaystyle K}$

${\displaystyle T(f)(x)=\int _{\mathbb {R} ^{n}}K(x,y)f(y)\mathrm {d} y}$

singular integral operator. The name comes from the fact that the operator has for a singularity. Because of this singularity, the integral does not generally converge absolutely. Therefore, the expression must be used as ${\displaystyle x=y}$ ${\displaystyle T(f)}$

${\displaystyle T(f)(x):=\lim _{\epsilon \searrow 0}\int _{|x-y|>\epsilon }K(x,y)f(y)\mathrm {d} y}$

be understood. This expression exists for everyone with . ${\displaystyle f\in L^{p}(\mathbb {R} ^{n})}$ ${\displaystyle 1\leq p<\infty }$

Distributions as integral kernels

Also distributions can be used as integral kernels. A central sentence from this area is Schwartz's core sentence . This says that there is a linear operator for every distribution ${\displaystyle K\in {\mathcal {D}}'(\Omega _{1}\times \Omega _{2})}$

${\displaystyle {\mathcal {K}}:{\mathcal {D}}(\Omega _{2})\to {\mathcal {D}}'(\Omega _{1})}$

there that for everyone and through ${\displaystyle \psi \in {\mathcal {D}}(\Omega _{1})}$ ${\displaystyle \phi \in {\mathcal {D}}(\Omega _{2})}$

${\displaystyle ({\mathcal {K}}\phi )(\psi )=K(\phi \otimes \psi )}$

given is. The reverse direction also applies. There is a unique distribution for each operator so that applies. This distribution is called the Schwartz core, named after the mathematician Laurent Schwartz , who was the first to formulate the core sentence. However, these operators cannot be represented as integral operators with the Lebesgue integral . As the representation as an integral operator seemed desirable, Lars Hörmander introduced the term oscillating integral . With this new concept of integral the integral core can through ${\displaystyle {\mathcal {K}}}$ ${\displaystyle K,}$ ${\displaystyle ({\mathcal {K}}\phi )(\psi )=K(\phi ,\psi )}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {K}}}$

${\displaystyle K(x,y)={\frac {1}{2\pi }}\int _{\mathbb {R} ^{n}}p(x,\xi )e^{\mathrm {i} (x-y)\xi }\mathrm {d} \xi }$

and then the operator is an integral operator of the shape ${\displaystyle {\mathcal {K}}}$

${\displaystyle {\mathcal {K}}\phi =\int _{\mathbb {R} ^{n}}K(x,.)\phi (x)\mathrm {d} x=\int _{\Omega _{2}\times \mathbb {R} ^{n}}p(x,\xi )e^{\mathrm {i} (x-.)\xi }\phi (x)\mathrm {d} (x,\xi )}$

given, where the integrals are again oscillating integrals. The equal signs are to be understood in the sense of distributions, what

${\displaystyle ({\mathcal {K}}\phi )(\psi )=K(\phi \otimes \psi )=\int _{\Omega _{1}\times \Omega _{2}}K(x,.)\phi (x)\psi (y)\mathrm {d} (x,y)=\int _{\Omega _{1}\times \Omega _{2}\times \mathbb {R} ^{n}}p(x,\xi )e^{\mathrm {i} (x-.)\xi }\mathrm {d} \xi \phi (x)\psi (y)\mathrm {d} (x,y,\xi )}$

means.

Nonlinear integral operators

A nonlinear (Urysohn) integral operator has the form

${\displaystyle (Tf)(x)=\int _{\Omega }K(t,x,f(t))\mathrm {d} t}$

with a suitable domain of definition of the core function K and integration domain Ω.

literature

• MA Krasnoselski: Topological Methods in the Theory of nonlinear Integral Equations . Oxford 1964.
• Dirk Werner : Functional Analysis . Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .
• Elias M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals . Princeton University Press, 1993, ISBN 0-691-03216-5 .