Integral transform

A linear integral operator is a mathematical object from the functional analysis. This object is a linear operator, which can be represented by a definite integral notation with an integral kernel.

• 4.1 Standard integral kernel
• 4.2 Singular integral operator

Definition

Let and be open subsets and is a measurable function. A linear operator between function spaces is called integral operator as it passes through

Can be displayed. The function is called the integral kernel or core of short. At certain course Regularitätsanforderungen must be provided so that the integral exists. These requirements depend on the domain of definition of the integral operator. Many times the integral kernels of the space of continuous functions, or from the space of square integrable functions. Applies to an integral kernel and for all, then is called the integral kernel symmetrical.

Examples

Tensor - integral kernel

Be two square integrable functions. The tensor product of these functions is defined as

Where the complex conjugation is. The tensor can be seen as integral kernel of the operator with

Be used. This integral operator is on well-defined.

Volterraoperator

The integral operator

Is defined, for example, for all functions. His name is Volterraoperator and can be used to determine a primitive of. Its integral kernel is given by

As is true, a Hilbert-Schmidt operator.

Fredholm integral operator

Let be a continuous function then is an integral operator by

For all and defined. This operator is continuous and forms between the function spaces from. This integral operator is an example of a Fredholm integral operator and is its core, which is also called Fredholm kernel. A general Fredholm integral operator is characterized by the fact that the integral boundaries in contrast to Voltara operator are fixed and the integral operator is a linear compact operator.

Cauchy's integral formula

The Cauchy integral formula is defined as

Wherein a closed curve is to the point. Then a holomorphic function, then the derivative of this function. But this integral operator is not used also for the study of holomorphic functions in the theory of partial differential equations. The integral kernel of the Cauchy integral formula.

Integral transforms

Some integral operators is called traditionally more integral transforms. They play for example in signal processing a major role and are used for better handling and analysis of the information content of a signal. Essential for integral transforms is the integral kernel, which is a function of the target variables and the time variable. By multiplying the signal by the integral kernel and subsequent integration over the base space in the time domain, the so-called image function is formed in the image area:

Meets the integral kernel, the reciprocity, that is to say, there is a "inverse core ", the signal can be reconstructed from the image function. In practical use in the field of signal processing, the group of cores selbstreziproken plays an essential role. A core is then selbstreziprok if:

With the complex conjugation of the integration core. An example of an integral transform is the core with selbstreziprokem Fourier transform.

A more in signal processing significant form represent the convolution kernels, which depend only on the difference or of. The transformation and inverse transformation can then be expressed as the convolution:

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

The following table lists some known, invertible integral transformations with the corresponding integral kernel, the integration field and " inverse integral kernel " listed.

Integral transforms can be extended to higher dimensions, for example, play in the two-dimensional image processing integral transforms an essential role. In extension to two dimensions, the functions of one variable be set to functions of two variables, the integral kernels are then functions of four variables. In the case of independent variables, the cores can be factored and then sat down as a product of two simple cores together.

Singular integral

Singular integrals are integral operators that have an integral kernel with singularity. That is, the integral kernel is on the diagonal is not Lebesgue integrable. Therefore, the integral term must be adjusted for the integral kernels defined below.

Standard integral kernel

Be the diagonal in. Then is called a standard core is a continuous function

With the following two properties:

The gradients are to be understood in the distributional sense.

Singular integral operator

Be a standard integral kernel. Then called the operator

Singular integral operator. The name comes from the fact that the operator has a singularity. Due to this singularity, the integral generally does not converge absolutely. Therefore, the expression must be as

Be understood. This expression exists for all with.

Distributions as integral kernels

Distributions that can be used as integral kernels. A central proposition in this area is the core set of Schwartz. This means that there will be any distribution is a linear operator

Are designed for all and by

Is given. In addition, the reverse direction is true. Thus, for every operator a unique distribution so that applies. This distribution is called the Schwartz kernel, named after the mathematician Laurent Schwartz, who formulated the core set first. However, these operators can not be represented as integral operators with the Lebesgue integral. However, since the integral operator View as seemed desirable Lars Hörmander introduced the concept of the oscillating integral. With this new integral concept of integral kernel can be obtained by

Be specified and then the operator is as an integral operator of the form

Given, where the integrals are oscillatory integrals again. The equal sign are to be understood in the sense of distributions, which

Means.

Nonlinear integral operators

A non-linear ( Urysohn ) integral operator has the form

With a suitable domain of the kernel function K and integration area Ω.