Oscillatory integral

An oscillating integral is an object from the mathematical branch of functional analysis or from the micro- local analysis. It is a generalized integral term, which has particular application in the field of distribution theory. Since the phase function can oscillate the integrand, the integral was appropriately called oscillatory integral. Was introduced this concept by Lars Hörmander.

  • 3.1 Fourier transform on L2
  • 3.2 Area of ​​symbol classes
  • 4.1 Oscillating integral
  • 4.2 Oscillating integral operator
  • 6.1 Fourier transform
  • 6.2 pseudo-differential operator

Phase function

Definition

A function is called a phase function if for all

  • The imaginary part is negative, that is
  • The function is homogeneous, that is to say
  • The differential is not zero, that is,

Example

  • The images, with the standard scalar thinks are phase functions which appear in the Fourier transform and its inverse transform.

Motivation

Be a phase function for example and be a symbol. Then we have

And the mapping

Is continuous. These types of parameter integrals are used in the field of functional analysis. For example, the Fourier transform and Laplace transform, these two -sided shape. Or the solution of the Bessel differential equation

Can be noted.

Continuation records

Fourier transform on L2

The Fourier transform on the can Schwartz space by the integral operator

Be defined. Using a tightness argument can continue on this operator, however, the Fourier integral does not converge for each function. The operator must be presented differently so.

Space of the symbol classes

With the space of distributions is called. Be a phase function and is. Then there is a possibility a picture

Defined so that for the integral

Exist and the map is continuous.

Definition

Show the two continuation records mentioned above that it is desirable to have an integral term, so that one can express the sequels in the integral notation. For the defined hereinafter oscillating integral can be used.

Oscillating integral

Be a cut-off function with all generations. Furthermore, it is a phase function and a symbol of class. Now it is

Where the limit is understood in the sense of distributions. This means that the limit is by

Declared for all the test functions. The integral term is called oscillatory integral.

Oscillating integral operator

Be back a phase function and a symbol of class. The figure

Is an oscillatory integral operator.

Boundedness on L2

Lars Hörmander showed that oscillatory integral operators under certain conditions bounded operators on the space of square integrable functions.

Be a phase function and the symbol class is a smooth function with compact support. Then there exists a constant such that

Holds, which means that the linear operator on limited, so steadily, is. Moreover, it follows from the Banach - Steinhaus, that the family of operators is uniformly bounded.

Examples

Fourier transformation

Let be a smooth function with compact support and with and is the phase function. By rescaling can see the oscillatory integral operator

In

Transform. This family of operators on uniformly bounded and we obtain the Fourier transform

Pseudo- differential operator

With the help of the oscillating integral defining a special continuous and linear operator

By the Schwartz - space, which

Is given. The function is a function symbol and the operator is called pseudo- differential operator. It is a generalization of a differential operator. The integral kernel of this operator is

And Schwartz is a typical core.

Microlocal analysis Differentiation under the integral sign Two-sided Laplace transform Bump function Differentialoperator
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