An oscillating integral is an object from the mathematical sub-area of functional analysis or from micro-local analysis . It is a generalized integral term, which is used especially in the field of distribution theory . Since the phase function causes the integrand to oscillate , the integral was called the oscillating integral. This term was introduced by Lars Hörmander .
Phase function
definition
A function is called a phase function, if for all
- the imaginary part is nonnegative, that is
-
.
-
for everyone .
- the differential is non-zero, that is
-
.
example
- The images , where the standard scalar product denotes, are phase functions which occur during the Fourier transformation and its inverse transformation.
motivation
Be a phase function like for example and be a symbol with . Keep defining
-
.
The image
is steady . These types of parameter integrals are common in the field of functional analysis. For example, the Fourier transform and the bilateral Laplace transform have this shape. Or the solution of Bessel's differential equation
can be noted like this.
Continuation sentences
Fourier transform on L 2
The Fourier transformation can be carried out on the Schwartz space by the integral operator
To be defined. This operator can be continued using a density argument , but the Fourier integral does not converge for every function. The operator must therefore be represented differently.
Space of symbol classes
With the space of the distributions on and with the space of the symbol classes . Be a phase function and is , . Then there is exactly one possibility of a mapping
to define so that for the integral
exists and the mapping is continuous.
definition
The two continuation clauses mentioned above show that it is desirable to have an integral term so that the continuations can also be expressed in the integral notation. The oscillating integral defined below can be used for this.
Oscillating integral
Be a clipping function with for and for . In addition, let it be a phase function and a symbol class . Now you bet
where the limit value is to be understood in terms of distributions . That means the limit is through
explained for all test functions . The integral term is called the oscillating integral.
Oscillating integral operator
Be again a phase function and a symbol of class . The image
is an oscillating integral operator .
Boundedness to L 2
Lars Hörmander showed that oscillating integral operators are bounded operators on the space of square integrable functions under certain conditions .
Let be a phase function and let the symbol class be a smooth function with compact support. Then there is a constant such that
holds, which means that the linear operator is bounded, i.e. continuous. In addition, it follows from the Banach-Steinhaus theorem that the family of operators is uniformly bounded .
Examples
Bessel function
The Bessel function
is an oscillating integral with the phase function and the symbol .
Fourier transform
Let be a smooth function with compact support and with and be the phase function. By rescaling, one can find the oscillating integral operator
in
transform. This family of operators is equally restricted to and for one obtains the Fourier transform
-
.
Pseudo differential operator
With the help of the oscillating integral one defines a special continuous and linear operator
on the Schwartz room , which through
given is. The function is a symbolic function and the operator is called the pseudo differential operator. It is a generalization of a differential operator . The integral kernel of this operator is
and is a typical Schwartz core .
literature
-
Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).
-
Elias M. Stein : Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton NJ 1993, ISBN 0-691-03216-5 ( Princeton mathematical Series 43 = Monographs in harmonic Analysis 3).
- Alain Grigis, Johannes Sjöstrand: Microlocal analysis for differential operators. An introduction. Cambridge University Press, Cambridge et al. 1994, ISBN 0-521-44986-3 ( London Mathematical Society lecture note series 196).
Individual evidence
-
↑ L. Hörmander: Fourier integral operators , Acta Math. 127 (1971), 79-183. doi : 10.1007 / BF02392052
-
^ Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals . Princeton University Press, 1993. ISBN 0-691-03216-5 , p. 377.
-
↑ Christopher D. Sogge: Fourier integrals in classical analysis . Cambridge University Press, 1993, ISBN 0-521-06097-4 , pp. 41 .