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Wikipedia / Oscillatory integral

In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals.

Definition

is written formally as

with the following properties:

  1. is real-valued, positive-homogeneous of degree does not have any critical points on the support is usually called a phase function. In some contexts more general functions are considered and still referred to as phase functions.

  2. belongs to one of the symbol classes is taken to be in more general, or just different, classes.

may be approximated by functions that have exponential decay. One possible way to do this is by setting

where the limit is taken in the sense of tempered distributions. Using integration by parts, it is possible to show that this limit is well defined, and that there exists a differential operator in the Schwartz space is given by

sufficiently large. This is the main purpose of the definition of the symbol classes.

Examples

Many familiar distributions can be written as oscillatory integrals.

The Fourier inversion theorem implies that the delta function is equal to

If we apply the first method of defining this oscillatory integral from above, as well as the Fourier transform of the Gaussian, we obtain a well known sequence of functions which approximate the delta function:

in this case is given for example by

is the Laplacian we have

and this integral converges absolutely.

The Schwartz kernel of any differential operator can be written as an oscillatory integral. Indeed if

is given by

Relation to Lagrangian distributions

Any Lagrangian distribution can be represented locally by oscillatory integrals, see . Conversely, any oscillatory integral is a Lagrangian distribution. This gives a precise description of the types of distributions which may be represented as oscillatory integrals.

See also

References