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.

is written formally as where

ϕ ( x , ξ )

are functions defined on

, the formal integral defining

, and there is no need for any further discussion of the definition of

, the oscillatory integral is still defined as a distribution on

is defined by using the fact that

may be approximated by functions that have exponential decay in

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

such that the resulting distribution

in the Schwartz space is given by where this integral converges absolutely.

is not uniquely defined, but can be chosen in such a way that depends only on the phase function

In fact, given any integer

, it is possible to find an operator

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

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: An operator

is any integer greater than

we have and this integral converges absolutely.

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

is given by Any Lagrangian distribution[clarification needed] can be represented locally by oscillatory integrals, see Hörmander (1983).

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.