Oscillatory integral

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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.


An oscillatory integral is written formally as

where and are functions defined on with the following properties:

  1. The function is real-valued, positive-homogeneous of degree 1, and infinitely differentiable away from . Also, we assume that does not have any critical points on the support of . Such a function, is usually called a phase function. In some contexts more general functions are considered and still referred to as phase functions.
  2. The function belongs to one of the symbol classes for some . Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree . As with the phase function , in some cases the function is taken to be in more general, or just different, classes.

When , the formal integral defining converges for all , and there is no need for any further discussion of the definition of . However, when , the oscillatory integral is still defined as a distribution on , even though the integral may not converge. In this case the distribution 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 acting on any in the Schwartz space is given by

where this integral converges absolutely. The operator is not uniquely defined, but can be chosen in such a way that depends only on the phase function , the order of the symbol , and . In fact, given any integer , it is possible to find an operator so that the integrand above is bounded by for sufficiently large. 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 in this case is given for example by

where is the Laplacian with respect to the variables, and is any integer greater than . Indeed, with this we have

and this integral converges absolutely.

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

where , then the kernel of is given by

Relation to Lagrangian distributions[edit]

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.

See also[edit]


  • Hörmander, Lars (1983), The Analysis of Linear Partial Differential Operators IV, Springer-Verlag, ISBN 0-387-13829-3
  • Hörmander, Lars (1971), "Fourier integral operators I", Acta Math., 127: 79–183, doi:10.1007/bf02392052