Definition
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 Schwarz 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.
Read more about this topic: Oscillatory Integral
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