In mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 an oscillatory integral is a type of distribution

Distribution (mathematics)

In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

. 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

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

Support (mathematics)

In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...

 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

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

  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.

Examples

Many familiar distributions can be written as oscillatory integrals.

1) The Fourier inversion theorem

Fourier inversion theorem

In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.Sometimes the following expression is used as the definition of the Fourier transform:...

 implies that the delta function

Dirac delta function

The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...

, 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

Laplace operator

In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

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

and this integral converges absolutely.

2) The Schwartz kernel

Schwartz kernel theorem

In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz himself have a two-variable theory that includes all reasonable...

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

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.