Oscillatory integral

Short description: Type of distribution in mathematical analysis

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.

## Contents

* 1 Definition
* 2 Examples
* 3 Relation to Lagrangian distributions
* 4 See also
* 5 References

## Definition

An oscillatory integral f(x) is written formally as

f(x)=∫eiϕ(x,ξ)a(x,ξ)dξ,

where ϕ(x,ξ) and a(x,ξ) are functions defined on ℝxn×RξN with the following properties:

1. The function ϕ is real-valued, positive-homogeneous of degree 1, and infinitely differentiable away from {ξ=0}. Also, we assume that ϕ does not have any critical points on the support of a. Such a function, ϕ is usually called a phase function. In some contexts more general functions are considered and still referred to as phase functions.
2. The function a belongs to one of the symbol classes S1,0m(ℝxn×RξN) for some m∈ℝ. Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree m. As with the phase function ϕ, in some cases the function a is taken to be in more general, or just different, classes.

When m<−N, the formal integral defining f(x) converges for all x, and there is no need for any further discussion of the definition of f(x). However, when m≥−N, the oscillatory integral is still defined as a distribution on ℝn, even though the integral may not converge. In this case the distribution f(x) is defined by using the fact that a(x,ξ)∈S1,0m(ℝxn×RξN) may be approximated by functions that have exponential decay in ξ. One possible way to do this is by setting

f(x)=lim⁡ϵ->0+∫eiϕ(x,ξ)a(x,ξ)e−ϵ|ξ|2/2dξ,

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 L such that the resulting distribution f(x) acting on any ψ in the Schwartz space is given by

⟨f,ψ⟩=∫eiϕ(x,ξ)L(a(x,ξ)ψ(x))dxdξ,

where this integral converges absolutely. The operator L is not uniquely defined, but can be chosen in such a way that depends only on the phase function ϕ, the order m of the symbol a, and N. In fact, given any integer M, it is possible to find an operator L so that the integrand above is bounded by C(1+|ξ|)−M for |ξ| 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, δ(x) is equal to

1(2π)n∫ℝneix⋅ξdξ.

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:

δ(x)=limε->0+1(2π)n∫ℝneix⋅ξe−ε|ξ|2/2dξ=limε->0+1(2πε)ne−|x|2/(2ε).

An operator L in this case is given for example by

L=(1−Δx)k(1+|ξ|2)k,

where Δx is the Laplacian with respect to the x variables, and k is any integer greater than (n−1)/2. Indeed, with this L we have

⟨δ,ψ⟩=ψ(0)=1(2π)n∫ℝneix⋅ξL(ψ)(x,ξ)dξdx,

and this integral converges absolutely.

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

L=∑|α|≤mpα(x)Dα,

where Dα=∂xα/i|α|, then the kernel of L is given by

1(2π)n∫ℝneiξ⋅(x−y)∑|α|≤mpα(x)ξαdξ.

## Relation to Lagrangian distributions

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

* Riemann–Lebesgue lemma
* van der Corput lemma
* Oscillatory integral operator

## References

* 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

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