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 [math]\displaystyle{ f(x) }[/math] is written formally as

[math]\displaystyle{ f(x) = \int e^{i \phi(x, \xi)}\, a(x, \xi) \, \mathrm{d}\xi, }[/math]

where [math]\displaystyle{ \phi(x, \xi) }[/math] and [math]\displaystyle{ a(x, \xi) }[/math] are functions defined on [math]\displaystyle{ \mathbb{R}_x^n \times \mathrm{R}^N_\xi }[/math] with the following properties:

1. The function [math]\displaystyle{ \phi }[/math] is real-valued, positive-homogeneous of degree 1, and infinitely differentiable away from [math]\displaystyle{ \\{\xi = 0\\} }[/math]. Also, we assume that [math]\displaystyle{ \phi }[/math] does not have any critical points on the support of [math]\displaystyle{ a }[/math]. Such a function, [math]\displaystyle{ \phi }[/math] is usually called a phase function. In some contexts more general functions are considered and still referred to as phase functions.
2. The function [math]\displaystyle{ a }[/math] belongs to one of the symbol classes [math]\displaystyle{ S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) }[/math] for some [math]\displaystyle{ m \in \mathbb{R} }[/math]. Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree [math]\displaystyle{ m }[/math]. As with the phase function [math]\displaystyle{ \phi }[/math], in some cases the function [math]\displaystyle{ a }[/math] is taken to be in more general, or just different, classes.

When [math]\displaystyle{ m \lt -N }[/math], the formal integral defining [math]\displaystyle{ f(x) }[/math] converges for all [math]\displaystyle{ x }[/math], and there is no need for any further discussion of the definition of [math]\displaystyle{ f(x) }[/math]. However, when [math]\displaystyle{ m \geq -N }[/math], the oscillatory integral is still defined as a distribution on [math]\displaystyle{ \mathbb{R}^n }[/math], even though the integral may not converge. In this case the distribution [math]\displaystyle{ f(x) }[/math] is defined by using the fact that [math]\displaystyle{ a(x, \xi) \in S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) }[/math] may be approximated by functions that have exponential decay in [math]\displaystyle{ \xi }[/math]. One possible way to do this is by setting

[math]\displaystyle{ f(x) = \lim\limits_{\epsilon \to 0^+} \int e^{i \phi(x, \xi)}\, a(x, \xi) e^{-\epsilon |\xi|^2/2} \, \mathrm{d}\xi, }[/math]

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 [math]\displaystyle{ L }[/math] such that the resulting distribution [math]\displaystyle{ f(x) }[/math] acting on any [math]\displaystyle{ \psi }[/math] in the Schwartz space is given by

[math]\displaystyle{ \langle f, \psi \rangle = \int e^{i \phi(x, \xi)} L\big(a(x, \xi) \, \psi(x)\big) \, \mathrm{d}x \, \mathrm{d}\xi, }[/math]

where this integral converges absolutely. The operator [math]\displaystyle{ L }[/math] is not uniquely defined, but can be chosen in such a way that depends only on the phase function [math]\displaystyle{ \phi }[/math], the order [math]\displaystyle{ m }[/math] of the symbol [math]\displaystyle{ a }[/math], and [math]\displaystyle{ N }[/math]. In fact, given any integer [math]\displaystyle{ M }[/math], it is possible to find an operator [math]\displaystyle{ L }[/math] so that the integrand above is bounded by [math]\displaystyle{ C(1 + |\xi|)^{-M} }[/math] for [math]\displaystyle{ |\xi| }[/math] 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, [math]\displaystyle{ \delta(x) }[/math] is equal to

[math]\displaystyle{ \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} \, \mathrm{d}\xi. }[/math]

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:

[math]\displaystyle{ \delta(x) = \lim_{\varepsilon \to 0^+}\frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} e^{-\varepsilon |\xi|^2/2} \mathrm{d}\xi = \lim_{\varepsilon \to 0^+} \frac{1}{(\sqrt{2\pi \varepsilon})^n} e^{-|x|^2/(2 \varepsilon)}. }[/math]

An operator [math]\displaystyle{ L }[/math] in this case is given for example by

[math]\displaystyle{ L = \frac{(1 - \Delta_x)^k}{(1 + |\xi|^2)^k}, }[/math]

where [math]\displaystyle{ \Delta_x }[/math] is the Laplacian with respect to the [math]\displaystyle{ x }[/math] variables, and [math]\displaystyle{ k }[/math] is any integer greater than [math]\displaystyle{ (n - 1)/2 }[/math]. Indeed, with this [math]\displaystyle{ L }[/math] we have

[math]\displaystyle{ \langle \delta, \psi \rangle = \psi(0) = \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} e^{i x \cdot \xi} L(\psi)(x, \xi)\, \mathrm{d}\xi \, \mathrm{d}x, }[/math]

and this integral converges absolutely.

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

[math]\displaystyle{ L = \sum \limits_{|\alpha| \leq m} p_\alpha(x) D^\alpha, }[/math]

where [math]\displaystyle{ D^\alpha = \partial^\alpha_{x}/i^{|\alpha|} }[/math], then the kernel of [math]\displaystyle{ L }[/math] is given by

[math]\displaystyle{ \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} e^{i \xi \cdot (x - y)} \sum \limits_{|\alpha| \leq m} p_\alpha(x) \, \xi^\alpha \, \mathrm{d}\xi. }[/math]

## 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

## 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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