User:Holmansf/oscillatoryintegrals

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.

Definition
An oscillatory integral $$ f(x) $$ is written formally as


 * $$ f(x) = \int e^{i \phi(x,\xi)}\, a(x,\xi) \, \mathrm{d} \xi $$

where $$ \phi(x,\xi) $$ and $$ a(x,\xi) $$ are functions defined on $$ \mathbb{R}_x^n \times \mathrm{R}^N_\xi $$ with the following properties.


 * 1) The function $$ \phi $$ is real valued, positive homogeneous of degree 1, and infinitely differentiable away from $$ \{\xi = 0 \} $$. Also, we assume that $$ \phi $$ does not have any critical points on the support of $$ a $$. Such a function, $$ \phi $$ 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 $$ S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) $$ for some $$ m \in \mathbb{R}$$. Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree $$ m $$. As with the phase function $$ \phi $$, in some cases the function $$ a $$ is taken to be in more general, or just different, classes.

When $$ m < -n + 1 $$ 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 $$ m \geq - n+1$$ the oscillatory integral is still defined as a distribution on $$ \mathbb{R}^n $$ even though the integral may not converge. In this case the distribution $$ f(x) $$ is defined by using the fact that $$ a(x,\xi) \in S^m_{1,0}(\mathbb{R}_x^n \times \mathrm{R}^N_\xi) $$ may be approximated by functions that have exponential decay in $$ \xi$$. One possible way to do this is by setting


 * $$ f(x) = \lim \limits_{\epsilon \rightarrow 0^+} \int e^{i \phi(x,\xi)}\, a(x,\xi) e^{-\epsilon |\xi|^2/2} \, \mathrm{d} \xi $$

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 $$ \psi $$ in the Schwarz space is given by


 * $$ \langle f, \psi \rangle = \int e^{i \phi(x,\xi)} L \left ( a(x,\xi) \, \psi(x) \right ) \, \mathrm{d} x \, \mathrm{d} \xi $$

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 $$ \phi $$, the order $$ m $$ of the symbol $$ a $$, and $$ N$$. In fact, given any integer $$ N $$ it is possible to find an operator $$ L $$ so that the integrand above is bounded by $$ C (1 + |\xi|)^{-N} $$ for $$ |\xi| $$ 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 implies that the delta function, $$ \delta(x) $$ is equal to


 * $$ \frac{1}{(2 \pi)^n}\int_{\mathbb{R}^n} e^{i x \cdot \xi} \mathrm{d} \xi $$.


 * 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:


 * $$ \delta(x) = \lim \limits_{\epsilon \rightarrow 0^+}\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} e^{i x\cdot \xi} e^{-\epsilon |\xi|^2/2} \mathrm{d} \xi = \lim \limits_{\epsilon \rightarrow 0^+} \frac{1}{(\sqrt{2 \pi \epsilon})^n} e^{-|\xi|^2/(2 \epsilon)} $$.


 * An operator $$ L $$ in this case is given for example by


 * $$ L = \frac{(1 - \Delta_x)^k}{(1 + |\xi|^2)^k} $$


 * where $$ \Delta_x $$ 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


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


 * and this integral converges absolutely.


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


 * $$ L = \sum \limits_{|\alpha| \leq m} p_\alpha(x) D^\alpha $$


 * where $$ D^\alpha = \partial^\alpha_{x}/i^{|\alpha|} $$, then the kernel of $$ L $$ is given by


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

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.