# Oscillatory integral

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 $\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}$ with the following properties:
* 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.
* The function $a$ belongs to one of the symbol classes $S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})$ 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 $m<-N$, 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 $m\geq -N$, 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 $a(x,\xi )\in S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})$ may be approximated by functions that have exponential decay in $\xi$. One possible way to do this is by setting
    $f(x)=\lim \limits _{\epsilon \to 0^{+}}\int e^{i\phi (x,\xi )}\,a(x,\xi )e^{-\epsilon |\xi |^{2}/2}\,\mathrm {d} \xi ,$
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 Schwartz space is given by
    $\langle f,\psi \rangle =\int e^{i\phi (x,\xi )}L{\biga(x,\xi )\,\psi (x){\big )}\,\mathrm {d} x\,\mathrm {d} \xi ,$
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 $M$, it is possible to find an operator $L$ so that the integrand above is bounded by $C(1+|\xi |)^{-M}$ 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.
The Fourier inversion theorem implies that the delta function, $\delta (x)$ is equal to
    ${\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }\,\mathrm {d} \xi .$
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:
    $\delta (x)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }e^{-\varepsilon |\xi |^{2}/2}\mathrm {d} \xi =\lim _{\varepsilon \to 0^{+}}{\frac {1}{({\sqrt {2\pi \varepsilon }})^{n}}}e^{-|x|^{2}/(2\varepsilon )}.$
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
    $\langle \delta ,\psi \rangle =\psi (0)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }L(\psi )(x,\xi )\,\mathrm {d} \xi \,\mathrm {d} x,$
and this integral converges absolutely.
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 $D^{\alpha }=\partial _{x}^{\alpha }/i^{|\alpha |}$, 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 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.