# 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 ${\displaystyle f(x)}$ is written formally as

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

where ${\displaystyle \phi (x,\xi )}$ and ${\displaystyle a(x,\xi )}$ are functions defined on ${\displaystyle \mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}}$ with the following properties:

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

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

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

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

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

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

${\displaystyle {\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }\,\mathrm {d} \xi .}$

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:

${\displaystyle \delta (x)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }e^{-\varepsilon |\xi |^{2}/2}\mathrm {d} \xi =\lim _{\varepsilon \to 0^{+}}{\frac {1}{({\sqrt {2\pi \varepsilon }})^{n}}}e^{-|x|^{2}/(2\varepsilon )}.}$

An operator ${\displaystyle L}$ in this case is given for example by

${\displaystyle L={\frac {(1-\Delta _{x})^{k}}{(1+|\xi |^{2})^{k}}},}$

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

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

and this integral converges absolutely.

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

${\displaystyle L=\sum \limits _{|\alpha |\leq m}p_{\alpha }(x)D^{\alpha },}$

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

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

## 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.