Oscillatory integral

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: • 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^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 , 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^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 \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\big(a(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^{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_{\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)}. 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. The Schwartz kernel of any differential operator can be written as an oscillatory integral. Indeed if : L = \sum \limits_ , 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. == See also ==