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 ) {\displaystyle f(x)} is written formally as
f ( x ) = ∫ e i ϕ ( x , ξ ) a ( x , ξ ) d ξ , {\displaystyle f(x)=\int e^{i\phi (x,\xi )}\,a(x,\xi )\,\mathrm {d} \xi ,}
where ϕ ( x , ξ ) {\displaystyle \phi (x,\xi )} and a ( x , ξ ) {\displaystyle a(x,\xi )} are functions defined on R x n × R ξ N {\displaystyle \mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}} with the following properties:
The function ϕ {\displaystyle \phi } is real-valued, positive-homogeneous of degree 1, and infinitely differentiable away from { ξ = 0 } {\displaystyle \{\xi =0\}} . Also, we assume that ϕ {\displaystyle \phi } does not have any critical points on the support of a {\displaystyle a} . Such a function, ϕ {\displaystyle \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 {\displaystyle a} belongs to one of the symbol classes S 1 , 0 m ( R x n × R ξ N ) {\displaystyle S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})} for some m ∈ R {\displaystyle m\in \mathbb {R} } . Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree m {\displaystyle m} . As with the phase function ϕ {\displaystyle \phi } , in some cases the function a {\displaystyle a} is taken to be in more general, or just different, classes. When m < − N {\displaystyle m<-N} , the formal integral defining f ( x ) {\displaystyle f(x)} converges for all x {\displaystyle x} , and there is no need for any further discussion of the definition of f ( x ) {\displaystyle f(x)} . However, when m ≥ − N {\displaystyle m\geq -N} , the oscillatory integral is still defined as a distribution on R n {\displaystyle \mathbb {R} ^{n}} , even though the integral may not converge. In this case the distribution f ( x ) {\displaystyle f(x)} is defined by using the fact that a ( x , ξ ) ∈ S 1 , 0 m ( R x n × R ξ N ) {\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 ξ {\displaystyle \xi } . One possible way to do this is by setting
f ( x ) = lim ϵ → 0 + ∫ e i ϕ ( x , ξ ) a ( x , ξ ) e − ϵ | ξ | 2 / 2 d ξ , {\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 ,}
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