Fourier integral operator

The concept of Fourier integral operators stems from mathematical analysis. They have become an important tool in the theory of partial differential equations. The class of Fourier integral Operators contains differential operators as well as classical integral operators as special cases.

A Fourier integral operator T is given by::$(Tf)(x)=int\_\{mathbb\{R\}^n\}e^\{2pi\; i\; Phi(x,xi)\}a(x,xi)hat\{f\}(xi),dxi$

where $hat\; f$ denotes the Fourier transform of "f", "a"("x","ξ") is a standard symbol of order 0 which is compactly supported in "x" and Φ is real valued and homogeneous of degree 1 in ξ. It is also necessary to require that $operatorname\{det\}left(frac\{partial^2\; Phi\}\{partial\; x\_ipartial\; xi\_j\}\; ight)\; eq\; 0$ on the support of "a." Under these conditions it is possible to show that "T" defines an "L"² bounded operator harv|Stein|1993.

References

* F. Treves, Introduction to Pseudo Differential and Fourier Integral Operators, (University Series in Mathematics), Plenum Publ. Co. 1981. ISBN 0-306-40404-4
* J.J. Duistermaat, Fourier Integral Operators, (Progress in Mathematics), Birkhäuser 1995. ISBN 0-817-63821-0
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