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By Symes W.

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2 )n d p(x )eix u^( ) =: p(x D)u(x) de nes a map from smooth functions of bounded support to smooth functions (and between many other function classes as well). Such an operator is called pseudodi erential. (It is conventional to denote the operator associated with the p symbol by replacing the Fourier vector with the derivative vector D = ; ;1r. e. an integral operator with an in nitely smooth kernel. Smoothing operators yield small results when applied to oscillatory functions, so the entire importance of pseudodi erential operators for the theory of wave imaging lies in their ability to describe approximately the behaviour of highfrequency signals.

J; m2 ;1 : IRm (x ) More is true: one can actually develop an asymptotic series 01 1 Z X m dx g(x)ei! ;j A m IR j =0 where the gj are explicitly determined in terms of derivatives of g and associated quantities. We shall make explicit use only of the rst term g0, given above. n;1 @ r (x x x) ! j jr (xs xr x)j @x (xs xr x) det r0r (x sx rx) j jn;1 s r n ^x 0 0 i! j ) : In this and succeeding formulas, xr = xr (xs x ^) as determined by the stationarity conditions. ;1 @ 0 0 0 0 0 (x xr x ) = 1(x xr (xr x)) (x xr (xr x)) @y (xr x) : n By more reasoning of the sort of which the reader has become tired, it is possible to show that @x@ n remains positive.

4) With minor further restriction on support, the class of pseudodi erential operators is closed under composition. Moreover, if p and q are symbols with principal parts p0 and q0, then p(x D)q(x D) = r(x D) and the principal part of r(x ) is p0 (x )q0(x ) | so far as principal parts go, one composes pseudodi erential operators simply by multiplying their symbols! This and some related facts give a calculus of pseudodi erential operators. (5) Di erential operators with smoothly varying coe cients are naturally pseudodifferential operators.

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