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Unformatted text preview: 1 Lecture 8 Definition 1 Let G be an open and bounded set in R n and x,y G . A kernel K is weakly singular if K is smooth and for x 6 = y there exists constants b > and < n such that  K ( x,y )  b  x + y  Examples: 1) Xray tomography: Consider a circular region with radius R , centered at zero and let ( x,y ) be the attenuation coefficient. Then for each x we have ln I I ( x ) = Z y ( x ) y ( x ) ( x,y ) dy Suppose ( x,y ) is radially symmetric i.e. ( x,y ) = ( r ) with r = ( x 2 + y 2 ) 1 / 2 . A change of variables gives p ( x ) = ln I I ( x ) = Z R x 2 r r 2 x 2 ( r ) dr Changing variables again with z = R 2 r 2 ( r 2 = R 2 z and dz = 2 rdr ) and = R 2 x 2 , ( x ) = ( R 2 z ) results in P ( ) = p ( R 2 z ) = Z ( z )  z dz Then  k ( ,z )  1   z  1 / 2 for all ,z [0 ,R ] 2) Optimal Tomography The field U satisfies the Lipmann  Schwinger equation u ( x ) = u ( x ) k 2 4 Z d e ik  x y   x y  q ( y ) u ( y...
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This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
 Fall '08
 Staff

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