This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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) X-ray 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...
View Full Document
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