Generalized Principal Value Integrals and Related Random Processes

In previous chapters the importance of the existence of an iterative solution of the majorant equation was emphasized many times. For a lot of problems of mathematical physics, including radiation transport problems, the situation is simplified since a majorant operator coincides with an initial one. Meanwhile it is easy to list many examples where the investigation of majorant equations is very useful. The most simple one is the matrix operator $$ A = \left( {\begin{array}{*{20}c} a & a \\ a & { - a} \\ \end{array} } \right)$$ An iterative procedure $$X_{n + 1} = AX_n + F,X_0 = F $$ converges for $$|a| < 1/\sqrt 2 $$ but it it is easy to see that a majorant iterative procedure converges only for $$ |a| < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}$$.