Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods

Let m be a polynomial satisfying some mild conditions. Given a set R ⊂ C, a continuous function f on R and its best approximation p*_{n- 1} from II_n with respect to the maximum norm, we show that p*_n p*_n is a best approximation to f o _m on the inverse polynomial image 5 of R, i.e. _m(S) = R, where the extremal signature is given explicitly. A similar result is presented for constrained Chebyshev polynomial approximation. Finally, we apply the obtained results to the computation of the convergence rate of Krylov subspace methods when applied to a preconditioned linear system. We investigate pairs of preconditioners where the eigenvalues are contained in sets 5 and R, respectively, which are related by _m(S) = R.