THE NEAR-BEST SOLUTION OF A POLYNOMIAL MINIMIZATION PROBLEM BY THE CARATHEODORY-FEJER METHOD

For a compact set Ω{square image of or equal to}C, 1∉Ω, we consider the Chebyshev problem {Mathematical expression}, where n is a fixed nonnegative integer and IIm denotes the space of all complex polynomials of degree m. This problem is of importance for the construction of semi-iterative methods for singular systems of linear algebraic equations. In the case when Ω is a Jordan region whose boundary is sufficiently smooth, we determine the asymptotic behavior of {norm of matrix}pm*{norm of matrix}Ω, where Pm*, denotes the solution of the above Chebyshev problem. Moreover, using the Carathéodory-Fejér method, we construct a "near minimal" solution {Mathematical expression} of this problem if CΩ is simply connected. © 1990 Springer-Verlag New York Inc.