On polynomial approximations to fa(Z)(z-a)−1 with complex a and some applications to certain non-hermitian matrices

AbstractWe are concerned with the problem1 $$\mathop {min}\limits_{p \in P_n } \mathop {max}\limits_{z \in [ - 1,1]} |w(z)(f_a (z) - p(z))|,a \in C/[ - 1,1],n = 0 \cdots $$ of best polynomial approximation of degree n to fa(z)=(z−a)−1 on the unit interval. Here Pn denotes the class of complex polynomials of degree at most n, and ω belongs to a certain classical family of weight functions. For real a the solution of this approximation problem is known. In this paper, we obtain the best approximations for purely imaginary a. For general a, close approximations to the optimal polynomials are derived by solving the approximation problem expli citly for a certain subclass of Pn. We then use these polynomials to devise an iterative method for the solution of linear systems Ax=b with coefficient matrices of the form A=cI+dT where T=TH and c, d ∈C. Finally, as a further appication of our results, we derive bounds for the decay rates of the inverses of banded matrices A=cI+dT.