An Estimate of Approximation of a Matrix-Valued Function by an Interpolation Polynomial

Abstract

Let A be a square complex matrix; z1, . . . , zn ∈ C be (possibly repetitive) points of interpolation; f be a function analytic in a neighborhood of the convex hull of the union of the spectrum of A and the points z1, . . . , zn; and p be the interpolation polynomial of f constructed by the points z1, . . . , zn. It is proved that under these assumptions ‖f(A)− p(A)‖ ≤ 1 n! max t∈[0,1] µ∈co{z1,z2,...,zn} ∥∥Ω(A)f (n) ( (1− t)µ1 + tA )∥∥, where Ω(z) = ∏n k=1(z − zk) and the symbol co means the convex hull.