An Estimate of Approximation of an Analytic Function of Two Matrices by a Polynomial

Let U, V subset of C be open convex sets, and z(1), z(2),..., z(N) is an element of U and w(1), w(2),..., w(M) is an element of V be (maybe repetitive) points. Let f : U x V -> C be an analytic function. Let the interpolating polynomial p be determined by the values of f on the rectangular grid (z(i), w(j)), i = 1, 2,..., N, j = 1, 2,..., M. Let A and B be matrices of the sizes n x n and m x m, respectively. The function f of A and B can be defined by the formula f(A, B) = 1/(2 pi i)(2) integral(Gamma 1)integral(Gamma 2) f(lambda, mu)(lambda 1 - A)(-1) circle times (mu 1 - B)(-1) d mu d lambda, where Gamma(1) and Gamma(2) surround the spectra sigma(A) and sigma(B), respectively; p(A, B) is defined in the same way. An estimate of || f(A, B) - p(A, B)|| is given.