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

Abstract

Let \(U,V\subseteq\mathbb{C}\) be open convex sets, and \(z_{1}\), \(z_{2}\), \(\dots,z_{N}\in U\) and \(w_{1}\), \(w_{2}\), \(\dots,w_{M}\in V\) be (maybe repetitive) points. Let \(f:\,U\times V\to\mathbb{C}\) be an analytic function. Let the interpolating polynomial \(p\) be determined by the values of \(f\) on the rectangular grid \((z_{i},w_{j})\), \(i=1,2,\dots,N\), \(j=1,2,\dots,M\). Let \(A\) and \(B\) be matrices of the sizes \(n\times n\) and \(m\times m\), respectively. The function \(f\) of \(A\) and \(B\) can be defined by the formula

$$f(A,B)=\frac{1}{(2\pi i)^{2}}\int\limits_{\Gamma_{1}}\int\limits_{\Gamma_{2}}f(\lambda,\mu)(\lambda\mathbf{1}-A)^{-1}\otimes(\mu\mathbf{1}-B)^{-1}\,d\mu\,d\lambda,$$

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.