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 ananalytic function. Let the interpolating polynomial \(p\) bedetermined 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 bythe 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 sameway. An estimate of \(||f(A,B)-p(A,B)||\) is given.
