An Estimate of Approximation of an Analytic Function of Two Matrices by a Polynomial
Vitalii Gennad'evich Kurbatov, I. V. Kurbatova · Lobachevskii Journal of Mathematics · 2024
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.