Given a sequence $\{A_n\}$ of matrices $A_n$ of increasing dimension $d_n$ with $d_k>d_q$ for $k>q$, $k,q\in \mathbb{N}$, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that $\{\{f(B_{n,m})\}\:\ m\in\mathbb{N}\}$ is an a.c.s. for $\{f(A_n)\}$, if $\{\{B_{n,m}\}\:\ m\in\mathbb{N}\}$ is an a.c.s. for $\{A_n\}$, $\{A_n\}$ is sparsely unbounded, and $f$ is a suitable continuous function defined on $\mathbb{R}$. We also discuss the potential impact and future developments of such a result.
Approximating classes of sequences: The Hermitian case
SERRA CAPIZZANO, STEFANO;
2011-01-01
Abstract
Given a sequence $\{A_n\}$ of matrices $A_n$ of increasing dimension $d_n$ with $d_k>d_q$ for $k>q$, $k,q\in \mathbb{N}$, we recently introduced the concept of approximating class of sequences (a.c.s.) in order to define a basic approximation theory for matrix sequences. We have shown that such a notion is stable under inversion, linear combinations, and product, whenever natural and mild conditions are satisfied. In this note we focus our attention on the Hermitian case and we show that $\{\{f(B_{n,m})\}\:\ m\in\mathbb{N}\}$ is an a.c.s. for $\{f(A_n)\}$, if $\{\{B_{n,m}\}\:\ m\in\mathbb{N}\}$ is an a.c.s. for $\{A_n\}$, $\{A_n\}$ is sparsely unbounded, and $f$ is a suitable continuous function defined on $\mathbb{R}$. We also discuss the potential impact and future developments of such a result.
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