Approximating classes of sequences: The Hermitian case

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.