may be obtained by varying $\chi$ for different values of $i$ and $j$. See [@BBR13] for examples and additional discussion of this issue.

The bounds in Theorem \[thm\_h2\] essentially state that as long as the possible singularities of $f$ remain bounded away from the interval $[-1,1]$ or, slightly more generally, from a (fixed) interval $[\alpha, \beta]$ containing the union of all the spectra $\sigma (A_n)$, $n\in \mathbb{N}$, then the entries of $f(A_n)$ decay exponentially fast away from the main diagonal, at a rate that is bounded below by a positive constant that does not depend on $n$. As a consequence, for every $\varepsilon >0$ one can determine a bandwidth $M = M(\varepsilon)$ (independent of $n$) such that $$\|f(A_n) - [f(A_n)]_{M}\| < \varepsilon$$ holds for all $n$, where $[B]_{M}$ denotes the matrix with entries $b_{ij}$ equal to those of $B$ for $|i-j|\le M$, zero otherwise. It is precisely this fact that makes exponential decay an important property in applications; see, e.g., [@BBR13]. As a rule, the closer the possibile singularities of $f$ are to the spectral interval $[\alpha, \beta]$, the slower the decay is (that is, the larger is $c$ and the closer $\zeta$ is to the upper bound $1$, or 