M(f)}{\chi-1}\right\}$ and $\zeta =\left(\frac{1}{\chi}\right)^{1/m}$.

Letting $\theta := - \ln \zeta >0$ the decay bound can be rewritten in the form $\|[f(A)]_{ij}\|\leq c\, {\rm e}^{-\theta |i-j|}$, which is sometimes more convenient.

\[thm\_h2\] Let ${\mathcal A}_0$ be a $C^*$-algebra and let $\{A_n\}_{n\in\mathbb{N}}\subset {\mathcal A}_0^{n\times n}$ be a sequence of Hermitian matrices of increasing size, with bandwidths uniformly bounded by $m\in\mathbb{N}$ and spectra all contained in $[-1,1]$. Let the complex function $f(z)$ be continuous on a Bernstein ellipse $\mathcal{E}_\chi$ and analytic in the interior of $\mathcal{E}_\chi$, and assume $f(z)\in \mathbb{R}$ for $z\in \mathbb{R}$. Then there exist constants $c>0$ and $0<\zeta <1$, independent of $n$, such that $$\|[f(A_n)]_{ij}\|\leq c\,\zeta^{|i-j|} = c\, {\rm e}^{-\theta |i-j|}, \quad
     \theta = - \ln \zeta\,,$$ for all indices $i,j$. Moreover, one can choose $c=\max\left\{\|f(A)\|,
\frac{2 M(f)}{\chi-1}\right\}$ and $\zeta =\left(\frac{1}{\chi}\right)^{1/m}$.

It is worth noting that the decay bounds in the above results are actually [*families*]{} of bounds; different choices of the ellipse $\mathcal{E}_\chi$ will result in different bounds. If $\chi$ and $\chi'$, with $\chi < \chi'$, are both admissible values, choosing $\chi'$ will result in a smaller value of $\zeta$, thus yielding a faster asymptotic decay rate, but possibly a larger value of the prefactor $c$; in general, tighter bounds 