-   the spectra $\sigma(A_n)$ are all contained in a common interval $\cal I$ (independent of $n$), say ${\cal I}=[-1,1]$,

then the bound (\[decay\_bound\]) holds independent of $n$. In particular, the entries of $f(A_n)$ will actually decay to zero away from the main diagonal as $|i-j|$ and $n$ tend to infinity, at a rate that is uniformly bounded below by a positive constant independent of $n$.

More specifically, Bernstein’s Theorem yields the values $c_0=\frac{2 \chi M(f)}{\chi-1}$ and $\xi=1/\chi$, where $\chi$ is the sum of the semi-axes of an ellipse $\mathcal{E}_\chi$ with foci in $1$ and $-1$, such that $f(z)$ is continuous on $\mathcal{E}_\chi$ and analytic in the interior of $\mathcal{E}_\chi$ (and $f(z)\in\mathbb{R}$ whenever $z\in\mathbb{R}$); furthermore, we have set $M(f)=\max_{z\in\mathcal{E}_\chi}|f(z)|$.

Summarizing, we have established the following results:

\[thm\_h1\] Let $\mathcal A = {\mathcal A}_0^{n\times n}$ where ${\mathcal A}_0$ is a $C^*$-algebra and let $A\in\mathcal{A}$ be Hermitian with bandwidth $m$ and spectrum 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 $0<\zeta<1$ such that $$\|[f(A)]_{ij}\|\leq c\,\zeta^{|i-j|}$$ for all $1\leq i,j\leq n$. Moreover, one can choose $c=\max\left\{\|f(A)\|,
\frac{2 