exist constants $c_0>0$ and $0<\xi<1$ such that $E_k(f)\leq c_0\, \xi^{k+1}$. From this we can deduce exponentially decaying bounds for $\|[f(A)]_{ij}\|$ with respect to $|i-j|$, by observing that $|i-j|>km$ implies $k+1 < \frac{|i-j|}{m} + 1$ and therefore $$\label{decay_bound}
\| [f(A)]_{ij}\| \le c_0\, \xi^{\frac{|i-j|}{m} + 1} = c\, \zeta^{|i-j|},
\quad c = c_0\, \xi, \quad \zeta = \xi^{\frac{1}{m}} \in (0,1).$$

The above bound warrants further discussion. Indeed, as it is stated it is a trivial bound, in the sense that for any [*fixed*]{} matrix $A$ and function $f$ such that $f(A)$ is defined one can always find constants $c_0 > 0$ and $0 < \xi < 1$ such that (\[decay\_bound\]) holds for all $i,j=1,\ldots ,n$; all one has to do is pick $c_0$ large enough. Thus, the entries of $f(A)$ may exhibit no actual decay behavior! However, what is important here is that the constants $c_0$ and $\xi$ (or at least bounds for them) can be given explicitly in terms of properties of $f$ and, indirectly, in terms of the bounds $\alpha$ and $\beta$ on the spectrum of $A$. If we have a sequence $\{A_n\}$ of $n\times n$ matrices such that

-   the $A_n$ are banded with bounded bandwidth (independent of $n$);

