$\theta$ to $0$).

For entire functions, such as the exponential function $f(z) = {\rm e}^z$, the above exponential decay results are not optimal; indeed, in such cases [*superexponential*]{} decay bounds can be established, exploiting the fact that the coefficients in the Chebyshev expansion of $f$ decay superexponentially. For an example of this type of result, see [@iserles]; see also Example \[example4\] below. Also, in some cases improved decay bounds can be obtained by using different tools from polynomial approximation theory, or exploiting additional structure in $f$ or in the spectra $\sigma (A_n)$; see [@BBR13].

Bounds for the normal case {#normal}
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We briefly discuss the case when the banded matrix $A\in \mathcal{A}$ is normal, but not necessarily Hermitian. As usual, we denote by $m$ the bandwidth of $A$.

The main difference with respect to the previously discussed Hermitian case consists in the fact that $\sigma(A)$ is no longer real. Let ${\cal F}\subset\mathbb{C}$ be a compact, connected region containing $\sigma(A)$, and denote by $\mathbb{P}_k$, as before, the set of complex polynomials of degree at most $k$. Then the argument in (\[step1\]-\[step5\]) still holds, except that now polynomial approximation is no longer applied on a real interval, but on the complex region 