${\cal F}$. Therefore, the following bound holds for all indices $i$, $j$ such that $|i-j|>km$: $$\|[f(A)]_{ij}\|\leq\sup |(f-p)(\sigma (A))|\leq E_k(f,{\cal F}),
\label{bound_nonhermitian}$$ where $$E_k(f,{\cal F}):=\min_{p\in\mathbb{P}_k}\max_{z\in {\cal F}}|f(z)-p(z)|.$$ Unless more accurate estimates for $\sigma(A)$ are available, a possible choice for $\cal F$ is the disk of center $0$ and radius $\rho(\hat{A})$: see Remark \[remark2\].

If $f$ is analytic on $\cal F$, bounds for $E_k(f,{\cal F})$ that decay exponentially with $k$ are available through the use of Faber polynomials: see [@BR07 Theorem 3.3] and the next section for more details. More precisely, there exist constants $\tilde{c}>0$ and $0<\tilde{\lambda}<1$ such that $E_k(f,{\cal F})\leq \tilde c\, \tilde{\lambda}^k$ for all $k\in\mathbb{N}$. This result, together with , yields for all $i$ and $j$ the bound $$\|[f(A)]_{ij}\|\leq c\, \lambda^{|i-j|} = c\, {\rm e}^{-\theta |i-j|}$$ (where $\theta = -\ln \lambda$) for suitable constants $c>0$ and $0<\lambda<1$, which do not depend on $n$, although they generally depend on $f$ and $\cal F$.

Bounds for the general case {#general}
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If $A$ is not normal, then the equality between (\[step2\]) and (\[step3\]) does not hold. We therefore need other explicit bounds on the norm of a function of a matrix.

The field of values and bounds for complex matrices
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Given 