  We propose and investigate two new methods to approximate $f({\bf A}){\bf b}$
for large, sparse, Hermitian matrices ${\bf A}$. The main idea behind both
methods is to first estimate the spectral density of ${\bf A}$, and then find
polynomials of a fixed order that better approximate the function $f$ on areas
of the spectrum with a higher density of eigenvalues. Compared to
state-of-the-art methods such as the Lanczos method and truncated Chebyshev
expansion, the proposed methods tend to provide more accurate approximations of
$f({\bf A}){\bf b}$ at lower polynomial orders, and for matrices ${\bf A}$ with
a large number of distinct interior eigenvalues and a small spectral width.
