  We study AAK as well as Pad\'e approximants to functions f, where f is a sum
of a Cauchy transform of a complex measure \mu supported on a real interval
included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine
distribution on its support is Dini-continuous, non-vanishing and with and
argument of bounded variation, and of a rational function with no poles on the
support of \mu. It is shown that the approximants converge to f locally
uniformly in the domain of holomorphy of f, intersected with the unit disk in
the case of AAK approximants. In the case of Pad\'e approximants we need to
assume that the interpolation scheme is "nearly" conjugate-symmetric.
