I'd like to have some informations about Markov-type functions (or Cauchy-type): \[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\] $\gamma$ is a positive measure with compact support $\subset \bar{R} $.
I'm wondering if there is a reference that explains the study of such functions and gives some examples ($\exp(z)$,$z^{-1/2}$,$\frac{log(1+z)}{z}$,...) I'm interested in rational approximation of Markov functions to bound an error estimation in the domain of matrix functions.
Rational approximations of Markov functions are expressed in terms of orthogonal polynomials with respect to $\gamma$ and related functions, and as Alexandre Eremenko pointed out you'll find more details by looking at Padé approximation; the error terms are somehow explicit. An excellent book for the topic is "Rational approximations and orthogonality" of Nikishin and Sorokin.
Btw, these functions are moment generating functions, since $$ \int \frac{d\gamma(x)}{z-x}=\sum_{k=0}^\infty z^{-k-1}\int x^k d\gamma(x), $$ and encodes all the information concerning $\gamma$.
Theorem 3 in Chapter VI of Akhiezer and Glazman “Theory of Linear Operators in Hilbert Space,” Ungar, New York, 1963, gives a characterization, a holomorphic function $f$ that maps the upper half plane to the lower half plane can be represented as $\int_{\mathbb{R}} \frac{1}{z-x} \gamma({\rm d}x)$ with a finite measure $\gamma$ if and only if $$ \limsup_{y\to\infty}y|f(iy)|<\infty. $$ The $\limsup$ then is equal to $\gamma(\mathbb{R})$.
