all. I am now struggling with a approximation problem. Suppose we have a matrix-valued measure $\mathrm{d}\Lambda(\omega)$, with compact support $[a,b]$, then its Cauchy transform is a well-defined analytic function $$ F(z) = \int_a^b \frac{\mathrm{d}\Lambda(\omega)}{\omega - z}, z\in\mathbb C\backslash[a,b]. $$ Unfortunately, this measure $\mathrm{d}\Lambda(\omega)$ does not admits an analytic expression. Instead, only numerical value of $F(z)$ at each discrete point $z\in\mathbb C\backslash[a,b]$ can be calculated very precisely.
My problem is to find a good approximation of $F(x+iy_0)$ for $x\in(-\infty,\infty)$, given a constant $y_0>0$. It seems that the problem boils down to decompose the following complex valued real rational function $$ g(\omega, x) = \frac{1}{\omega - (x + iy_0)}, $$ into the following separable form with $N$ as small as possible $$ g(\omega, x) \approx \sum_{j=1}^N \frac{f_j(x)}{\omega - x_j}, $$ where the point $x_j\in\mathbb C\backslash[a,b]$ and the functional form of the function $f_j(x)$ are to be determined. How can we find these $x_j$ and $f_j(x)$?