Approximation of functions analytic in a simply connected domain and representable with the help of Cauchy type integral by sequences of rational fractions with poles prescribed by a given matrix

G. Ts. Tumarkin

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 254-273

Résumé

Let $G$ and $\{x_{kj}\}$ be the domain and the matrix mentioned in the title, the boundary of $G$ being rectifiable. A general scheme of approximation of functions $f$ in $G$ representable in the form $f(z)=(2\pi i)^{-1}\int g(\zeta)(\zeta-z)^{-1}d \zeta$ with $g\in Z_2(\partial G)$ by a sequence of rational fractions $\{r_k\}$ is described. A specific feature of this scheme is that the poles of $r_k$ are all in the $k$-th row of $\{x_{kj}\}$. A necessary and sufficient condition on $\{x_{kj}\}$ is given for all functions $f$ as above to be approximable, uniformly inside $G$, with the help of the scheme in question. In the case when this condition is not satisfied, all approximable functions are described, provided $\mathbb{C}\setminus G$ is a Smirnov domain.