Expansion in some systems of rational functions

A. A. Kitbalyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (1985), pp. 10-15

Résumé

Let $\{ \alpha_k\}_t^\infty$ be an arbitrary sequence of complex numbers lying in the unit disk: $|\alpha_k|1 (k\in N)$. We consider a system of rational functions with poles at the points $z=\alpha_k$ and $z=1/\sqrt{\alpha_k} (k\in N)$: $$P_n(z)=\dfrac{z^2}{( 1-z)^4}[\pi_n(z)+\pi_n^{-1}(z)-2]^2,$$ where $\pi_n(z)=\prod\limits_{k=1}^n\dfrac{z-\alpha_k}{1-\overline{\alpha_k}z}\cdot\dfrac{1-\overline{\alpha_k }}{1-\alpha_k} (n\in N).$ The following theorem is proved. If the series $\sum\limits_{n=1}^\infty (1-|\alpha_n|)$ diverges, then for any function $f(z)$ continuous on the unit circle, the sequence rational functions $f_n(z)$ converges to $f(z)$ uniformly on the unit circle.