Fejér-type positive operator based on Takenaka--Malmquist system on unit circle

Abstract

Let $\phi =\{{\phi }_k{\}}_{k=-\infty }^{\infty }$ denote the extended Takenaka--Malmquist system on unit circle $\mathbb{T}$ and let ${\sigma }_{n,\phi }(f),$ $f\in L^1(\mathbb{T})$, be the Fejér-type operator based on $\phi$, introduced by V. N. Rusak. We give the convergence criteria for ${\sigma }_{n,\phi }(f)$ in Banach space $X(\mathbb{T}):=L^p(\mathbb{T})\vee C(\mathbb{T})$, $p\ge 1$. Also we prove the Voronovskaya-type theorem for ${\sigma }_{n,\phi }(f)$ on class of holomorphic functions representable by Cauchy-type integrals with bounded densities.