Shift basic sequences in the Wiener disc algebra

Abstract:

Let $W(D)$ denote the set of functions $f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}$ for which $\sum \nolimits _{n = 0}^\infty {\left | {{a_n}} \right | < + \infty }$. It is shown that for any positive integer $k$ the $k$-shifted sequence $\left \{ {{z^{kn}} \cdot f(z)} \right \}_{n = 0}^\infty$ is a basic sequence in $W(D)$ equivalent to the basis $\left \{ {{z^n}} \right \}_{n = 0}^\infty$ if and only if $f(z)$ has no set of $k$ symmetrically distributed zeros on the circle $\left | z \right | = 1$.