No Entire Inner Functions

Abstract

We study generalized inner functions on a large family of Reproducing Kernel Hilbert Spaces. We show that the only inner functions that are entire are the normalized monomials.

Analogues of finite Blaschke products as inner functions

“…In a recent paper, Cobos and Seco [13] showed that under several additional assumptions, the only entire inner functions on 𝐻 2 𝑤 are multiples of monomials. It turns out that this result remains true on all 𝐻 2 𝑤 , as we show here.…”

“…Consider a weighted Hardy space $$H^2_{w}$$, where $$w=\lbrace w_k\rbrace _{k\geqslant 0}$$ consists of positive real numbers with $$\begin{equation*} \lim _{k\rightarrow \infty }\frac{w_k}{w_{k+1}}=1. \end{equation*}$$In a recent paper, Cobos and Seco [13] showed that under several additional assumptions, the only entire inner functions on $$H^2_w$$ are multiples of monomials. It turns out that this result remains true on all $$H^2_w$$, as we show here.…”