Interpolating sequences for the Nevanlinna and Smirnov classes

Interpolation and peak functions for the Nevanlinna and Smirnov classes

It is known (implicit in [HMNT]) that when $Λ$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_λ$ in these spaces, with uniform control of their growth and attaining values 1 on $λ$ and 0 in all other $λ'\neqλ$. We provide an example showing that, contrary to what happens in other algebras of holomorphic functions, the existence of such functions does not imply that $Λ$ is an interpolating sequence.

Ideals in some Rings of Nevanlinna-Smirnov Type

Let $N^p$ $(1<p<\infty)$ denote the algebra of holomorphic functions in the open unit disk, introduced by I.~I.~Privalov with the notation $A_q$ in [8]. Since $N^p$ becomes a ring of Nevanlinna--Smirnov type in the sense of Mortini [7], the results from [7] can be applied to the ideal structure of the ring $N^p$. In particular, we observe that $N^p$ has the Corona Property. Finally, we prove the $N^p$-analogue of the Theorem 6 in [7], which gives sufficient conditions for an ideal in $N^p$, generated by a finite number of inner functions, to be equal to the whole algebra $N^p$.

Nevanlinna和Smirnov类及相关空间中的随机插值

Random interpolation in the Nevanlinna and Smirnov classes and related spaces

本文研究了Nevanlinna和Smirnov类及相关空间中随机插值序列的性质，揭示了其与Blaschke条件的关系，并探讨了大Hardy-Orlicz空间中插值条件的刻画问题。

The Corona Property in Nevanlinna quotient algebras and Interpolating sequences

Let $I$ be an inner function in the unit disk $\mathbb D$ and let $\mathcal N$ denote the Nevanlinna class. We prove that under natural assumptions, Bezout equations in the quotient algebra $\mathcal N/I\mathcal N$ can be solved if and only if the zeros of $I$ form a finite union of Nevanlinna interpolating sequences. This is in contrast with the situation in the algebra of bounded analytic functions, where being a finite union of interpolating sequences is a sufficient but not necessary condition. An analogous result in the Smirnov class is proved as well as several equivalent descriptions of…

Factoring derivatives of functions in the Nevanlinna and Smirnov classes

We prove that, given a function $f$ in the Nevanlinna class $N$ and a positive integer $n$, there exist $g\in N$ and $h\in BMOA$ such that $f^{(n)}=gh^{(n)}$. We may choose $g$ to be zero-free, so it follows that the zero sets for the class $N^{(n)}:=\{f^{(n)}: f\in N\}$ are the same as those for $BMOA^{(n)}$. Furthermore, while the set of all products $gh^{(n)}$ (with $g$ and $h$ as above) is strictly larger than $N^{(n)}$, we show that the gap is not too large, at least when $n=1$. Precisely speaking, the class $\{gh': g\in N, h\in BMOA\}$ turns out to be the smallest ideal space containing …

Traces of the Nevanlinna class on discrete sequences

We show that a discrete sequence $Λ$ of the unit disk is the union of $n$ interpolating sequences for the Nevanlinna class $N$ if and only if the trace of $N$ on $Λ$ coincides with the space of functions on $Λ$ for which the divided differences of order $n - 1$ are uniformly controlled by a positive harmonic function.