Interpolation in $H^{p}$ spaces over the right half-plane

Abstract

For a certain class of sequences with multiple terms $\{\underbrace{\lambda _1,\lambda _1,\ldots ,\lambda _1}_{\mu _1 - times}, \underbrace{\lambda _2,\lambda _2,\ldots ,\lambda _2}_{\mu _2 - times},\ldots \}$ in the right half-plane $\mathbb {C}_+$, and a doubly-indexed sequence $\{d_{n,k}{:}\, n\in \mathbb {N},\, k=0,1,\ldots ,\mu _n-1\}$ of complex numbers satisfying certain growth conditions, we consider an interpolation problem

$$\begin{aligned} f^{(k)}(\lambda _n)=d_{n,k}\qquad n\in \mathbb {N},\quad k=0,1,\ldots ,\mu _n-1, \end{aligned}$$

where f is a bounded analytic function in $\mathbb {C}_+$, belonging to the Hardy spaces $H^1 (\mathbb {C}_+)$ and $H^2 (\mathbb {C}_+)$.