On Interpolation by Functions in $\ell^p_A$

TL;DR

This paper investigates interpolation in the space of analytic functions with p-summable coefficients, providing characterizations of interpolating sequences, weak separation, and Carleson measures through a Carlesonian perspective.

Contribution

It introduces new boundedness conditions for Gramian matrices and characterizes universal interpolating sequences and weak separation in $\, ext{ell}^p_A$.

Findings

Boundedness of Gramian matrices characterizes interpolation.

Universal interpolating sequences are characterized as Riesz systems.

Weak separation is characterized via a generalized pseudohyperbolic metric.

Abstract

This work explores several aspects of interpolating sequences for $\ell^p_A$, the space of analytic functions on the unit disk with $p$-summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We investigate various analogues of Gramian matrices, for which we show boundedness conditions are necessary and sufficient for interpolation, including a characterization of universal interpolating sequences in terms of Riesz systems. We also discuss weak separation, giving a characterization of such sequences using a generalization of the pseudohyperbolic metric. Lastly, we consider Carleson measures and embeddings.