Uniform minimality, unconditionality and interpolation in backward shift invariant spaces

TL;DR

This paper explores the relationships between uniform minimality, unconditionality, and interpolation in backward shift invariant spaces, highlighting differences from classical Hardy spaces and emphasizing the role of space size adjustments.

Contribution

It establishes new connections between these properties in backward shift invariant spaces and shows how changing the space's size affects these relationships.

Findings

Uniform minimality does not imply interpolation or unconditionality in Paley-Wiener spaces.

Adjusting the space size can lead to unconditionality or interpolation from minimality.

Khinchin's inequalities are crucial in proving the main results.

Abstract

We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and other scales of spaces), changing the size of the space seems %in this context necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two directions: lowering the power of integration, or "increasing" the defining inner function (e.g. increasing the type in the case of Paley-Wiener space). Khinchin's inequalities play a substantial r\^ole in the proofs of our main results.