Minimality, (Weighted) Interpolation in Paley-Wiener Spaces and Control Theory

TL;DR

This paper explores the relationship between minimality, interpolation, and control in Paley-Wiener spaces, showing that under certain conditions, weak controllability implies exact controllability with a slight time extension.

Contribution

It establishes new links between minimality, interpolation, and controllability in Paley-Wiener spaces, extending classical results from Hardy spaces.

Findings

Carleson condition plus minimality implies interpolation in slightly larger Paley-Wiener spaces.

Results connect weak controllability to exact controllability with a small time increase.

Application of interpolation results to control theory enhances understanding of controllability in signal spaces.

Abstract

It is well known from a result by Shapiro-Shields that in the Hardy spaces, a sequence of reproducing kernels is uniformly minimal if and only if it is an unconditional basis in its span. This property which can be reformulated in terms of interpolation and so-called weak interpolation is not true in Paley-Wiener spaces in general. Here we show that the Carleson condition on a sequence $\Lambda$ together with minimality in Paley-Wiener spaces $PW_{\tau}^{p}$ of the associated sequence of reproducing kernels implies the interpolation property of $\Lambda$ in $PW_{\tau+\epsilon}^{p}$, for every $\epsilon>0$. With the same technics, using a result of McPhail, we prove a similary result about minimlity and weighted interpolation in $PW_{\tau+\epsilon}^{p}$.. We apply the results to control theory, establishing that, under some hypotheses, a certain weak type of controllability in time $\tau>0$ implies exact controllability in time $\tau+\epsilon$, for every $\epsilon>0$.