Random Interpolating Sequences in the Polydisc and the Unit Ball

TL;DR

This paper investigates the probabilistic properties of random sequences in the polydisc and unit ball, establishing laws and conditions for their interpolation capabilities across various function spaces.

Contribution

It provides new probabilistic laws and conditions for random sequences to be interpolating in complex domains, highlighting differences between the unit ball and polydisc.

Findings

Almost sure interpolating property follows a 0-1 law in the unit ball.

Necessary and sufficient conditions for interpolation in the polydisc are identified.

Differences between the unit ball and polydisc regarding interpolation are analyzed.

Abstract

We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^{\sigma}\left(\mathbb{B}_{d}\right)$, in the range $0 < \sigma\leq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $\mathrm{H}^2(\mathbb{D}^d)$ and its multiplier algebra $\mathrm{H}^\infty(\mathbb{D}^d)$: in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $\mathrm{H}^\infty(\mathbb{D}^d)$-interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $0-1$ law for random interpolating sequences for $\mathrm{H}^2(\mathbb{D}^d)$.