Adaptative decomposition: the case of the Drury-Arveson space

TL;DR

This paper extends the maximum selection principle and adaptive decomposition techniques from the Hardy space of the disk to the Drury-Arveson space in several complex variables, introducing algorithms and convergence analysis.

Contribution

It generalizes the maximum selection principle to the Drury-Arveson space and develops an adaptive algorithm with convergence properties for this higher-dimensional setting.

Findings

Extended maximum selection principle to the Drury-Arveson space.

Developed an adaptive greedy algorithm for function expansion.

Studied convergence of infinite Blaschke products in this context.

Abstract

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space $\mathbf H_2$ of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of $\mathbb C^N$, and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury-Arveson space of functions analytic in the unit ball of $\mathbb C^N$. This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.