Adaptative Decomposition: The Case of the Drury–Arveson Space

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.