  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.
