Helson operators and coinvariant subspaces

We give an explicit expression of the functions in the finite-dimensional coinvariant subspace of the Hardy and weighted Bergman spaces over the infinite polydisc. As an application, an alternative form of the Perfekt-Pushnitski theorem characterizing Helson operators with finite rank is obtained. Interestingly, it turns out that a Helson operator is automatically bounded if it is of finite rank.