A lower bound in Nehari's theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari's theorem (i.e., if H-psi is a bounded Hankel form on H-2(D-d) with analytic symbol psi, then there is a function phi in L-infinity(T-d) such that psi is the Riesz projection of phi) is known to hold on the polydisc D-d for d > 1. A method proposed in Helson's last paper is used to show that the constant C-d in the estimate parallel to phi parallel to(infinity) <= C-d parallel to H-psi parallel to grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.