A lower bound in Nehari's theorem on the polydisc

Abstract

By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_ψis a bounded Hankel form on H^2(D^d) with analytic symbol ψ, then there is a function ϕin L^\infty(\T^d) such that ψis the Riesz projection of ϕ. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|ϕ\|_\infty\le C_d \|H_ψ\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.

Cited by in corpus (8)

• The multiplicative Hilbert matrix
• Helson's problem for sums of a random multiplicative function
• On Helson matrices: moment problems, non-negativity, boundedness, and finite rank
• Volterra operators on Hardy spaces of Dirichlet series
• Weak product spaces of Dirichlet series
• Linear functions and duality on the infinite polytorus
• The Nehari problem for the Paley--Wiener space of a disc
• Spectral measure of large random Helson matrices