Cohen Type Inequalities for Jacobi, Laguerre and Hermite Expansions

Abstract

Cohen’s inequality was the first result on the way to the solution of Littlewood's conjecture. It is an estimate from below for the norm of a trigonometric polynomial in terms of the number of its nonzero coefficients. Inequalities of this type have been established in various other contexts, e.g., on compact groups or for Jacobi expansions. The purpose of this paper is to prove such inequalities for the classical orthogonal expansions in the appropriate weighted $L^p $ spaces, here in terms of the highest coefficient. The results are best possible, apart from certain limiting values of the space parameter p.