Estimates of entropy for multiplier operators of systems of orthonormal functions

Abstract

We obtain upper and lower estimates for ɛ -entropy and entropy numbers of multiplier operators of systems of orthonormal functions bounded from L p to L q . Upper estimates in our study require that a Marcinkiewicz-type multiplier theorem is available for the system. As application we obtain estimates for ɛ -entropy and entropy numbers of the multiplier operators associated with the sequences ( k − γ ln k − ξ ) k = 2 ∞ and ( e − γ k r ) k = 0 ∞ where γ > 0 , ξ ≥ 0 and 0 < r ≤ 1 . Some of these estimates are order sharp. We verify that the trigonometric system on the circle, the Vilenkin system and the Walsh system satisfy the conditions of our study. We also study analogous results for the Haar system and the Walsh systems on spheres.