An Inequality for Trigonometric Polynomials and Its Application for Estimating the Entropy Numbers

Abstract We prove in the two-dimensional case an inequality for trigonometric polynomials with frequencies between two hyperbolic crosses. This inequality is an analog of Talagrand′s inequality for the Haar polynomials. We use this inequality to prove some new estimates of the entropy numbers of classes of functions with bounded mixed difference or derivative in the most difficult case of the uniform norm.