Holomorphic Jackson's theorems in polydiscs

Abstract

The purpose of this article is to establish Jackson-type inequality in the polydiscs U N of C N for holomorphic spaces X , such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely, E k ⇒ ( f , X ) ≲ ω 1 / k → , f , X , where E k ⇒ ( f , X ) is the deviation of the best approximation of f ∈ X by polynomials of degree at most k j about the jth variable z j with respect to the X -metric and ω ( 1 / k → , f , X ) is the corresponding modulus of continuity.