Jackson's Theorem on Bounded Symmetric Domains

Abstract

Polynomial approximation is studied on bounded symmetric domain Ω in ℂ n for holomorphic function spaces \({\fancyscript X}\) , such as Bloch-type spaces, Bergman-type spaces, Hardy spaces, Ω algebra and Lipschitz space. We extend the classical Jackson’s theorem to several complex variables: $$ E_{k} {\left( {f,{\fancyscript X}} \right)} \leqslant \omega {\left( {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k,f,{\fancyscript X}} \right)}, $$ where E k (f,\({\fancyscript X}\) ) is the deviation of the best approximation of f ∈ \({\fancyscript X}\) by polynomials of degree at most k with respect to the \({\fancyscript X}\) -metric and ω(1/k, f,\({\fancyscript X}\) ) is the corresponding modulus of continuity.