Jackson’s Theorem in Hardy-Sobolev Type Spaces in the Unit Polydiscs

Abstract

The purpose of this article is to establish Jackson type inequality in the unit polydiscs U ⁿ of ℂⁿ for Hardy-Sobolev type spaces $F_{\alpha}^{p,q}(U^n)$. Namely,

$$E_{\overrightarrow{k}}(f, F_{\alpha}^{p,q}(U^n))\lesssim\omega_r \left(\overrightarrow{1/k},f, F_{\alpha}^{p,q}(U^n)\right),$$

where $E_{\overrightarrow{k}}(f, F_{\alpha}^{p,q}(U^n))$ is the deviation of the best approximation of $f\in F_{\alpha}^{p,q}(U^n)$ by polynomials of degree at most k _j about the j-th variable z _j with the corresponding moduli of smoothness.