Academic Journal
Holomorphic Jackson's theorems in polydiscs
| Title: | Holomorphic Jackson's theorems in polydiscs |
|---|---|
| Authors: | Mingzhi Wang, Guangbin Ren |
| Source: | Journal of Approximation Theory. 134:175-198 |
| Publisher Information: | Elsevier BV, 2005. |
| Publication Year: | 2005 |
| Subject Terms: | Mathematics(all), Polynomial approximation, Bergman-type space, polydisc, Bergman spaces of functions in several complex variables, Multidimensional problems, Hardy space, polydisc algebra, 01 natural sciences, Jackson-type inequality, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Modulus of continuity, modulus of continuity, 0101 mathematics, Sobolev spaces and other spaces of ''smooth' functions, embedding theorems, trace theorems, Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities), Numerical Analysis, Applied Mathematics, Sobolev space, Lipschitz space, modulus of smoothness, generalized Jackson's kernel, Bergman spaces, Jackson's theorem, polynomial approximation, Bergman space, Analysis, Algebras of holomorphic functions of several complex variables |
| Description: | The article is devoted to obtaining Jackson-type inequalities for several function spaces. The authors consider functions defined on the unit polydisc \(U^n\) of \({\mathbb C}^n\) with the Shilov boundary \(T^n\). The authors' approach is to construct the best approximation of polynomials with a kind of complex measures on \(T^n\), whose total variations are given by generalized Jackson's kernels: \[ T_k^\beta\left(\theta\right):= \left({\sin k\theta/2\over{\sin\theta/2}}\right)^{2\beta}. \] The authors obtain some results on these kernels, introduce the required normalized complex measure \(d\mu_k^\rho\left(\varphi\right)\) on \(\left[-\pi,\pi\right)^n\), and investigate some operators \[ P_k\left[f\right]\left(z\right)= \int_{\left[-\pi,\pi\right)^n} f\left(\rho e^{i\varphi}z\right)\,d\mu_k^\rho\left(\varphi\right), \quad z\in U^n, \] that provide the best approximation of polynomials. The next step is to introduce the modulus of continuity and the modulus of smoothness on a semi-normed space \(X\) of functions defined on \(U^n\). These moduli are used in right-hand sides of Jackson-type inequalities for the spaces considered. The technique developed is successfully applied to various function spaces: Bergman-type spaces, Sobolev spaces, Hardy spaces, polydisc algebra, and Lipschitz spaces. The Jackson-type theorems are obtained as well as some interesting properties of the moduli of continuity and the moduli of smoothness are derived. The Jackson-type theorems, obtained in the article, generalize well-known results of \textit{W.~E.~Sewell} [Degree of approximation by polynomials in the complex domain (1942; Zbl 0063.08342)], \textit{D.~Jackson} [``On approximation by trigonometric sums and polynomials'', American M. S. Trans. 13, 491--515; Amer. M. S. Bull. (2) 18, 333--334, 492 (1912; JFM 43.0499.02)], \textit{L.~Colzani} [``Jackson theorems in Hardy spaces and approximation by Riesz means'', J. Approximation Theory 49, 240--251 (1987; Zbl 0657.42024)]. The article is recommended for all researchers in multivariate approximation theory as well as in the theory of functions of several complex variables. |
| Document Type: | Article |
| File Description: | application/xml |
| Language: | English |
| ISSN: | 0021-9045 |
| DOI: | 10.1016/j.jat.2005.02.005 |
| Access URL: | https://zbmath.org/2183805 https://doi.org/10.1016/j.jat.2005.02.005 https://dblp.uni-trier.de/db/journals/jat/jat134.html#RenW05 https://www.sciencedirect.com/science/article/pii/S0021904505000481 https://www.sciencedirect.com/science/article/abs/pii/S0021904505000481 https://core.ac.uk/display/82289213 |
| Rights: | Elsevier Non-Commercial |
| Accession Number: | edsair.doi.dedup.....9cc94dfa9cbfb965a2b6feb1fa20bb4d |
| Database: | OpenAIRE |
| ISSN: | 00219045 |
|---|---|
| DOI: | 10.1016/j.jat.2005.02.005 |