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Jackson's Theorem on Bounded Symmetric Domains

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Abstract

Polynomial approximation is studied on bounded symmetric domain Ω in ℂn 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.

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Authors and Affiliations

  1. School of Science, Huzhou Teachers College, Huzhou, 313000, P. R. China

    Ming Zhi Wang

  2. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, P. R. China

    Guang Bin Ren*

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  1. Ming Zhi Wang
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  2. Guang Bin Ren*
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Correspondence to Ming Zhi Wang.

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*Partially supported by the NNSF of China (No. 10471134), SRFDP, and NCET

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Wang, M.Z., Ren*, G.B. Jackson's Theorem on Bounded Symmetric Domains. Acta Math Sinica 23, 1391–1404 (2007). https://doi.org/10.1007/s10114-007-0953-5

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