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Rational Approximation in a Class of Weighted Hardy Spaces

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Abstract

This study aims at rational approximation of a class of weighted Hardy spaces, including the classical Bergman space, the weighted Bergman spaces, the Hardy space, the Dirichlet space and the Hardy–Sobolev spaces. We will mainly concentrate in the Bergman cases in the unit disc context. The methodology of the approximation is a pre-orthogonal method, called Pre-Adaptive Fourier Decomposition. The new idea is that a function is not expanded into a basis but an orthonormal system adapted to the given function. In such way by using a unified method we obtain efficient approximations in our sequence of spaces while avoiding discussions on basis and uniqueness sets, etc. The type of function decompositions is related to direct sum decompositions of the underlying spaces into the closure of the span of a sequence of repeating reproducing kernels and the corresponding zero-based invariant subspaces that arises deep studies.

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Acknowledgements

The authors would like to express their thankfulness to Prof. T. Qian and Dr. W. X. Mai for their suggestions and inspired discussions.

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Authors and Affiliations

  1. Faculty of Information Technology, University of Science and Technology, Macau, China

    Wei Qu & Pei Dang

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  1. Wei Qu
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  2. Pei Dang
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Correspondence to Wei Qu.

Additional information

Communicated by Tao Qian.

This work was supported in part by NSFC Grant 11701597; Macao Science and Technology Development Fund, MSAR. Ref. 154/2017/A3.

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Qu, W., Dang, P. Rational Approximation in a Class of Weighted Hardy Spaces. Complex Anal. Oper. Theory 13, 1827–1852 (2019). https://doi.org/10.1007/s11785-018-0862-x

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