Authors:
П. Бородин
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P. Borodin
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Abstract  

A subspaceY of a Banach spaceX is called a Chebyshev one if for everyxX there exists a unique elementPY(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH1=H1 (|z|<1) be Chebyshev ones, and also the properties of the operatorPY are studied. These results show that the theory of Chebyshev subspaces inH1 differs sharply from the corresponding theory inL1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL1(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the Chebyshev subspacesY with a linear operatorPY inH1 (in contrast toL1(C)) is exhausted by that minimum which is necessary for any Banach space.

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Analysis Mathematica
Language English
Size B5
Year of
Foundation
1975
Volumes
per Year
1
Issues
per Year
4
Founder Akadémiai Kiadó
Founder's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0133-3852 (Print)
ISSN 1588-273X (Online)