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Abstract  

A subspaceY of a Banach spaceX is called a Chebyshev one if for everyxX there exists a unique elementP Y(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH 1=H 1 (|z|<1) be Chebyshev ones, and also the properties of the operatorP Y are studied. These results show that the theory of Chebyshev subspaces inH 1 differs sharply from the corresponding theory inL 1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH 1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL 1(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 operatorP Y inH 1 (in contrast toL 1(C)) is exhausted by that minimum which is necessary for any Banach space.

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Blatter, J., Reflexivity and the existence of best approximations, Approximation Theory, Proc. Internat. Sympos. , Univ. Texas, Austin, Tex., Academic Press, New York (1976), 299-301. MR 54 #13423 Approximation

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460 Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory 103 (2000), 292-301. MR 2001a :54023

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polynomial inequalities with doubling and A ∞ weights, Constr. Approx. , 16 , 37–71. Mastroianni, G. and Totik, V. , Best approximations and moduli of smothness for doubling weights, J. Approx. Theory , 110 , 180

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