Mendoza Casas, José and Pakhrou, Tijani (2007) Best simultaneous approximation in L-1 (mu, X). Journal of Approximation Theory, 145 (2). pp. 212-220. ISSN 0021-9045
Restricted to Repository staff only until 31 December 2020.
Let X be a Banach space, (Omega, Sigma, mu) a finite measure space, and L-1 (mu, X) the Banach space of X-valued Bochner mu-integrable functions defined on Omega endowed with its usual norm. Let us suppose that Sigma(0) is a sub-sigma-algebra of Sigma, and let mu(0) be the restriction of mu to Sigma(0). Given a natural number n, let N be a monotonous norm in R-n. It is shown that if X is reflexive then L-1 (mu(0), X) is N-simultaneously proximinal in L-1 (mu, X) in the sense of Fathi et al. [Best simultaneous approximation in L-p(I, E), J. Approx. Theory 116 (2002), 369-379]. Some examples and remarks related with N-simultaneous proximinality are also given.
|Uncontrolled Keywords:||Banach space.|
|Subjects:||Sciences > Mathematics > Differential geometry|
P. Cembranos, J. Mendoza, Banach spaces of vector-valued functions, Lecture Notes in Mathematics, vol. 1676, Springer, Berlin, 1997.
M.M. Day, Normed Linear Spaces, third ed., Springer, New York, 1973.
J. Diestel, J.J. Uhl Jr., Vector measures, Math. Surveys Monographs, vol. 15, American Mathematical Society, Providence, RI, 1977.
R. Durier, Optimal locations and inner product spaces, J. Math. Anal. Appl. 207 (1997) 220–239.
B. Fathi, D. Hussein, R. Khalil, Best simultaneous approximation in Lp (I,E), J. Approx. Theory 116 (2002) 369–379.
A.L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962) 87–106 (in Russian), translation in Amer. Math. Soc. Transl. 39 (1964) 111–132.
S.V. Konyagin, A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1988) 81–82 (in Russian), translation in Moscow Univ. Math. Bull. 43 (2) (1988) 55–56.
N.S. Papageorgiou, On best approximations in the Lebesgue–Bochner space L1 (X), Tamkang J. Math. 24 (1993) 303–307.
T. Shintani, T. Ando, Best approximants in L1 -space, Z. Wahrschein. Verw. Gebiete 33 (1975) 33–39.
L. Veselý, A characterization of reflexivity in the terms of the existence of generalized centers, Extracta Math. 8 (1993) 125–131.
L. Veselý, Generalized centers of finite sets in Banach spaces, Acta Math. Univ. Comenian. (N.S.) 66 (1997) 83–115.
|Deposited On:||24 Oct 2012 10:44|
|Last Modified:||13 Nov 2013 17:27|
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