Best simultaneous approximation in orlicz spaces
27 May 2007-International Journal of Mathematics and Mathematical Sciences (Hindawi Publishing Corporation)-Vol. 2007, Iss: 5, pp 1-7
TL;DR: A distance formula dist(f1, f2, LΦ(I,G))Φ, where G is a closed subspace of X, and f1,f2∈LΦ (I,X) is the space of Orlicz X-valued integrable functions on the unit interval I equipped with the Luxemburg norm.
Abstract: Let X be a Banach space and let LΦ(I,X) denote the space of Orlicz X-valued integrable functions on the unit interval I equipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, where G is a closed subspace of X, and f1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation in LΦ(I,X) are presented.
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TL;DR: In this article, the distance of f ∈ L ϕ (A, X) from a generalized Musielak-Orlicz function on a complete complete measure space is characterized.
Abstract: Let (A, A, μ )b e aσ -finite complete measure space and let Y be a subspace of a Banach space X.L etϕ be a generalized � -function on (A, A, μ). Denote by L ϕ (A, Y) and L ϕ (A, X) the Musielak-Orlicz spaces whose functions take values in Y and X, respectively. Firstly, let f ∈ L ϕ (A, X), we characterize the distance of f from L ϕ (A, Y). Then, if Y is weakly K-analytic and proximinal in X, we show that L ϕ (A, Y) is proximinal in L ϕ (A, X). Finally, we give the connection between the proximinality of L ϕ (A, Y )i n L ϕ (A, X) and the proximinality of L 1 (A, Y )i nL 1 (A, X).
4 citations
Journal Article•
TL;DR: A characterization of best simultaneous approximation of Kothe spaces of vector-valued functions is given in this paper, which is a generalization of some analogous theorems for Orlicz Bochner spaces.
Abstract: A characterization of best simultaneous approximation of Kothe spaces of vector-valued functions is given. This characterization is a generalization of some analogous theorems for Orlicz Bochner spaces.
2 citations
01 Jan 2012
TL;DR: In this paper, the authors give a characterization for some subspaces of Bochner integrable functions space to be simultaneously proximinal, i.e., the subspace of a continuous function can be simultaneously approximated by a constant number of functions.
Abstract: The problem of simultaneous approximation in function spaces has attracted many researchers recently. Major results on the space of vector-valued continuous functions started to appear early nineties. In 2002, results on simultaneous approximation in p-Bochner integrable function spaces were published. The objective of this paper is to give a characterization for some subspaces of Bochner integrable functions space to be simultaneously proximinal.
2 citations
TL;DR: In this article, the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X is discussed.
Abstract: For a Banach space X, L?(T,X) denotes the metric space of all X-valued
?-integrable functions f : T ? X, where the measure space (T,?,?) is a
complete positive ?-finite and ? is an increasing subadditive continuous
function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality
problem for the monotonous norm on best simultaneous approximation from the
closed subspace Y?X to a finite number of elements in X.
1 citations
TL;DR: In this article , the problem of nonlinear best simultaneous approximation in K?the Bochner function spaces with respect to Minkowski norms in Euclidean spaces was studied.
Abstract: This paper is concerned with the problem of nonlinear best simultaneous
approximations in K?the Bochner function spaces with respect to Minkowski?
norms in Euclidean spaces. Characterization results of the generalized best
simultaneous approximation are established. These results are considered a
generalization of the results concerning the Lebesgue Bochner spaces and the
Orlicz Bochner spaces.
References
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"Best simultaneous approximation in ..." refers methods in this paper
...This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited....
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TL;DR: In this paper, a family of modulars depending on a parameter is described, and some applications of modular spaces are discussed, including orlicz spaces and countably modulared spaces.
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"Best simultaneous approximation in ..." refers background or methods in this paper
...In this paper, we present a distance formula distΦ( f1, f2,LΦ(I ,G)), where G is a closed subspace of X , and f1, f2 ∈ LΦ(I ,X)....
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...This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited....
[...]
TL;DR: It is proved that Lp(I,G) is N-simultaneously proximinal in Lp (I,E), the space of all p-Bochner integrable functions on the interval I=0,1 with values in E, 1?p?∞.
42 citations
"Best simultaneous approximation in ..." refers methods in this paper
...This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited....
[...]
TL;DR: In this paper, the problem of simultaneous approximation of a subset F of a Banach space B by elements of another subset S ⊂ B was considered and results on the existence, uniqueness, and characterization of best simultaneous approximations were obtained.
39 citations