Best approximation in polyhedral Banach spaces

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (∗) (a geometric property stronger than polyhedrality) and Y⊂X is any proximinal subspace, then the metric projection P_Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y_n}⊂Y, x∈X and , then ).One of the main results of a different nature is the following: if X satisfies (∗) and Y⊂X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^⊥ attains its norm. Moreover, in this case the quotient X/Y is polyhedral.The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.     

Best approximation mappings in Hilbert spaces

Mathematical Programming - The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to...     

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

Best approximation with wavelets in weighted Orlicz spaces

Democracy functions of wavelet admissible bases are computed for weighted Orlicz Spaces L ^??(w) in terms of the fundamental function of L ^??(w). In particular, we prove that these bases are greedy in L ^??(w) if and only if L ^??(w) =?L ^p (w), 1?<?p?<???. Also, sharp embeddings for the approximation spaces are given in terms of weighted discrete Lorentz spaces. For L ^p (w) the approximation spaces are identified with weighted Besov spaces.     

Best approximation properties in spaces of measurable functions

We research proximinality of μ‐sequentially compact sets and μ‐compact sets in measurable function spaces. Next we show a correspondence between the Kadec–Klee property for convergence in measure and μ‐compactness of the sets in Banach function spaces. Also the property S is investigated in Fréchet spaces and employed to provide the Kadec–Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fréchet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one‐dimensional subspace in sequence Lorentz spaces .     

Best approximation by polynomial operators in banach spaces

The problem of approximation in the space of bounded linear operators ? (E;G) between normed spaces E and G by compact operators has been extensively studied in the last few years.

Recently Deutsch, Mach and Saatkamp ([2]) have considered the problem of approximating elements of ?(E;G) by the subset K _N(E;G) of operators whose range is at most N dimensional. We consider in this paper the problem of approximating operators (not necessarily linear) beteen normed spaces E and G by continuous homogeneous polynomials, and in particular by such polynomials which have finite-dimensional range.     

Shape preserving approximation in vector ordered spaces

The aim of this note is to extend some classical results on the shape preserving approximation of real functions (of real variables) to functions with values in ordered vector spaces.     

Best proximity pair theorem in metrizable topological vector spaces

The object of this paper is to prove an existence result on best proximity pair. For this purpose, the class of factorizable multifunctions in approximately weakly compact, convex subset of metrizable topological vector space is used. As consequence, our theorem generalizes the result of Basha and Veeramani^[10]. Finally, certain known results have also been obtained as corollaries in this work.     

Best simultaneous approximation to totally bounded sequences in Banach spaces

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. The first author is supported in part by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 06C651); the second author is supported in part by the National Natural Science Foundation of China (Grant Nos. 10671175, 10731060) and Program for New Century Excellent Talents in University; the third author is supported in part by Projects MTM2006-13997-C02-01 and FQM-127 of Spain     

复赋范空间中复RS集的最佳同时逼近

研究了复赋范空间中具限制系数的广义多项式集G对无穷序列的最佳同时逼近问题,得到了特征定理;当G是复RS集时还得到了惟一性定理．     

Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces

In hyperconvex metric spaces we consider best approximation, invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive.     

Best and coupled best approximation theorems in abstract convex metric spaces

In this paper, we first give a best approximation theorem in abstract convex metric spaces. As applications, we then derive some best and coupled best approximations and coupled coincidence point results in normed spaces and hyperconvex metric spaces.     

Best approximation of reproducing kernels of spaces of analytic functions

We obtain exact values for the best approximation of a reproducing kernel of a system of p-Faber polynomials by functions of the Hardy space H_q, p^-1 + q^-1 = 1 and a Szegö reproducing kernel of the space E²(Ω) in a simply connected domain Ωwith rectifiable boundary.     

Best proximity points: approximation and optimization in partially ordered metric spaces

The purpose of this article is to provide the existence of a unique best proximity point for non-self-mappings by using altering distance function in the setting of partially ordered set which is endowed with a metric. Further, our result provides an extension of a result due to Harjani and Sadarangani to the case of non-self-mappings.     

On the best approximation of vector valued functions by polynomials with coefficients in vector spaces

In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials. This paper was written during the 2005 Spring Semester when the second author (S.G. Gal) was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, USA. Mathematics Subject Classification (2000) 41A65, 41A17, 41A27     

Best simultaneous approximation in

Let X be a Banach space, (Ω,Σ,μ) a finite measure space, and L₁(μ,X) the Banach space of X-valued Bochner μ-integrable functions defined on Ω endowed with its usual norm. Let us suppose that Σ₀ is a sub-σ-algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in . It is shown that if X is reflexive then L₁(μ₀,X) is N-simultaneously proximinal in L₁(μ,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.