Approximation in weighted Orlicz spaces

We prove some direct and converse theorems of trigonometric approximation in weighted Orlicz spaces with weights satisfying so called Muckenhoupt’s A _p condition.     

On approximation in Weighted Orlicz spaces

An inverse theorem of the trigonometric approximation theory in Weighted Orlicz spaces is proved and the constructive characterization of the generalized Lipschitz classes defined in these spaces is obtained.     

Trigonometric approximation in reflexive Orlicz spaces

The Lipschitz classes Lip(α, M) , 0 α≤ 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f ∈ Lip (α, M) is estimated by n-α .     

On approximation by polynomials in Orlicz spaces

In this paper, we prove Jackson theorem of derivative type in Orlicz spaces, and the results of this paper are generalization of the results of A.-R.K. Ramazanov in [2].     

The uniform approximation property in Orlicz spaces

It is proved that for every reflexive Orlicz spaceX there is a functionn(k,ε) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT _1E=identity, ‖T‖≦1+ε and dimTX≦n(k,ε). Some general facts concerning the uniform approximation property are also presented. Research of the first named author was partially supported by NSF Grant MPS 74-07509-A01.     

Best approximation in vector spaces

In this paper, we obtain some results on best f-approximation in quotient spaces of vector spaces and determine under what conditions f-proximinality can be transmitted to and from quotient spaces. Also, in conclusion, we consider the relationship between f-approximation subsets and linear functionals on X.     

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 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.     

On approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces

This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f(x) ∈ LM*[0,1], changes its sign at most once in (0,1), then there exists x0 ∈ (0,1) and a polynomial Pn ∈ Πn(+) such that f (x) -Pn (x)x-x0 M ≤ Cω( f,n-1/2 )M, where Πn(+) indicates the set of all polynomials of degree n with positive coefficients.     

On weak uniform normal structure in weighted Orlicz sequence spaces

It is proved that a weighted Orlicz sequence space ?M(w), equipped with Luxemburg or Amemiya norm has weak uniform normal structure iff ?M(w)≅hM(w) for wide class of weight sequences . An example is constructed, where M has not Δ₂-condition but by choosing a suitable weight sequence limn→∞wn=∞ we get that ?M(w) has weak uniform normal structure.     

On approximation by polynomials and rational functions in Orlicz spaces

Для пространств Орли ча получен аналог изв естного неравенства С. Б. Стечк ина об оценке наименьших по линомиальных уклоне ний через модуль гладкости про извольного порядка. Например, если?∈L* _Φ (I), то \(R_n (f,I)_\Phi \leqq E(f,I)_\Phi \leqq C(\Phi ,r)\omega _r \left( {\tfrac{1}{n},f,I} \right)_\Phi \) при всех натуральныхr иn≧r (теорема I). Доказана неулучшаем ость этой теоремы, ее а налог для случая приближения т ригонометрическими полиномами и тригоно метрическими рацион альными функциями. Установлена связьΔ ₂-условия на функциюΦ(u) со свойствами аппрокси мации соответствующ их классов функций (теор ема 3).     

Simultaneous approximation and interpolation in weighted spaces

In this paper we present a result about simultaneous approximation and interpolation in weighted spaces. It generalizes a result of Prolla in the space of continuous functionsC(X;E) whereX is a compact Hausdorff space andE is a normed space. As a consequence, we prove that simultaneous approximation and interpolation is possible from certain vector subspaces.     

关于Bernstein-Durrmeyer-Bzier算子在Orlicz空间内的逼近

在连续函数空间和L_p空间内研究算子逼近方法的基础上,利用一阶DitzianTotik积分模与不等式技巧研究了Bernstein-Durrmeyer-Bzier算子在Orlicz空间内的逼近性质.得到了Bernstein-Durrmeyer-Bezier算子在Orlicz空间内的逼近正定理和逼近等价定理.由于Orlicz空间比连续函数空间和L_p空间都"大",其拓扑结构也比L_p空间复杂得多,所以本文的结果具有一定的拓展意义.     

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.     

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