WEIGHTED APPROXIMATION OF RANDOM FUNCTIONS

Let (Ω, A, P) be a probability space, X(t, ω) a random function continuous in probability for t∈[0,+∞) or (-∞,+∞)(ω∈Ω), and F(t) a positive function continuous for t∈[0,+∞) or (-∞, +∞). If X(t, ω) and F(t) verify certain conditions, then there exists a sequence {Qn(t,ω)} of random polynomials such that we have almost surely: for t∈[0,+∞) or (-∞, +∞), lim|X(t, ω)-Qn(t, ω)|/F(t)=0.

Weighted approximation of functions with singularities by Bernstein operators

As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.

WEIGHTED INEQUALITIES AND APPLICATIONS TO BEST LOCAL APPROXIMATION IN LUXEMBURG NORM

In this paper we establish a result about uniformly equivalent norms and the convergence of best approximant pairs on the unitary ball for a family of weighted Luxemburg norms with normalized weight functions depending on ε, when ε→ 0. It is introduced a general concept of Pade approximant and we study its relation with the best local quasi-rational approximant. We characterize the limit of the error for polynomial approximation. We also obtain a new condition over a weight function in order to obtain inequalities in Lp norm, which play an important role in problems of weighted best local Lp approximation in several variables.

On Weighted L^p-Approximation by Weighted Bernstein-Durrmeyer Operators

In the present paper, we establish direct and converse theorems for weight-ed Bernstein-Durrmeyer operators under weighted L^p-norm with Jacobi weight w（x）=x^a（1-x）β.All the results involved have no restriction a,β〈1-1/p,which indicates that the weighted Bemstein-Durrmeyer operators have some better approxi- mation properties than the usual Bernstein-Durrmeyer operators.

ON WEIGHTED L^p INTEGRABILITY AND APPROXIMATION OF TRIGONOMETRIC SERIES

We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.

APPROXIMATION BY GENERALIZED MKZ-OPERATORS IN POLYNOMIAL WEIGHTED SPACES

We prove some approximation properties of generalized Meyer-Konig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [1-3,7-16].

WEIGHTED L_p-APPROXIMATION OF DERIVATIVES BY THE METHOD OF BETA OPERATORS

The author consides Beta operators βnf on suitable Sobolev type subspace of Lp[0, ∞) and characterizes the global rate of approximation of derivatives f(r) through corresponding derivatives (βnf)(r) in an appropriate weighted Lp-metric by the rate of Ditzian and Totik's r-th order weighted modulus of Smoothness.

WEIGHTED BEST LOCAL APPROXIMATION IN ORLICZ SPACE

We present a common fixed point theorem for generalized asymptotically non- expansive and noncommuting mappings in normed linear spaces.

Concise review of relaxations and approximation algorithms for nonidentical parallel-machine scheduling to minimize total weighted completion times

A class of nonidentical parallel machine scheduling problems are considered in which the goal is to minimize the total weighted completion time. Models and relaxations are collected. Most of these problems are NP-hard, in the strong sense, or open problems, therefore approximation algorithms are studied. The review reveals that there exist some potential areas worthy of further research.

Weighted Approximation by Left Quasi-interpolants of Derivatives of Gamma Operators

In order to obtain much faster convergence, Miiller introduced the left Gamma quasi- interpolants and obtained an approximation equivalence theorem in terms of 2r wφ （f,t）p. Cuo extended the MiiUer＇s results to wφ^24 （f, t）∞. In this paper we improve the previous results and give a weighted approximation equivalence theorem.

On Weighted Approximation by Modified Bernstein Operators for Functions with Singularities

Della Vecchia et al. （see [2]） introduced a kind of modified Bernstein operators which can be used to approximate functions with singularities at endpoints on [0,1]. In the present paper, we obtain a kind of pointwise Stechkin-type inequalities for weighted approximation by the modified Bemsetin operators.

WEIGHTED APPROXIMATION ON SZASZ-TYPE OPERATORS

In this paper, we use weighted modules ω(?)(f,t)w to study the pointwise approximation on Szasz-type operators, and obtain the direct and converse theorem, as well as characterizations of the pointwise approximation of Jacobi-weighted Szasz-type operators.

Weighted Approximation of Kantorovich-Type Operator of Butzer-Hahn

A kind of Kantorovich-type operator of Butzer-Hahn B*_n(f; x) on the bounded and integrable function space was introduced, and the properties of B*_n(f; x) were studied. Positive theorem, converse theorem, and the order of weighted ~approximation of B*_n(f; x) were obtained.

A Study of Weighted Polynomial Approximations with Several Variables (II)

In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii- type inequality.

Weighted Best Local Approximation

In this survey the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the LP spaces and recent results in Orlicz spaces. The notion of balanced point, introduced by Chui et al. in 1984 are extensively used.

STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS

In this article,we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a={a j}j≥1 and b={b j}j≥1 of positive numbers.We obtain strong equivalences of the approximation numbers,and necessary and sufficient conditions on a,b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.

Weighted Best Local Approximation

In this survey, the notion of a balanced best multipoint local approximation is fully exposed since they were treated in the LP spaces and recent results in Orlicz spaces. The notion of balanced point, which was introduced by Chui et al. in 1984 are extensively used.

A Note on Pad Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.

WEIGHTED APPROXIMATION OF r-MONOTONE FUNCTIONS ON THE REAL LINE BY BERNSTEIN OPERATORS

In this paper, we give error estimates for the weighted approximation of rmonotone functions on the real line with Freud weights by Bernstein-type operators.

The Approximation of Hermite Interpolation on the Weighted Mean Norm

We research the simultaneous approximation problem of the higher-order Hermite interpolation based on the zeros of the second Chebyshev polynomials under weighted Lp-norm. The estimation is sharp.