CHARACTERIZATION OF HLDER SPACES CORRESPONDING TO DITZIAN-TOTIK MODULUS OF SMOOTHNESS

For the step-weight function , we prove that the Holder spaces ∧a,p on the interval [-1,1], defined in terms of moduli of smoothness with the step-weight function ,are linearly isomorphic to some sequence spaces, and the isomorphism is given by the cofficients of function with respect to a system of orthonormal splines with knots uniformly distributed according to the measure with density . In case ∧a,p is contained in the space of continuous functions, we give a discrete characterization of this space, using only values of function at the appropriate knots. Application of these results to characterize the order of polynomial approximation is presented.

On a New Geometric Constant Related to the Modulus of Smoothness of a Banach Space

We shall introduce a new geometric constant A（X） of a Banach space X, which is closely related to the modulus of smoothness px （T）, and investigate it in relation with the constant As （X） by Baronti et al., the von Neumann-Jordan constant CNj（X） and the James constant J（X）. A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.

Approximation Theorems of Bernstein-Durrmeyer Operators in L_M^（Ba） Spaces

With the weighted modulus of smoothness as a metric,we prove the direct and the inverse theorems of approximation by Bernstein-Durrmeyer operators in LBa M spaces. Especially an approximation equivalent theorem of the operators is also obtained.

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0,1]

In this paper, we will use the 2r-th Ditzian-Totik modulus of smoothness wψ2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn[2r-1]f for functions of the space Lp[0,1] (1≤ p≤ +∞).

On the rate of convergence of two generalized Bernstein type operators

In this paper,we introduce the Bézier variant of two new families of generalized Bernstein type operators.We establish a direct approximation by means of the Ditzian-Totik modulus of smoothness and a global approximation theorem in terms of second order modulus of continuity.By means of construction of suitable functions and the method of Bojanic and Cheng,we give the rate of convergence for absolutely continuous functions having a derivative equivalent to a bounded variation function.

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.

APPROXIMATION BY RATIONAL FUNCTIONS WITH POLYNOMIALS OF POSITIVE COEFFICIENTS AS THE DENOMINATORS

The present paper proves that if（x） ∈ C[0,1], changes its sign exactly l times at 0 〈 y1〈 y2 … 〈 y1 〈 1 in （0, 1）, then there exists a pn（x） ∈ Пn（＋）, such that ｜f（x）- p（x）/pn（x）｜≤ Cωφ（f,n^（-1/2））, where ρ（x） is defined by ρ（x）={^lПi=1（x-yi）,if f （x）≥0 for x ∈（y1,1）, {-^lПi=1（x-yi）,if f （x）〈0 for x ∈（y1,1）, which improves and generalizes the result of .

STRONG CONVERSE INEQUALITY FOR SZASZ-DURRMEYER OPERATORS

For Szasz-Durrmeyer operators L.(f,x),1<p ∞ we prove that, forsome m,where(x)2=x,M>0,is Ditzian-Totik modulus of smoothness.

Convergence of the q Analogue of Szász-Beta-Stancu Operators

In the present paper, we propose the q analogue of Szasz-Beta-Stancu operators. By estimate the moments, we establish direct results in terms of the modulus of smoothness. Investigate the rate of point-wise convergence and weighted approximation properties of the q operators. Voronovskaja type theorem is also obtained. Our results generalize and supplement some convergence results of the q-Szasz-Beta operators, thus they improve the existing results.

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.

SIMULTANEOUS APPROXIMATION BY BERNSTEIN-SIKKEMA OPERATORS

In this paper, we investigate the simultaneous approximation of Bernstein- Sikkema operators, and establish the direct and equivalent theorems by using the Ditzian-Totik modulus of smoothness.

Pointwise Approximation for the Iterated Boolean Sums of Bernstein Operators

In this paper, with the help of modulus of smoothness ω2r(f,t), we discuss the pointwise approximation properties for the iterated Boolean sums of Bernstein operator B n and obtain direct and inverse theorems when 1-1/r ≤λ≤ 1, r ∈N.

Entropy Numbers of Besov Classes of Generalized Smoothness on the Sphere

We investigate the asymptotic behavior of the entropy numbers of Besov classes BBΩp,θ（Sd 1）of generalized smoothness on the sphere inL q（Sd 1）for 1≤p,q,θ≤∞,and get their asymptotic orders.We also obtain the exact orders of entropy numbers of Sobolev classesBWr p（Sd 1）inL q（Sd 1）whenpand/orqis equal to 1 or∞.This provides the last piece as far as exact orders of entropy numbers ofBWr p（Sd 1）inL q（Sd 1）are concerned.

On Approximation by Neural Networks with Optimized Activation Functions and Fixed Weights

Recently,Li[16]introduced three kinds of single-hidden layer feed-forward neural networks with optimized piecewise linear activation functions and fixed weights,and obtained the upper and lower bound estimations on the approximation accuracy of the FNNs,for continuous function defined on bounded intervals.In the present paper,we point out that there are some errors both in the definitions of the FNNs and in the proof of the upper estimations in[16].By using new methods,we also give right approximation rate estimations of the approximation by Li’s neural networks.

The Pointwise Estimate for Stancu-Sikkema Operators

In this paper, we construct Stancu-Sikkema operators and obtain the point-wise direct and converse theorems in terms of pointwise modulus of smoothness. Some results about Bernstein-Sikkema operators and Stancu operators are extended.

The essential order of approximation for nearly exponential type neural networks

For the nearly exponential type of feedforward neural networks （neFNNs）, it is revealed the essential order of their approximation. It is proven that for any continuous function defined on a compact set of Rd, there exists a three-layer neFNNs with fixed number of hidden neurons that attain the essential order. When the function to be approximated belongs to the α-Lipschitz family （0 〈α≤ 2）, the essential order of approxi- mation is shown to be O（n^-α） where n is any integer not less than the reciprocal of the predetermined approximation error. The upper bound and lower bound estimations on approximation precision of the neFNNs are provided. The obtained results not only characterize the intrinsic property of approximation of the neFNNs, but also uncover the implicit relationship between the precision （speed） and the number of hidden neurons of the neFNNs.

Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.

Approximation by semigroup of spherical operators

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the （n ＋ 1）- dimensional Euclidean space for n ≥2. We prove that such operators form a strongly continuous contraction semigroup of class （l0） and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ＋Vt^γ and the rth Boolean of the generalized spherical Weierstrass operator ＋Wt^k for integer r ≥ 1 and reals γ, k∈ （0, 1] have errors ｜｜＋r Vt^γ- f｜｜X ω^rγ（f, t^1/γ）X and ｜｜＋rWt^kf - f｜｜X ω^2rk（f, t^1/（2k））X for all f ∈ X and 0 ≤t ≤2π, where X is the Banach space of all continuous functions or all L^p integrable functions, 1 ≤p ≤＋∞, on S^n with norm ｜｜·｜｜X, and ω^s（f,t）X is the modulus of smoothness of degree s 〉 0 for f ∈X. Moreover, ＋r^Vt^γ and ＋rWt^k have the same saturation class if γ= 2k.

Pointwise Approximation Theorems for Combinations and Derivatives of Bernstein Polynomials

We establish the pointwise approximation theorems for the combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness wФ^r（f, t） where Ф is an admissible step-weight function. An equivalence relation between the derivatives of these polynomials and the smoothness of functions is also obtained.

Direct Result for a Summation-Integral Type Modification of Szasz–Mirakjan Operators

In this paper,study of direct result for a summation-integral type modification of Szasz-Mirakjan operators is carried out.Calculation of moments,density result and a Voronvskaja-type result are also obtained.