Weighted Approximation by Analogues of Bernstein Operators for Rational Functions

Weighted modifications of generalized Bernstein operators in rational functions (Videnskii operators) are introduced. Their convergence in weighted spaces is studied.     

Weighted Approximation by Rational Operators

The author constructs rational operators for the weighted uniform approximation of functions with endpoints singularities by algebraic weights in [?1,1]. New direct and converse results not possible by polynomials are proved.     

On Approximation Properties of Phillips Operators Preserving Exponential Functions

In the present paper, we study a modification of the Phillips operators, which reproduces constant and the exponential functions. We obtain the moments using the concept of moment-generating function for the Phillips operators. Here we discuss a uniform convergence estimate for this modified forms. Also some direct estimates, which also involve the asymptotic-type result are established.     

Approximation Properties by q-Durrmeyer-Stancu Operators

In this paper, we are dealing with q-Bemstein-Durrmeyer-Stancu operators. Firstly, we have estimated moments of these operators. Then we have discussed some approximation properties and asymptotic formulas. We have obtained better estimations by using King type approach and given statistical convergence for the operators.     

Approximation of Unbounded Functions by Linear Positive Operators

A unified class of linear positive operators has been defined. Using these operators some approximation estimates have been obtained for unbounded functions. For particular linear positive operators these results sharpen and improve the earlier estimates due to Fuhua Cheng (J. Approx. Theory, 1984) and Xiehua Sun (J. Approx. Theory, 1988).     

Approximation by Entire Functions on a Countable Union of Segments on the Real Axis: 3. Further Generalization

In this paper, an approximation of functions of extensive classes set on a countable unit of segments of a real axis using the entire functions of exponential type is considered. The higher the type of the approximating function is, the higher the rate of approximation near segment ends can be made, compared with their inner points. The general approximation scale, which is nonuniform over its segments, depending on the type of the entire function, is similar to the scale set out for the first time in the study of the approximation of the function by polynomials. For cases with one segment and its approximation by polynomials, this scale has allowed us to connect the so-called direct theorems, which state a possible rate of smooth function approximation by polynomials, and the inverse theorems, which give the smoothness of a function approximated by polynomials at a given rate. The approximations by entire functions on a countable unit of segments for the case of Hölder spaces have been studied by the authors in two preceding papers. This paper significantly expands the class of spaces for the functions, which are used to plot an approximation that engages the entire functions with the required properties.     

Discrete l1 Approximation by Rational Functions

The problem of finding a best approximation by a rational functionto discrete data, using the l₁ norm, is considered. An algorithmis developed which is frequently convergent in a finite numberof steps, and failing this usually has a second-order convergencerate. Details are given of the application of the algorithmto a number of rational approximation problems.     

Approximation of Continuous Functions by de La Vallee-Poussin Operators

We study the asymptotic (as σ → ∞) behavior of upper bounds of the deviations of functions belonging to the classes and from the so-called de la Vallee-Poussin operators. We obtain asymptotic equalities that, in some important cases, give a solution of the Kolmogorov-Nikol’skii problem for the de la Vallee-Poussin operators on the classes and .__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 230–238, February, 2005.     

Uniform Approximation of Nonperiodic Functions Defined on the Entire Axis

Using the following notation: C is the space of continuous bounded functions f equipped with the norm , V is the set of functions f such that , the set E consists of fCV and possesses the following property:

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

The Approximation of The Continuous Functions by Means of Some Linear Positive Operators

In this paper we give positive answers to some problems posed by H.Gonska, respectively A.Lupa?. We show here that there exist positive linear operators H _n : C[0, 1] → π_n satisfying      

多元推广的Bernstein算子的逼近性质

构造了一类一致收敛于被逼近函数的多元序列,以此序列为基础,运用多元函数的全连续模及部分连续模来刻画这种多元推广的Bernstein算子的逼近性质,不仅得出了理论逼近结果,而且给出了数值逼近的例子.     

关于Fejer点上的一种复有理型插值的一致逼近阶

设D是一个Jordan,Г为其边界，并设Г满足Aльпер条件。本文得到了一种基于Fejer点的有理型插值算子对于f(z)∈C（Г）的一致逼近阶。     

覆盖下近似算子的拓扑性质

在不限制U为有限论域的情况下,研究了覆盖下近似算子XL和CL的拓扑性质。证明了覆盖下近似算子XL是内部算子,而且由XL生成的拓扑TXL为包含由覆盖C本身作为子基生成的拓扑TC的最小Alexandrov拓扑。特别地,当U为有限论域时,TXL=TC.然而,覆盖下近似算子CL不是内部算子。当覆盖C为某拓扑的基时,CL是内部算子,且此时由CL生成的拓扑TCL与TC是同一个拓扑。若进一步要求U为有限论域,则TCL=TXL=TC,进而CL=XL.     

Approximation Properties and Some Classes of Operators

The following questions and close problems are studied.(i) Is it true that T is p-nuclear provided that T ^** is p-nuclear? (ii) Is it true that Tis dually p-nuclear provided that T ^* is p-nuclear? (iii) Is it true that if T ^*is compactly factorable in the space l _p, then T is (strictly) factorable in the space l _p'? Here, T ^* is the adjoint operator of a bounded operator T:X Yin Banach spaces X and Y. Bibliography: 30 titles.     

Bernstein算子对具有奇性函数的加权同时逼近

利用在端点用Lagrange插值代替函数值的方法构造了一种新的Bernstein算子,这种新的算子可以用以逼近端点具有奇性的函数,并给出了它同时逼近的正定理.     

Rational Approximation of Functions in Hardy Spaces

Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces \(H^p({\mathbb {R}})\) for the index range \(1\le p\le \infty ,\) in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions for the Hardy spaces \(H^p({\mathbb {R}}), 0 < p\le \infty ,\) with particular interest in the index range \( 0< p \le 1.\) We show that the set of rational functions in \( H^p({\mathbb {C}}_{+1}) \) with the single pole \(-i\) is dense in \( H^p({\mathbb {C}}_{+1}) \) for \(0<p<\infty .\) Secondly, for \(0<p<1\), through rational function approximation we show that any function f in \(L^p({\mathbb {R}})\) can be decomposed into a sum \(g+h\), where g and h are, in the \(L^p({\mathbb {R}})\) convergence sense, the non-tangential boundary limits of functions in, respectively, \( H^p({\mathbb {C}}_{+1})\) and \(H^{p}({\mathbb {C}}_{-1}),\) where \(H^p({\mathbb {C}}_k)\ (k=\pm 1) \) are the Hardy spaces in the half plane \( {\mathbb {C}}_k=\{z=x+iy: ky>0\}\). We give Laplace integral representation formulas for functions in the Hardy spaces \(H^p,\) \(0<p\le 2.\) Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces \(H^p\) for \(0<p\le 1\).