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1.
Daniela P. Kacsó 《Results in Mathematics》1999,35(1-2):89-102
In the present paper we use piecewise linear functions in order to obtain representations and estimates for the remainder in approximating continuous functions by positive linear operators. Applications of these results for Bernstein and Stancu’s operators are also presented. In addition, we give some partial results concerning the best constant problem for Bernstein operators with respect to the second order modulus of continuity. 相似文献
2.
Note on Bernstein polynomials and Kantorovich polynomials 总被引:2,自引:0,他引:2
Wang Xiaochun 《分析论及其应用》1991,7(2):99-105
We obtain two asymptotic representations of remainder of approximation of derivable functions by Bernstein polynomials and
Kantorovich polynomials separately. 相似文献
3.
Gancho T. Tachev 《Numerical Functional Analysis & Optimization》2013,34(2):206-215
The aim of this note is to study the impact of the weight numbers in two concrete cases on the rate of approximation by rational Bernstein functions. The approximation is measured in terms of second order moduli of continuity. We consider also the case of rational Bernstein curves and the role of the weights to modify the shape of the curves. 相似文献
4.
Ana Maria Acu Tuncer Acar Carmen‐Violeta Muraru Voichia Adriana Radu 《Mathematical Methods in the Applied Sciences》2019,42(16):5551-5565
Starting with the well‐ known Bernstein operators, in the present paper, we give a new generalization of the bivariate type. The approximation properties of this new class of bivariate operators are studied. Also, the extension of the proposed operators, namely, the generalized Boolean sum (GBS) in the Bögel space of continuous functions is given. In order to underline the fact that in this particular case, GBS operator has better order of convergence than the original ones, some numerical examples are provided with the aid of Maple soft. Also, the error of approximation for the modified Bernstein operators and its GBS‐type operator are compared. 相似文献
5.
Based upon a new error bound for the linear interpolant to a function defined on a triangle and having continuous partial derivatives of second order, the related error bound for n-th Bernstein triangular approximation is obtained. The order of approximation is 1/n. 相似文献
6.
Yafang Gong 《分析论及其应用》2006,22(4):329-338
Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Hoelder norm when a HSlder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Hoeder norm or the weighted HSlder norms. 相似文献
7.
It is known that Szász–Durrmeyer operator is the limit, in an appropriate sense, of Bernstein–Durrmeyer operators. In this paper, we adopt a new technique that comes from the representation of operator semigroups to study the approximation issue as mentioned above. We provide some new results on approximating Szász–Durrmeyer operator by Bernstein–Durrmeyer operators. Our results improve the corresponding results of Adell and De La Cal (Comput Math Appl 30:1–14, 1995). 相似文献
8.
This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically. 相似文献
9.
In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions
which give on (−∞,∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions
with an arbitrary spectrum. 相似文献
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11.
N.I. Mahmudov 《Applied Mathematics Letters》2011,24(7):1231-1238
In this paper, the order of simultaneous approximation and Voronovskaja-type theorems with quantitative estimate for complex Bernstein–Durrmeyer-type polynomials attached to analytic functions on compact disks are obtained. Our results show that extension of the complex Bernstein–Durrmeyer-type polynomials from real intervals to compact disks in the complex plane extends approximation properties. 相似文献
12.
S. A. Mohiuddine Tuncer Acar Abdullah Alotaibi 《Mathematical Methods in the Applied Sciences》2017,40(18):7749-7759
In the present paper, we construct a new sequence of Bernstein‐Kantorovich operators depending on a parameter α. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first‐ and second‐order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein‐Kantorovich operators and their approximation behaviors. 相似文献
13.
A. R. Mirotin 《Mathematical Notes》2013,93(1-2):257-265
A multidimensional generalization of the class of Bernstein functions is introduced and the properties of functions belonging to this class are studied. In particular, a new proof of the integral representation of Bernstein functions of several variables is given. Examples are considered. 相似文献
14.
We introduce a new type of modified Bernstein quasi-interpolants, which can be used to approximate functions with singularities. We establish direct, inverse, and equivalent theorems of the weighted approximation of this modified quasi-interpolants. Some classical results on approximation of continuous functions are generalized to the weighted approximation of functions with singularities. 相似文献
15.
This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain. 相似文献
16.
Nursel etin 《Mathematical Methods in the Applied Sciences》2019,42(16):5582-5594
In this paper, we investigate approximation properties of the complex form of an extension of the Bernstein polynomials, defined by Stancu by means of a probabilistic method. We obtain quantitative upper estimates for simultaneous approximation and the exact order of approximation by these operators attached to analytic functions in closed disks. Also, we prove that the new generalized complex Stancu operators preserve the univalence, starlikeness, convexity, and spirallikeness in the unit disk. 相似文献
17.
《Optimization》2012,61(4):345-357
In this Paper the approximation of continuous functions by positive linear operators of Bernstein type is investigated. The consideered operators are constructed using a system of rational functions with prescribed matrix of real poles. A certain general problem of S Bernstein concerning a scheme of construction of a sequence of positive linear operators is discussed. The answer on the Bernstein's hypothesis is given. The optimal limiting relations for the norm of the second central moment of our sequence of operators are established. 相似文献
18.
Dan Miclu 《Mathematical Methods in the Applied Sciences》2019,42(16):5317-5330
In this article, we try to assign a place on the map of the closed Newton–Cotés quadrature formulas to a new approximation formula based on the classical Bernstein polynomials. We create a procedure for a computer implementation that allows us to verify the accuracy of the new approximation formula. In order to get a complete image of this kind of approximation, we compare some well‐known quadrature formulas. Although effective in most situations, there are instances when the composite quadrature formulas cannot be applied, as they use equally‐spaced nodes. We present also an adaptive method that is used to obtain better approximations and to minimize the number of function evaluations. Numerical examples are given to increase the validity of the theoretical aspects. 相似文献
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