共查询到20条相似文献,搜索用时 15 毫秒
1.
Sofiya Ostrovska 《Czechoslovak Mathematical Journal》2008,58(4):1195-1206
Due to the fact that in the case q > 1 the q-Bernstein polynomials are no longer positive linear operators on C[0, 1], the study of their convergence properties turns out to be essentially more difficult than that for q < 1. In this paper, new saturation theorems related to the convergence of q-Bernstein polynomials in the case q > 1 are proved. 相似文献
2.
Sofiya Ostrovska 《Numerical Algorithms》2007,44(1):69-82
Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc
{z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown
that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.
相似文献
3.
Nazim I. Mahmudov 《Czechoslovak Mathematical Journal》2012,62(2):557-566
In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate q-Bernstein polynomials for a function analytic in the polydisc ${D_{{R_1}}} \times {D_{{R_2}}} = \{ z \in C:\left| z \right| < {R_1}\} \times \{ z \in C:\left| z \right| < {R_1}\} $ for arbitrary fixed q > 1. We give quantitative Voronovskaja type estimates for the bivariate q-Bernstein polynomials for q > 1. In the univariate case the similar results were obtained by S.Ostrovska: q-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255, and S.G.Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009. 相似文献
4.
Xueyan Xiang Qian He Wenshan Yang 《分析论及其应用》2007,23(3):243-254
Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein polynomials and their Boolean sum are estimated. Moreover, the saturation of {Bn(., qn)} when n → ∞ and convergence rate of Bn(f,q;x) when f ∈ C^n-1 [0, 1], q → ∞ are also presented. 相似文献
5.
Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C[0,1] has the order q−n (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞. 相似文献
6.
Sorin G. Gal 《Advances in Applied Clifford Algebras》2012,22(2):313-319
Defining for q > 1 the q-Bernstein polynomials of degree n of a quaternion variable, attached to a function f defined on a ball in the field of quaternions, the order of approximation ${\frac{1} {q^n}}$ is obtained when f is in some classes of analytic functions in the sense of Weierstrass. The result extends that in the case of approximation of analytic functions of a complex variable in disks, by q-Bernstein polynomials of complex variable. 相似文献
7.
An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define
a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix
representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier
patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B
n,m
(f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed.
AMS subject classification (2000) 65D17 相似文献
8.
The rate of convergence of q-Bernstein polynomials for 总被引:3,自引:0,他引:3
In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {Bn,q(f)} for 0<q<1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions , and the orders do not depend on α, unlike the classical case. 相似文献
9.
In the paper, we discuss Voronovskaya-type theorem and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0<q<1. We give explicit formulas of Voronovskaya-type for the q-Bernstein polynomials for 0<q<1. If , we show that the rate of convergence for the q-Bernstein polynomials is o(qn) if and only ifWe also prove that if f is convex on [0,1] or analytic on (-ε,1+ε) for some ε>0, then the rate of convergence for the q-Bernstein polynomials is o(qn) if and only if f is linear. 相似文献
10.
We introduce polynomials B
n
i
(x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as
q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating
B
n
i
(x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
Mehmet Aç?kgöz 《Applied mathematics and computation》2011,218(3):707-712
In this paper, we consider the modified q-Bernstein polynomials for functions of several variables on q-Volkenborn integral and investigate some new interesting properties of these polynomials related to q-Stirling numbers, Hermite polynomials and Carlitz’s type q-Bernoulli numbers. 相似文献
12.
Zhi Yuan HUANG Pei Yan LI Ying WU 《数学学报(英文版)》2008,24(2):201-214
The classical Levy-Meixner polynomials are distinguished through the special forms of their generating functions. In fact, they are completely determined by 4 parameters: c1, c2,γ and β. In this paper, for-1 〈q〈 1, we obtain a unified explicit form of q-deformed Levy-Meixner polynomials and their generating functions in term of c1, c2, γand β, which is shown to be a reasonable interpolation between classical case (q=1) and fermionic case (q=-1).In particular, when q=0 it's also compatible with the free case. 相似文献
13.
Nazim Mahmudov 《Numerical Algorithms》2010,53(4):439-450
In this note we give the estimates of the central moments for q-Bernstein operators (0 < q < 1) which can be used for studying the approximation properties of the operators. 相似文献
14.
In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q−1,…,q−n+1. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), qn(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if limn→∞qn=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. 相似文献
15.
Grzegorz Nowak 《Journal of Mathematical Analysis and Applications》2009,350(1):50-55
In this paper, we introduce the generalized q-Bernstein polynomials based on the q-integers and we study approximation properties of these operators. In special case, we obtain Stancu operators or Phillips polynomials. 相似文献
16.
Sorin G. Gal 《Complex Analysis and Operator Theory》2012,6(2):515-527
Attaching to a compact disk
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} in the quaternion field
\mathbbH{\mathbb{H}} and to some analytic function in Weierstrass sense on
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation
in
[`(\mathbbDr)]{\overline{\mathbb{D}_{r}}} for these operators, namely
\frac1n{\frac{1}{n}} if q = 1 and
\frac1qn{\frac{1}{q^{n}}} if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks
by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:, 2011). 相似文献
17.
The paper deals with positive linear operators based on q-integer. The rate of convergence of these operators is established. For these operators, we present Voronovskaya-type theorems
and apply them to q-Bernstein polynomials and q-Stancu operators. 相似文献
18.
Sofiya Ostrovska 《印度理论与应用数学杂志》2017,48(2):205-210
The limit q-Bernstein operator B q comes out naturally as the limit for the sequence of q-Bernstein operators in the case 0 < q < 1: Alternatively, it can be viewed as a modification of the Szász-Mirakyan operator related to the Euler distribution. In this paper, a necessary and sufficient condition for a function g to be an image of an entire function under B q is presented. 相似文献
19.
Sofiya Ostrovska 《Results in Mathematics》2016,69(3-4):275-295
The article exhibits a review of results on two popular q-versions of the Bernstein polynomials, namely, the Lupa? q-analogue and the q-Bernstein polynomials. Their similarities and distinctions are discussed. 相似文献