Sofiya Ostrovska

q -Bernstein polynomials and their iterates

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 + e} 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→ ∞.

Convergence of Generalized Bernstein Polynomials

Let f?C0, 1], q?(0, 1), and Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers. These polynomials were introduced by G. M. Phillips in 1997. We study convergence properties of the sequence {Bn(f, q; x)}∞n=1. It is shown that in general these properties are essentially different from those in the classical case q=1.

On the improvement of analytic properties under the limit q -Bernstein operator

Let Bn(f, q; x), n = 1,2,... be the q-Bernstein polynomials of a function f ∈ C[0, 1]. In the case 0 < q < 1, a sequence {Bn(f, q; x)} generates a positive linear operator B∞ = B∞,q on C[0,1], which is called the limit q-Bernstein operator. In this paper, a connection between the smoothness of a function f and the analytic properties of its image under B∞ is studied.

THE NORM ESTIMATES FOR THE q-BERNSTEIN OPERATOR IN THE CASE q > 1

The q-Bernstein basis with 0 1, the behavior of the q-Bernstein basic polynomials on [0,1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operator B n,q in the case q > 1. While for 0 1, the norm ∥B n,q ∥ increases rather rapidly as n → oo. We prove here that ∥B n,q ∥ ~ C q q n(n-1)/2 /n, n → oo with C q = 2(q -2 ;q -2 )∞/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.

q-Bernstein polynomials of the Cauchy kernel

Due to the fact that 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 1) is still open. In this paper, the q-Bernstein polynomials Bn,q(fa;z) of the Cauchy kernel fa=1/(z-a),a∈C⧹[0,1] are found explicitly and their properties are investigated. In particular, it is proved that if q>1, then polynomials Bn,q(fa;z) converge to fa uniformly on any compact set K⊂{z:|z| |a|}, the sequence {Bn,q(fa;z)} is not even uniformly bounded.

The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

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.

A Survey of Results on the Limit

The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

q

The aim of this paper is to present new results related to the -Bernstein polynomials of unbounded functions in the case and to illustrate those results using numerical examples. As a model, the behavior of polynomials is examined both theoretically and numerically in detail for functions on satisfying as , where and are real numbers.

-Bernstein Operator

The aim of this paper is to present new results related to the convergence of the sequence of the 𝑞-Bernstein polynomials {𝐵𝑛,𝑞(𝑓;𝑥)} in the case 𝑞g1, where 𝑓 is a continuous function on [0,1]. It is shown that the polynomials converge to 𝑓 uniformly on the time scale 𝕁𝑞={𝑞−𝑗}∞𝑗=0∪{0}, and that this result is sharp in the sense that the sequence {𝐵𝑛,𝑞(𝑓;𝑥)}∞𝑛=1 may be divergent for all 𝑥∈𝑅⧵𝕁𝑞. Further, the impossibility of the uniform approximation for the Weierstrass-type functions is established. Throughout the paper, the results are illustrated by numerical examples.

On the -Bernstein Polynomials of Unbounded Functions with

The limit q-Bernstein operator emerges naturally as a q-version of the Szász–Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. The limit q-Bernstein operator has been widely studied lately. It has been shown that is a positive shape-preserving linear operator on with Its approximation properties, probabilistic interpretation, the behaviour of iterates, eigenstructure and the impact on the smoothness of a function have been examined. In this article, we prove the following unicity theorem for operator: if f is analytic on [0, 1] and for then f is a linear function. The result is sharp in the following sense: for any proper closed subset of [0, 1] satisfying there exists a non-linear infinitely differentiable function f so that for all