Borislav R. Draganov
The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.
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In this paper, we introduce the higher order generalization of Bernstein type operators defined by (p,q)-integers. We establish some approximation results for these new operators by using the modulus of continuity.
In the present article, we have given a corrigendum to our paper "Some approximation results on Bernstein-Schurer operators defined by (p,q)-integers" published in Journal of In- equalities and Applications (2015) 2015:249.
We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the sufficient conditions under which those estimates hold are also necessary.
We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big $O$-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein polynomials with integer coefficients is saturated. We identify its saturation rate and the trivial class.
In this paper, First we have given the modified form of (p,q)-analogues of Bernstein and Bernstein operators [21-23] and then we introduce a new analogue of Bernstein-Kantorovich operators which we call as (p,q)-Bernstein-Kantorovich operators. We discuss approximation properties for these operators based on Korovkin's type approximation theorem and we compute the order of convergence using usual modulus of continuity and also the rate of convergence when f is a Lipschitz function. Moreover, we also study the local approximation property of the (p,q)-Kantorovich operators . We show comparisons…
We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator $B_{n}:C[a,b]\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\in U_{n}$. If $U_{n}\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negative Bernstein basis, then there exists a Bernstein operator $B_{…
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed via Mellin Transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the generalized Stirling numbers, and Bernoulli polynomials of higher orde…
Continuous solutions of the complex Kac--Bernstein functional equation on the integers and the real numbers
研究整数集与实数集上复卡茨 - 伯恩斯坦泛函方程连续解的分类，利用\(\mathbb{Z}\)上分类推广一维环面\(\mathbb{T}\)上卡茨 - 伯恩斯坦定理，类似地推广\(\mathbb{R}\)上原定理，均从概率博雷尔测度推广到复博雷尔测度。
On a specific family of orthogonal polynomials of Bernstein-Szegö type
本文研究 Bernstein-Szegö 型权函数及其扩展，发现特定次数的正交多项式具有简单形式和已知根，由此推导出显式求积公式，构造等矩测度，并找到 Glaisher 和 Ramanujan 反常积分的有限模拟。
We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from $\{ \pm 1\}$ only. A basic case of our results states that for any Lipschitz function $f:[0,1] \to [-1,1]$ and for any positive integer $n$, there are signs $σ_0,\dots,σ_n \in \{\pm 1\}$ such that $$\left |f(x) - \sum_{k=0}^n σ_k \, \binom{n}{k} x^k (1-x)^{n-k} \right | \leq \frac{C (1+|f|_{\mathrm{Lip}})}{1+\sqrt{nx(1-x)}} ~\mbox{ for all } x \in [0,1].$$ More generally, we show that higher accuracy is achievable for sm…
Recently, Mursaleen et al applied (p,q)-calculus in approximation theory and introduced (p,q)-analogue of Bernstein operators in [16]. In this paper, we construct and introduce a generalization of the bivariate Bleimann-Butzer-Hahn operators based on (p,q)-integers and obtain Korovkin type approximation theorem of these operators. Furthermore, we compute the rate of convergence of the operators by using the modulus of continuity and Lipschitz type maximalfunctions.
Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$\|f\|_A := \sup_{x \in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A \subset {\Bbb R}$. Let $$V_a^b(f) := \int_a^b{|f^{\prime}(x)| \, dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 \frac nk\leq \min_{P \…
Uniform Approximation by Polynomials with Integer Coefficients via the Bernstein Lattice
本文提出“Bernstein 格”这一整系数多项式子集，证明其在特定连续函数空间中稠密，并给出了优于经典结果的逼近误差定量上界。
Dimension-free Markov--Bernstein inequalities for product measures
针对乘积概率测度下的多项式，研究无维数Markov-Bernstein不等式，在Gauss情形下对p≥4得到与次数相关的界，并推广到均匀分布、单峰密度乘积及Freud测度乘积。
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