In the present paper, we consider (p, q)-analogue of the Beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also,... more
In the present paper, we consider (p, q)-analogue of the Beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also,... more
We study the asymptotic properties of the Bernstein estimator for unbounded density copula functions. We show that the estimator converges to infinity at the corner. We establish its relative convergence when the copula is unbounded and... more
In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of successive approximations are applied to obtain the approximate solution of nonlinear fuzzy Fredholm integral equations. The main idea of using... more
We define the concept of rough limit set of a triple sequence space of Bernstein-Stancu polynomials of fuzzy numbers and obtain the relation between the set of rough limit and the extreme limit points of a triple sequence space of... more
In this paper, a numerical approximation for the solution of linear fractional differential equations, based on Galerkin method and Bernstein polynomials, is proposed. A system of linear equations is obtained and the coefficients of... more
In the present article, we have given a corrigendum to our paper "On (p, q)-analogue of Bernstein operators" published in Applied Mathematics and Computation 266 (2015) 874-882.
We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in L p to all p ≥ 0. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic... more
The central focus of this paper is upon the alleviation of the boundary problem when the probability density function has a bounded support. Mixtures of beta densities have led to different methods of density estimation for data assumed... more
The goal of this research article is to introduce a sequence of α–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A... more
In this paper, we give a King-type modification of the Bernstein-Kantorovich operators and study the approximation properties of these operators. We prove that the error estimation of these operators is better than the classical... more
In this paper we put in evidence localization results for the socalled Bernstein max-min operators and a property of translation for the Bernstein max-product operators.
We use the Bernstein polynomials of degree d as the basis for constructing a uniform approximation to the rate of evolution (related to the fixation probability) of a species in a two-component finite-population frequency-dependent... more
We prove a multivariate version of Bernstein's inequality about the probability that degenerate U-statistics take a value larger than some number u. This is an improvement of former estimates for the same problem which yields an... more
This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform,... more
In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube [0, 1] d and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its... more
In this note, we introduce the Durrmeyer variant of Stancu operators that preserve the constant functions depending on non-negative parameters. We give a global approximation theorem in terms of the Ditzian-Totik modulus of smoothness, a... more
We give a simple, elementary, and at least partially new proof of Arestov's famous extension of Bernstein's inequality in L p to all p ≥ 0. Our crucial observation is that Boyd's approach to prove Mahler's inequality for algebraic... more
The mean value interpolation operators are extended to less smooth functions when the points de ning them are in general position, and the corresponding interpolation conditions are described. In particular, it is shown that Kergin... more
The Bernstein operator B n for a simplex in R d is naturally defined via the Bernstein basis obtained from the barycentric coordinates given by its vertices. Here we consider a generalisation of this basis and the Bernstein operator,... more
Here we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing... more
Quasi-interpolation is a important tool, used both in theory and in practice, for the approximation of smooth functions from univariate or multivariate spaces which contain Π m = Π m (IR d) the d-variate polynomials of degree ≤ m. In... more
We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in... more
We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in... more
We consider a sequence of polynomials appearing in expressions for the derivatives of the Lambert W function. The coefficients of each polynomial are shown to form a positive sequence that is log-concave and unimodal. This property... more
In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube [0, 1] d and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its... more
In this paper, we establish some Bernstein-type inequalities for rational functions with prescribed poles. These results refine prior inequalities on rational functions and strengthen many well-known polynomial inequalities.
In this paper, we consider a more general class of rational functions r(s(z)) of degree mn, where s(z) is a polynomial of degree m and prove some sharp results concerning to Bernstein type inequalities for rational functions.
In this paper, we establish some Bernstein-type inequalities for rational functions with prescribed poles. These results refine prior inequalities on rational functions and strengthen many well-known polynomial inequalities.
Rational functions of total degree $l$ in n variables have a representation in the Bernstein form defined over $n$ dimensional simplex. The range of a rational function is bounded by the smallest and the largest rational Bernstein... more
We study the control problem for polynomial continuous-time dynamical systems. We consider polynomial Lyapunov functions and controllers, both parameterised in Bernstein form. Specifically, we present necessary and sufficient conditions... more
A Note on the Bernstein Algorithm for Bounds for Interval Polynomials. In computing the range of values of a polynomial over an interval a_< x < b one may use polynomials of the form (~) (x-a)J (b-x) 1'-j called Bernstein polynomials of... more
In the papers dealing with derivation and applications of operational matrices of Bernstein polynomials, a basis transformation, commonly a transformation to power basis, is used. The main disadvantage of this method is that the... more
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We give an explicit formula for the Hodge filtration on the DX -module OX( * Z)f 1-α associated to the effective Q-divisor D = α • Z, where 0 < α ≤ 1 and Z = (f = 0) is an irreducible hypersurface defined by f , a weighted homogeneous... more
We derive the non-asymptotical non-uniform sharp error estimation for Bernstein's approximation of continuous function based on the modern probabilistic apparatus. We investigate also the convergence of derivative of these polynomials and... more
The purpose of the research is to find the numerical solutions to the system of time dependent nonlinear parabolic partial differential equations (PDEs) utilizing the Modified Galerkin Weighted Residual Method (MGWRM) with the help of... more
In this paper we introduce new bivariate Bernstein type operators B M,i n (f ; x, y), i = 1, 2, 3. The rates of approximation by these operators are calculated and it is shown that the errors are significantly smaller than those of... more
We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil... more
Nonlinear q-Bernstein operator of max-product kind was introduced and its approximation order was examined, and the order of approximation was found to be 1/ [n] q by Duman in [8]. In this paper, we found a better order of approximation... more
The aim of this work is application of Bernstein polynomials (BPs) for solving multi-order multi-dimensional fractional optimal control problem (MOMDFOCP). Firstly, by the Bernstein basis, we introduce operational matrices for... more
In this paper, we present a method for solving multi-dimensional fractional optimal control problems. Firstly, we derive the Bernstein polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been... more
In this paper, we present a method for solving time varying fractional optimal control problems by Bernstein polynomials. Firstly, we derive the Bernstein polynomials (BPs) operational matrix for the fractional derivative in the Caputo... more
We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this... more
This paper presents a numerical method for solving Abel's integral equation as singular Volterra integral equations. In the proposed method, the functions in Abel's integral equation are approximated based on Bernstein polynomials (BPs)... more
This paper presents a numerical method for solving Abel's integral equation as singular Volterra integral equations. In the proposed method, the functions in Abel's integral equation are approximated based on Bernstein polynomials (BPs)... more
In this Paper we give a scheme for the numerical solution of fractional order Logistic equations (FOLE) using operational matrices for fractional order integration and multiplications based on Bernstein Polynomials (BPs). By this method... more