Sharp estimates for the polynomial approximation in weighted Sobolev spaces

© 2015, Pleiades Publishing, Ltd. We obtain sharp estimates for the accuracy of the best approximation of functions by algebraic polynomials on an interval, the half-line, and the entire line in weighted Sobolev spaces with Jacobi, Laguerre, and Hermite weights, respectively. We show that the orthogonal polynomials associated with these weights form orthogonal bases in the respective weighted Sobolev spaces. We obtain sharp estimates of Markov–Bernstein type

A Modification of Bernstein-Durrmeyer Operators with Jacobi Weights on the Unit Interval

The present paper is devoted to the study of a sequence of positive linear operators, acting on the space of all continuous functions on [0, 1] as well as on some weighted spaces of integrable functions on [0, 1]. These operators are, as a matter of fact, a generalization of the Bernstein-Durrmeyer operators with Jacobi weights. In particular, we present qualitative and approximation properties of these operators, also providing estimates of the rate of convergence. Moreover, by means of their asymptotic formula, we compare our operators with the Bernstein-Durrmeyer ones and a suitable modification of theirs, showing that, in suitable intervals, they provide a lower approximating error estimate

Rational B\'ezier Curves Approximated by Bernstein-Jacobi Hybrid Polynomial Curves

In this paper, we propose a linear method for $C^{(r,s)}$ approximation of rational B\'{e}zier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for B\'{e}zier curve is used. A error bound between rational B\'{e}zier curve and B\'{e}zier curve is presented. Finally, some examples and figures were offered to demonstrate the efficiency, simplicity, and stability of our methods