An Answer to an Open Problem on the Multivariate Bernstein Polynomials on a Simplex

Among the applications of the Bernstein polynomials in one variable is their use in solving problems associated with stochastic computing. Taking as a starting point the notion of stochastic logic in the sense of Qian-Riedel-Rosenberg, the aim of this paper is to investigate some necessary and sufficient conditions for guaranteeing whether polynomial operations can be implemented with stochastic logic based on multivariate Bernstein polynomials with coefficients in the unit interval.

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous function and L p convergence for these operators are studied, respectively.

Division algorithms for Bernstein polynomials

An identity for multivariate Bernstein polynomials

It is well known that discretely defined operators such as Bernstein polynomials, Baskakov operators, Szasz–Mirakyan, Meyer-Konig–Zeller operators, Stancu operators, Jain operators, and their modified versions are not possible to approximate Lebesgue integrable functions. In the year 1930, Kantorovich proposed integral modification of the Bernstein polynomials. The more general integral modification of the Bernstein polynomial was given by Durrmeyer in the year 1967. After this several Durrmeyer-type modifications of different operators have been introduced and their approximation properties have been discussed. In the present note, to the best of our knowledge, we present the different Durrmeyer-type operators introduced in the last five decades. We also propose some open problems in the end.

Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order

A novel numerical technique to solve 2D Fredholm integral equations (2DFIEs) of first kind is proposed in this study. This technique is based on the discretization of 2DFIEs by replacing the unknown function with two-dimensional Bernstein polynomial basis functions. We formulate the convergence analysis which shows the fast converges of this technique to the actual solution. Some problems of 2D linear Fredholm integral equations are illustrated to show the efficiency of the proposed scheme.

V.9 - SUBDIVIDING SIMPLICES

The generalization of the Bernstein polynomials based on Polya distribution was first considered by Stancu [14]. Very recently Gupta and Rassias [6] proposed the Durrmeyer type modification of the Lupa? operators and established some results. Now we extend the studies and here we estimate the convergence estimates, which include quantitative asymptotic formula and rate of approximation bounded variation. We also give an open problem for readers to obtain the moments using hypergeometric function.