K-functionals and multivariate Bernstein polynomials

Abstract

This paper estimates upper and lower bounds for the approximation rates of iterated Boolean sums of multivariate Bernstein polynomials. Both direct and inverse inequalities for the approximation rate are established in terms of a certain K -functional. From these estimates, one can also determine the class of functions yielding optimal approximations to the iterated Boolean sums.

On Multivariate Bernstein Polynomials

“…The following theorem states for the pointwise and uniform convergence of our Bernstein polynomials with several variables. It should be mentionned that similar work has been done but only for two variables (see (Theorem 1.1 in Reference [26]), [30], (Theorems 3, 5 and 8 in [21]), (Theorem 6.7 in [33]), (Lemma 1 in [28]), (Theorem in [32]) and (Theorems 1 and 3 in [7]).…”

“…(b) the Bernstein polynomials on a m-dimensional simplex and very useful in stochastic analysis (see References [7,[23][24][25][26]):…”

“…For the multivariate Bernstein polynomials given by Equation (3): they have proven two identities for these polynomials on simplex, which are considered on a pointwise (see References [23][24][25]25]). They also showed that these polynomials converge to their corresponding functions f along with partial derivatives provided that the latter derivatives exist and are continuous (see References [7,26,33]). 3.…”

“…Some have tried to study the approximation by combination of Bernstein polynomials or operators (the rate of convergence of derivatives and the smoothness of the derivatives of functions) in order to approximate, under definite conditions, the function more closely than the Bernstein polynomials (see References [3,6,[37][38][39][40][41]). Moreover, they defined and also tried to study the q-Bernstein polynomials (see References [2,4,[42][43][44][45]), the generalized Bernstein polynomials (see References [10,26]) and the modified Bernstein polynomials (see Reference [25]).…”

A Note on the Generalized Bernstein Polynomials

“…This theory has many applications in different areas in mathematics and physics, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Throughout this paper we set I = [0, 1] and k ∈ N. Taking a k-dimensional simplex ∆ k :…”