Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

Abstract

Abstract. 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] → U n with strictly increasing nodes, fixing f 0 , f 1 ∈ U n . If U n ⊂ U n+1 and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 , f 1 , ..., f n is a basis of U n such that the linear span of f 0 , .., f k is an extended Chebyshev space over [a, b] for each k = 0, ..., n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold

A class of Markov type operators which preserveej,j⩾1

“…we get a discrete Markov type operator which preserves the above mentioned test function (see [2]). In this article we obtain a class of Markov type operators of the form (2) (some functionals k n;k ðf Þ are of integral form, but not all) which preserves the monomial e j ; j P 1.…”

Generalized Bernstein Operators on the Classical Polynomial Spaces

“…If these equalities can be satisfied, they uniquely determine the location of the nodes and the values of the coefficients, cf. [4,Lemma 5]; in other words, there is at most one Bernstein operator B n of the form (2) satisfying (3). We will consistently use the following notation.…”

“…Theorem 5.1) that given any polynomial f 1 (x), strictly increasing on [a, b], and of degree m, it is always possible to find a generalized Bernstein operator fixing 1 and f 1 , on the space P n [a, b] with the standard Bernstein basis, provided that n ≥ m and that n is "sufficiently large". The special case f 1 (x) = x j , [a, b] = [0, 1], had been previously solved in [4,Proposition 11]; on the other hand, it is known that such operators do not exist if we are required to fix f 0 (x) = x i and f 1 (x) = x j on [0, 1], 1 ≤ i < j, cf. [11,Theorem Within the line of research followed here, previous work has focused on spaces different or more general than spaces of polynomials (cf.…”

“…[11,Theorem Within the line of research followed here, previous work has focused on spaces different or more general than spaces of polynomials (cf. [15], [3], [4], [5], [13], [6], [14]), but convergence has also been studied in Müntz spaces, cf. [2], and for rational Bernstein operators, see [18].…”

“…Generally speaking, the situation regarding existence is well understood in the context of extended Chebyshev spaces (which generalize the space of polynomials of degree at most n, by retaining the bound on the number of zeros) cf. [4]: One considers a two dimensional extended Chebyshev space U 1 , for which a generalized Bernstein operator fixing it can always be defined, and inductively, this definition is extended to U 1 ⊂ U 2 ⊂ · · · ⊂ U n , where each U k is a k + 1-dimensional extended Chebyshev space.…”