Bernstein-Durrmeyer polynomials on a simplex

Berens, Hubert; Schmid, Hans; Xu, Yuan

doi:10.1016/0021-9045(92)90104-v

Cited by 44 publications

(39 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These operators were studied by several authors (see, for example, [7,11,12]); in the case of d = 1, the Jacobi series, they were studied in [4,8]. As far as we know, Proposition 4.10 appears to be new even for d = 1.…”

mentioning

confidence: 99%

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

2004

Constr Approx

View full text Add to dashboard Cite

Abstract. For weighted L 1 space on the unit sphere of R d+1 , in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de la Vallée Poussin means and the Cesàro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of R d .

show abstract

mentioning

confidence: 99%

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

2004

Constr Approx

View full text Add to dashboard Cite

show abstract

“…Multidimensional Bernstein Durrmeyer operators have been introduced and studied by Derriennic in [12]; some further results in this setting may also be found, for instance, in [9,29,31]; particular multidimensional weighted Bernstein Durrmeyer operators have also been considered by Sauer in [23].…”

Section: Introductionmentioning

confidence: 97%

Generalized Bernstein–Durrmeyer Operators and the Associated Limit Semigroup

Attalienti¹

1999

Journal of Approximation Theory

View full text Add to dashboard Cite

The aim of this paper is the study of a new sequence of positive linear approximation operators M n, * on C([0, 1]) which generalize the classical Bernstein Durrmeyer operators. After proving a Voronovskaja-type result, we show that there exists a strongly continuous positive contraction semigroup on C([0, 1]) which may be expressed in terms of powers of these operators. As a direct consequence, we are able to represent explicitly the solutions of the Cauchy problems associated with a particular class of second order differential operators. Academic Press

show abstract

“…The operators M a,b n , studied by Pȃltȃnea [14], [15], Berens and Xu [1] and others have attractive properties of approximation, very similar to those of the Durrmeyer-Lupaş operators, including the characteristic property of representation as modified partial Fourier sums. However, for positive linear operators this property is incompatible with that of preserving linear functions.…”

Section: )mentioning

confidence: 99%

“…Denote by L B [0, 1] the space of bounded Lebesgue integrable functions on [0,1] and by Π n the space of polynomials of degree at most n ∈ N. The operators U n : L B [0, 1] → Π n , n 1, given by U n (f, x) = (n − 1)…”

Section: Introductionmentioning

confidence: 99%