Best Constants in Global Smoothness Preservation Inequalities for Some Multivariate Operators

Abstract

dedicated to the memory of professor e. aparicioThe Bernstein operator on the standard k-simplex and other analogous k-variate operators allow for a probabilistic representation in terms of the successive increments of a real valued superstationary stochastic process (a notion introduced in the paper) starting at the origin and having nondecreasing paths. For this class of operators, we obtain estimates of the best constants in preservation of the first modulus of continuity corresponding to the l 1 -norm, and in preservation of classes of functions defined by concave moduli of continuity. We also show that, in some special cases, such best constants do not depend upon the dimension k. To show our results, we use probabilistic tools such as couplings and Wasserstein distances for multivariate probability distributions. The general results are applied to the computation of the aforementioned constants for several classical multivariate operators.

Best constants for tensor products of Bernstein type operators

“…Problems of this kind have been discussed in several works by using different approaches (see, for instance, [1][2][3][4][5][6][7][8][9][10]12] and references therein). The probabilistic approach developed in [1,2,[6][7][8][9]12] has proved to be suitable and fruitful when dealing with operators of probabilistic type (also called Bernstein-type operators), that is, operators allowing for a representation of the form…”

“…The probabilistic approach developed in [1,2,[6][7][8][9]12] has proved to be suitable and fruitful when dealing with operators of probabilistic type (also called Bernstein-type operators), that is, operators allowing for a representation of the form…”

Global smoothness preservation with second order modulus of smoothness

“…From (3.9) it is easy to observe that 10) which follows from the definition of the kth order modulus of continuity of f . It is well known (see [13]) that for all…”

“…In the multivariate setting, best constants in preservation inequalities for Bernstein-type operators can be found in de la Cal and Valle [10,11], de la Cal et al [9] and de la Cal and Cárcamo [8]. On the other hand, D. X. Zhou obtained in [22] …”