Best Constants in Global Smoothness Preservation Inequalities for Some Multivariate Operators

✍ Scribed by Jesús de la Cal; Ana M. Valle

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 175 KB
• Volume: 97
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3252

✦ Synopsis

The 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.