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Best constants for tensor products of Bernstein, Szász and Baskakov operators

Published online by Cambridge University Press:  17 April 2009

Jesús de la Cal  and
Ana M. Valle
Jesús de la Cal
Affiliation:
Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencias, Universidad del Paíis Vasco, Apartado 644, 48080 Bilbao, Spain e-mail: mepcaagj@lg.ehu.esmepvamaa@lg.ehu.es
Ana M. Valle
Affiliation:
Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencias, Universidad del Paíis Vasco, Apartado 644, 48080 Bilbao, Spain e-mail: mepcaagj@lg.ehu.esmepvamaa@lg.ehu.es
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Abstract

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We consider tensor product operators and discuss their best constants in preservation inequalities concerning the usual moduli of continuity. In a previous paper, we obtained lower and upper bounds on such constants, under fairly general assumptions on the operators. Here, we concentrate on the l∞-modulus of continuity and three celebrated families of operators. For the tensor product of k identical copies of the Bernstein operator Bn, we show that the best uniform constant coincides with the dimension k when k ≥ 3, while, in case k = 2, it lies in the interval [2, 5/2] but depends upon n. Similar results also hold when Bn is replaced by a univariate Szász or Baskakov operator. The three proofs follow the same pattern, a crucial ingredient being some special properties of the probability distributions involved in the mentioned operators, namely: the binomial, Poisson, and negative binomial distributions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 211 - 220
Copyright
Copyright © Australian Mathematical Society 2000

References

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