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Bernstein-type operators diminish the φ-variation

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Abstract

We show that a large class of Bernstein-type operators usually considered in approximation theory diminish the total and the fine φ-variation, thus extending a classical result on 1-variation diminution concerning the Bernstein polynomials. Also, the closely related problem of approximation in φ-variation is thoroughly discussed. For these purposes, we use a probabilistic approach in which coupling methods play a fundamental role.

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009, Zaragoza, Spain

    J. A. Adell

  2. Departamento de Matemática Aplicada y Estadística e Investigación Operativa Facultad de Ciencias, Universidad del País Vasco, Apdo. 644, 48080, Bilbao, Spain

    J. de la Cal

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  1. J. A. Adell
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  2. J. de la Cal
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Correspondence to J. de la Cal.

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Communicated by Vilmos Totik.

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Adell, J.A., de la Cal, J. Bernstein-type operators diminish the φ-variation. Constr. Approx 12, 489–507 (1996). https://doi.org/10.1007/BF02437505

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  • DOI: https://doi.org/10.1007/BF02437505

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