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A Weighted Generalization of the Classical Kantorovich Operator. II: Saturation

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Abstract

In [1], we have introduced a new weighted type of modification of the classical Kantorovich operator. The advantage of this operator is that there is no restriction on the parameters of the weight, and the class of functions is wider than in the earlier version of the weighted operator (cf. the monograph of Ditzian and Totik [3]). Direct and converse theorems and a Voronovskaya-type relation were proved. Here we solve the saturation problem of the operator (Theorem 2.1). We follow the method developed in [3], but the details are much more involved. A surprising fact emerges in determining the trivial class of saturation (Theorem 3.1).

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References

  1. Della Vecchia B., Mastroianni G., Szabados J.: A weighted generalization of the classical Kantorovich operator. Rend. Circ. Mat. Palermo (2) 82, 1–27 (2010)

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  2. Ditzian Z., May C. P.: L p saturation and inverse theorems for modified Bernstein polynomials. Indiana Univ. Math. J. 25, 733–751 (1976)

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, I 00185, Roma, Italy

    Biancamaria Della Vecchia

  2. Istituto per le Applicazioni del Calcolo ‘M. Picone’ CNR - Sede di Napoli, Via Pietro Castellino 111, I 80131, Napoli, Italy

    Biancamaria Della Vecchia

  3. Dipartimento di Matematica e Informatica, Università della Basilicata, Contrada Macchia Romana, I 85100, Potenza, Italy

    Giuseppe Mastroianni

  4. Alfréd Rényi Institute of Mathematics, P.O.B. 127, H 1364, Budapest, Hungary

    József Szabados

Authors
  1. Biancamaria Della Vecchia
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  2. Giuseppe Mastroianni
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  3. József Szabados
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Corresponding author

Correspondence to József Szabados.

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Dedicated to Professor Francesco Altomare on his 60th birthday

The work of Giuseppe Mastroianni was supported by University of Basilicata (local funds). Research of József Szabados was supported by OTKA No. T049196.

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Della Vecchia, B., Mastroianni, G. & Szabados, J. A Weighted Generalization of the Classical Kantorovich Operator. II: Saturation. Mediterr. J. Math. 10, 1–15 (2013). https://doi.org/10.1007/s00009-012-0210-8

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  • DOI: https://doi.org/10.1007/s00009-012-0210-8

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