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On the norms and minimal properties of de la Vallée Poussin’s type operators

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Abstract

In this paper we consider the de la Vallée Poussin’s type operators \(H_{rn,sn}\)

$$\begin{aligned} H_{rn,sn}:=\frac{F_{rn}+F_{rn+1}+...+F_{sn-1}}{(s-r)n}, \end{aligned}$$

where \(F_k\) are classical Fourier projections onto \(\varPi _k\) (the space of trigonometric polynomials of degree less than or equal to k). We determine when \(H_{n,sn}\) is the minimal generalized projection and provide the asymptotic behavior of the norm \(\Vert H_{n,sn}\Vert \). Additionally, we contrast the results obtained for the trigonometric system to the results obtained for the Rademacher system.

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Authors and Affiliations

  1. Institute of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084, Kraków, Poland

    Beata Deregowska

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Simon Foucart

  3. Faculty of Mathematics and Computer Science, Jagiellonian University, Lojasiewicza 6, 30-048, Kraków, Poland

    Barbara Lewandowska

  4. Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave., CMC 342, Tampa, FL, 33620-5700, USA

    Lesław Skrzypek

Authors
  1. Beata Deregowska
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  2. Simon Foucart
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  3. Barbara Lewandowska
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  4. Lesław Skrzypek
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Correspondence to Lesław Skrzypek.

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Communicated by A. Constantin.

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Deregowska, B., Foucart, S., Lewandowska, B. et al. On the norms and minimal properties of de la Vallée Poussin’s type operators. Monatsh Math 185, 601–619 (2018). https://doi.org/10.1007/s00605-018-1159-x

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  • DOI: https://doi.org/10.1007/s00605-018-1159-x

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