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Chebyshevskii Sbornik, 2020, Volume 21, Issue 1, Pages 247–258
DOI: https://doi.org/10.22405/2226-8383-2018-21-1-247-258
(Mi cheb871)
 

This article is cited in 6 scientific papers (total in 6 papers)

Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight

I. A. Martyanov

Tula State University
Full-text PDF (727 kB) Citations (6)
References:
Abstract: We study the Nikolskii constant (or the Jackson–Nikolskii constant) for complex trigonometric polynomials in the space Lpα(T) for p1 with the periodic Gegenbauer weight |sinx|2α+1:
Cp,α(n)=supTTn{0}TTp,
where p=Lpα(T). D. Jackson (1933) proved that Cp,1/2(n)cpn1/p for all n1. The problem of finding Cp,1/2(n) has a long history. However, sharp constants are known only for p=2. For p=1, the problem has interesting applications, e.g., in number theory. We note the results of Ja. L. Geronimus, L. V. Taikov, D. V. Gorbachev, I. E. Simonov, P. Yu. Glazyrina. For p>0, we note the results of I. I. Ibragimov, V. I. Ivanov, E. Levin, D. S. Lubinsky, M. I. Ganzburg, S. Yu. Tikhonov, in the weight case — V. V. Arestov, A. G. Babenko, M. V. Deikalova, Á. Horváth.
It is proved that the supremum here is achieved on a real even trigonometric polynomial with a maximum modulus at zero. As a result, a connection is established with the Nikolskii algebraic constant with weight (1x2)α, investigated by V. V. Arestov and M. V. Deikalova (2015). The proof follows their method and is based on the positive generalized translation operator in the space Lpα(T) with the periodic Gegenbauer weight. This operator was constructed and studied by D. V. Chertova (2009). As an application, we propose an approach to computing Cp,α(n) based on the Arestov–Deikalova duality relations.
Keywords: trigonometric polynomial, algebraic polynomial, the Nikolskii constant, the Gegenbauer weight.
Funding agency Grant number
Russian Foundation for Basic Research 19-31-90152
The reported study was funded by RFBR, project number 19-31-90152.
Document Type: Article
UDC: 517.5
Language: Russian
Citation: I. A. Martyanov, “Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight”, Chebyshevskii Sb., 21:1 (2020), 247–258
Citation in format AMSBIB
\Bibitem{Mar20}
\by I.~A.~Martyanov
\paper Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight
\jour Chebyshevskii Sb.
\yr 2020
\vol 21
\issue 1
\pages 247--258
\mathnet{http://mi.mathnet.ru/cheb871}
\crossref{https://doi.org/10.22405/2226-8383-2018-21-1-247-258}
Linking options:
  • https://www.mathnet.ru/eng/cheb871
  • https://www.mathnet.ru/eng/cheb/v21/i1/p247
  • This publication is cited in the following 6 articles:
    1. G. A. Akishev, “Neravenstvo raznykh metrik Nikolskogo dlya trigonometricheskikh polinomov v prostranstve so smeshannoi nesimmetrichnoi normoi”, Tr. IMM UrO RAN, 29, no. 4, 2023, 11–26  mathnet  crossref  elib
    2. I. A. Martyanov, “Reshenie zadachi Delsarta dlya 4-dizainov na sfere S2”, Chebyshevskii sb., 22:3 (2021), 154–165  mathnet  crossref
    3. D. V. Gorbachev, “Konstanty Nikolskogo dlya kompaktnykh odnorodnykh prostranstv”, Chebyshevskii sb., 22:4 (2021), 100–113  mathnet  crossref
    4. D. V. Gorbachev, “Tochnye neravenstva Bernshteina — Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2021), 58–110  mathnet  crossref
    5. D. V. Gorbachev, I. A. Martyanov, “Bounds of the Nikol'skii Polynomial Constants in Lp with Gegenbauer Weight”, Proc. Steklov Inst. Math. (Suppl.), 315, suppl. 1 (2021), S117–S127  mathnet  crossref  crossref  isi  elib
    6. D. V. Gorbachev, I. A. Martyanov, “Konstanty Markova–Bernshteina–Nikolskogo dlya polinomov v prostranstve Lp s vesom Gegenbauera”, Chebyshevskii sb., 21:4 (2020), 29–44  mathnet  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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