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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
Abstract:
We study the Nikolskii constant (or the Jackson–Nikolskii constant) for complex trigonometric polynomials in the space Lpα(T) for p⩾1 with the periodic Gegenbauer weight |sinx|2α+1:
Cp,α(n)=supT∈Tn∖{0}‖T‖∞‖T‖p,
where ‖⋅‖p=‖⋅‖Lpα(T). D. Jackson (1933) proved that Cp,−1/2(n)⩽cpn1/p for all n⩾1. 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 (1−x2)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.
Citation:
I. A. Martyanov, “Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight”, Chebyshevskii Sb., 21:1 (2020), 247–258
Linking options:
https://www.mathnet.ru/eng/cheb871 https://www.mathnet.ru/eng/cheb/v21/i1/p247
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