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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)=sup
where \|{ \cdot }\|_{p}=\|{ \cdot }\|_{L_{\alpha}^{p}(\mathbb{T})}. D. Jackson (1933) proved that \mathcal{C}_{p,-1/2}(n)\le c_{p}n^{1/p} for all n\ge 1. The problem of finding \mathcal{C}_{p,-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-x^{2})^{\alpha}, 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 L^{p}_{\alpha}(\mathbb{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 \mathcal{C}_{p,\alpha}(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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