Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight

Abstract

We study the Nikolskii constant (or the Jackson-Nikolskii constant) for
complex trigonometric polynomials in the space $L_{\alpha}^{p}(\mathbb{T})$ for
$p\ge 1$ with the periodic Gegenbauer weight $|\!\sin x|^{2\alpha+1}$:
$$
\mathcal{C}_{p,\alpha}(n)=\sup_{T\in \mathcal{T}_{n}\setminus \{0\}}
\frac{\|T\|_{\infty}}{\|T\|_{p}},
$$
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, A. Horvath.

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.