Nikolskii constant for trigonometric polynomials with periodic Gegenbauer weight

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, 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.