Some results for weighted Bernstein–Nikol'skii constants

In this short review paper, we present the latest results on the sharp Bernstein–Nikol'skii constants for polynomials on the multidimensional unit sphere in the space $L^{p}$ with the Dunkl weight and the Beltrami–Dunkl operator and related weight constants for polynomials and entire functions of exponential type and Gegenbauer and Bessel operators. For a long time, the classical trend in the theory of Bernstein–Nikol'skii inequalities was the establishment of an growth rate of constants depending on the growth of the degree of polynomials. The modern development of the theory is the proof of asymptotic equalities of Levin–Lubinsky-type, which refine the asymptotic equivalences. The main results here were obtained by F. Dai, M. Ganzburg, E. Levin, D. Lubinsky, S. Tikhonov, the authors of the work.
We start from the previously proven relations between the multidimensional Bernstein–Nikol'skii constant and the one-dimensional constant for algebraic polynomials with the Gegenbauer weight and the Gegenbauer differential operator. In the case of the reflection group of an octahedron and a multiplicity function $\kappa$ such that $\min \kappa=0$, these constants are equal. As a corollary, for $p\ge 1$ this allows one to write down the Levin–Lubinsky asymptotic equalities of the Bernstein–Nikol'skii constants with an integer power of the Beltrami–Dunkl operator. The case $\min \kappa>0$ is considered for the case of Nikol'skii constants and the circle. For the subspace of even polynomials with even harmonics, a connection is established with the sharp Nikol'skii constant for polynomials on compact homogeneous spaces of rank 1. This made it possible to write the Levin–Lubinsky equality for pointwise constants for all $p>0$ and general constants for $p\ge 1$, which agrees with the known asymptotic inequality.
The limit constants in the Levin–Lubinsky asymptotic equalities are expressed in terms of the Bernstein–Nikolskii constants for entire functions of exponential type on Euclidean space, half-axis with the power weight and Laplace, Laplace–Dunkl, Bessel operators. Further refinement of the values of the constants is connected with their estimation at large dimension of space or the weight exponent. In this paper, we present a scheme for obtaining such estimates for the case of the space $L^{1}$. This case is also interesting because it is related to the Remez extremal $L^{1}$-norm concentration problem.