Extremal Nikolskii–Bernstein- and Turán-type problems for Dunkl transform

We study the interrelation between the extremal Turán-type problems and Nikolskii – Bernstein problems for nonnegative functions on $\mathbb{R}^{d}$ with the Dunkl weight. The Turán problem is to find the supremum of a given moment of a positive definite (with respect to the Dunkl transform) function with a support in the Euclidean ball and a fixed value at zero. In the sharp $L^{1}$-Nikolskii–Bernstein inequality, the supremum norm of the Dankl Laplacian of an entire function of exponential spherical type with the unit $L^{1}$-norm is estimated. Extremal Feuér and Beaumann problems is also mentioned. The Dunkl transform covers the case of the classical Fourier transform in the case of unit weight.
Nikolskii–Bernstein inequalities are classical in approximation theory, and the Turán-type problems have applications in metric geometry. Nevertheless, we prove that they have the same answer, which is given explicitly. The easy proof is relied on our old results from the theory of solving extremal problems to the Dunkl transform.