Nikolskii–Bernstein constants in $L^{p}$ on the sphere with Dunkl weight

We study the sharp Nikol'skii–Bernstein constants for spherical polynomials in the space $L^{p}(\mathbb{S}^{d})$ with the Dunkl weight. An interrelationship with one-dimensional constants for algebraic polynomials in the space $L^{p}[-1,1]$ with the Gegenbauer weight is established.