aAlfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary;bBudapest University of Technology and Economics, Department of Analysis, Budapest, Hungary
Abstract:
Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the Lq-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in Lq-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.