𝐿^{𝑝}-Bernstein inequalities on 𝐶²-domains and applications to discretization

Abstract

We prove a new Bernstein type inequality in L p L^p spaces associated with the normal and the tangential derivatives on the boundary of a general compact C 2 C^2 -domain. We give two applications: Marcinkiewicz type inequality for discretization of L p L^p norms and positive cubature formulas. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.