Kroó András (Rényi): Homogeneous polynomials on convex bodies: Christoffel function, L^p Markov type inequalities, Marcinkiewicz-Zygmund type discretization

In this talk we shall consider several extremal problems related to multivariate homogeneous polynomials. The space of real homogeneous polynomials of d>=2 variables and degree n appears naturally as a fundamental tool in problems related to neural networks and approximation by ridge functions. Our main goal is to study the asymptotic behaviour of Christoffel functions and L^p Markov type estimates for homogeneous polynomials on convex bodies. In order to tackle these questions homogeneous ”needle” polynomials attaining value 1 at a given point and rapidly decreasing moving away from this point are introduced. These homogeneous needle polynomials play a crucial role in the study of the asymptotics of the Christoffel functions and L^p Markov type estimates, which in turn lead to Marcinkiewicz-Zygmund type discretization results.