On optimal polynomial meshes

Let P d n be the the space of real algebraic polynomials of d variables and degree at most n, K ⊂ R a compact set, ||p||K := supx∈K |p(x)| the usual supremum norm on K, card(Y ) the cardinality of a finite set Y . A family of sets Y = {Yn ⊂ K, n ∈ N} is called an admissible mesh in K if there exists a constant c1 > 0 depending only on K such that ||p||K ≤ c1||p||Yn , p ∈ P d n , n ∈ N, where the cardinality of Yn grows with n at most polynomially. If card(Yn) ≤ c2n, n ∈ N with some c2 > 0 depending only on K then we say that the admissible mesh is optimal. This notion of admissible meshes is related to norming sets which are widely used in the literature. In this talk we shall present some general families of sets possessing admissible meshes which are optimal or near optimal in the sense that the cardinality of Yn does not grow too fast. In particular, we shall see that graph domains bounded by polynomial graphs, convex polytopes and star like sets with C boundary possess optimal admissible meshes. In addition, graph domains with piecewise analytic boundary and arbitrary convex sets in R possess almost optimal admissible meshes in the sense that the cardinality of admissible meshes are larger than optimal only by a log n factor.