On optimal polynomial meshes

Let Pnd be the space of real algebraic polynomials of dd variables and degree at most nn, K⊂RdK⊂Rd a compact set, ‖p‖K:=supx∈K|p(x)| the usual supremum norm on KK, and card(Y)card(Y) the cardinality of a finite set YY. A family of sets Y={Yn⊂K,n∈N} is called an admissible mesh   in KK if there exists a constant c1>0c1>0 depending only on KK such that ‖p‖K≤c1‖p‖Yn,p∈Pnd,n∈N, where the cardinality of YnYn grows at most polynomially. If card(Yn)≤c2nd,n∈Ncard(Yn)≤c2nd,n∈N with some c2>0c2>0 depending only on KK 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 paper we present some general families of sets possessing admissible meshes which are optimal or near optimal in the sense that the cardinality of sets YnYn does not grow too fast. In particular, it will be shown that graph domains bounded by polynomial graphs, convex polytopes and star like sets with C2C2 boundary possess optimal admissible meshes. In addition, graph domains with piecewise analytic boundary and any convex sets in R2R2 possess almost   optimal admissible meshes in the sense that the cardinality of admissible meshes is larger than optimal only by a lognlogn factor.