Polynomial meshes on some classes of planar compact domains ∗

We construct low cardinality admissible meshes for polynomials on three classes of planar compact domains: cartesian graph domains, polar graph domains, and domains with piecewise C 2 boundary, that satisfy a Markov polynomial inequality. 1 Planar cartesian and polar graph domains Let K ⊂ R d be a polynomial determining compact domain (i.e., a polynomial vanishing there vanishes everywhere). We term family of (polynomial) norming sets for K any sequence of compact subsets N n ⊆ K, n ∈ N, such that the following polynomial inequality holds p K ≤ C p Nn , ∀p ∈ P d n , (1) where C > 0 is a constant and P d n denotes the space of real d-variate poly-nomials of total degree at most n. Such a property is invariant under affine transformations of K. Here and below, f X denotes the sup-norm of a function bounded on the set X. When the norming set N n is discrete and finite, and has cardinality O(n s) for some s ≥ d, the family is called an admissible mesh. An admissible mesh with s = d is called optimal; see [6, 9]. If in (1) we have a sequence C n instead * Supported the ex-60% funds of the University of Padova, and by the INdAM GNCS.