Suboptimal Polynomial Meshes on Planar Lipschitz Domains

Abstract

We construct norming meshes with cardinality O(n s ), s = 3, for polynomials of total degree at most n, on the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensional C 2 starlike domains, and that arising from a general construction on Markov compact sets due to Calvi and Levenberg (s = 4).2000 AMS subject classification: 41A10, 41A63, 65D05.

Pluripotential Numerics

“…Note that an algorithmic construction of an admissible mesh is available in the literature for several classes of sets [36,35,33], since the study of (weakly) admissible meshes is attracting certain interest during last years. • We assume n = 2 to hold the algorithm complexity growth.…”

Caratheodory-Tchakaloff Least Squares

“…For example, r = 2 on convex compact sets with nonempty interior. Construction of optimal admissible meshes has been carried out for compact sets with various geometric structures, but still the cardinality can be very large already for d = 2 or d = 3, for example on polygons/polyhedra with many vertices, or on star-shaped domains with smooth boundary; cf., e.g., [14,23].…”

“…In order to make an example, in Figure 2 we consider the (high cardinality) optimal polynomial mesh constructed on a smooth convex set (C 2 boundary), by the rolling ball theorem as described in [23] (the set boundary corresponds to a level curve of the quartic x 4 + 4y 4 ). The CATCH points have been computed by NNLS as in (5), and the LS and CATCHLS uniform operator norms have been numerically estimated on a fine control mesh via the corresponding discrete reproducing kernels, as discussed in [6, §2.1].…”

Polynomial fitting and interpolation on circular sections

“…When CðA n Þ is bounded we speak of an ''Admissible Mesh'' (AM), sometimes also called ''polynomial mesh'' in the literature (cf., e.g. [23,33,29,30]). …”