On the existence of optimal meshes in every convex domain on the plane

Kroó, András (2019) On the existence of optimal meshes in every convex domain on the plane. JOURNAL OF APPROXIMATION THEORY, 238. pp. 26-37. ISSN 0021-9045

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Abstract

In this paper we study the so called optimal polynomial meshes for domains in K⊂Rd,d≥2. These meshes are discrete point sets Yn of cardinality cnd which have the property that (norm of matrix)p(norm of matrix)K≤A(norm of matrix)p(norm of matrix)Yn for every polynomial p of degree at most n with a constant A≫1 independent of n. It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d=2.

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