Probabilistic spherical Marcinkiewicz-Zygmund inequalities

Authors:
Albrecht Böttcher;Stefan Kunis;Daniel Potts
Affiliations:
Chemnitz University of Technology, Department of Mathematics, 09107 Chemnitz, Germany;Chemnitz University of Technology, Department of Mathematics, 09107 Chemnitz, Germany;Chemnitz University of Technology, Department of Mathematics, 09107 Chemnitz, Germany
Venue:
Journal of Approximation Theory
Year:
2009

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Abstract

Recently, norm equivalences between spherical polynomials and their sample values at scattered sites have been proved. These so-called Marcinkiewicz-Zygmund inequalities involve a parameter that characterizes the density of the sampling set and they are applicable to all polynomials whose degree does not exceed an upper bound that is determined by the density parameter. We show that if one is satisfied by norm equivalences that hold with prescribed probability only, then the upper bound for the degree of the admissible polynomials can be enlarged significantly and that then, moreover, there exist fixed sampling sets which work for polynomials of all degrees.