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Gräf, M.* ; Schmid, D.

Probabilistic Marcinkiewicz-Zygmund inequalities on the rotation group.

Math. Geosci. 42, 731-746 (2010)
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Approximation problems on the rotation group SO(3) naturally arise in various fields, like crystallography, chemistry, and biology. For example, in crystallographic texture analysis one is confronted with the problem of evaluating so-called orientation density functions (ODFs). In many situations one only has a finite number of measurements at scattered sampling nodes. In order to reconstruct ODFs over all rotations, so-called Marcinkiewicz-Zygmund inequalities on the rotation group are an important tool. These inequalities provide norm equivalences between polynomials on SO(3) and their sample values. Recently shown equivalences depend on a density parameter of the sampling set and the proven inequalities hold true for polynomials on SO(3) whose degree does not exceed an upper bound which is determined by this density parameter. In this paper, we show that we can enlarge this upper bound for the polynomial degree significantly if we are satisfied by such norm equivalences that hold with a given probability only. Moreover, we show that there are fixed sampling sets for which we get probabilistic Marcinkiewicz-Zygmund inequalities that hold for polynomials on SO(3) of all degrees.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter Marcinkiewicz-Zygmund inequalities; Rotation group; Scattered data; Wigner D-functions; Random polynomials
ISSN (print) / ISBN 0882-8121
e-ISSN 1573-8868
Quellenangaben Band: 42, Heft: 7, Seiten: 731-746 Artikelnummer: , Supplement: ,
Verlag Springer
Begutachtungsstatus Peer reviewed