Marcinkiewicz–Zygmund Inequalities and Polynomial Approximation from Scattered Data on SO(3)

Abstract

We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L ^p -operator norm for all 1 ≤ p ≤ ∞. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattered data L ^p -Marcinkiewicz–Zygmund inequalities for these functions for all 1 ≤ p ≤ ∞. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).

Keywords:

• Marcinkiewicz–Zygmund inequalities
• Rotation group
• Scattered data approximation
• Wigner-D functions

AMS Subject Classification:

• 41A10
• 41A17
• 42C10
• 43A75
• 65T50

ACKNOWLEDGMENTS

It is a pleasure to thank W. Erb, F. Filbir, S. Kunis, H. N. Mhaskar, and D. Potts for many fruitful and enlightening discussions on this topic. The project is partially funded by Deutsche Forschungsgemeinschaft Grant FI 883/3-1 and PO 711/9-1.