Sampling scattered data with Bernstein polynomials: stochastic and deterministic error estimates

Authors:
Zongmin Wu;Xingping Sun;Limin Ma
Affiliations:
Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Science, Fudan University, Shanghai, China;Department of Mathematics, Missouri State University, Springfield, USA 65897;Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Science, Fudan University, Shanghai, China
Venue:
Advances in Computational Mathematics
Year:
2013

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Abstract

Viewing the classical Bernstein polynomials as sampling operators, we study a generalization by allowing the sampling operation to take place at scattered sites. We utilize both stochastic and deterministic approaches. On the stochastic side, we consider the sampling sites as random variables that obey some naturally derived probabilistic distributions, and obtain Chebyshev type estimates. On the deterministic side, we incorporate the theory of uniform distribution of point sets (within the framework of Weyl's criterion) and the discrepancy method. We establish convergence results and error estimates under practical assumptions on the distribution of the sampling sites.