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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beck, J., Chen, W.W.L.: Irregulariries of distributions. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press (1989)
Casella, G., Berger, R.: Statistical Reference, 2nd edn. Duxbury Advanced Series (2002)
Chazelle, B.: The Discrepancy Method, Randomness and Complexity. Cambridge University Press, Cambridge (2000)
Cheney, E.W.: Introduction to Approximation Theory, 2nd edn. Chelsea, New York (1982)
Chung, K.L.: Elementary Probability Theory with Stochastic Processes. Springer-Verlag, New York (1978)
Cucker, F., Smale, S.: On the mathematical foundations of learning. Bull. Amer. Math. Soc. (N. S.) 39, 1–49 (2001)
Cucker, F., Zhou, D.X.: Learning theory: an approximation theory viewpoint, with a foreword by Stephen Smale. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2007)
David, H.A., Nagaraja, H.N.: Order statistics, 3rd edn. Wiley Series in Probability and Statistics. John Wiley & Sons, Hoboken (2003)
DeVore, R.A., Lorentz, G.G.: Constructive approximation. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 303. Springer-Verlag, Berlin (1993)
Drmota, M., Tichy, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics, vol. 1651. Springer-Verlag, Berlin (1997)
Erdős, P., Turán, P.: On a problem in the theory of uniform distribution I and II. Indag. Math. 10, 370–378 and 406–413 (1948)
Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. John Wiley & Sons (1974)
Lorentz, G.G.: Approximation of Functions, 2nd edn. AMS Chelsea Publishing, American Mathematical Society, Providence (1986)
Lorentz, G.G.: Bernstein Polynomials. Mathematical Expositions, no. 8. University of Toronto Press, Toronto (1953)
Montgomery, H.L.: Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, no. 84. American Mathematical Society, Providence (1990)
Niederreiter, H.: Random Number Generation and Quasi Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadephia (1992)
Sikkema, P.C.: Der wert einiger konstanten in der theorie der approximation mit Bernstein–Polynomen. Numer. Math. 3, 107–116 (German, 1961)
Sikkema, P.C.: Über den grad der approximation mit Bernstein–Polynomen. Numer. Math. 1, 221–239 (German, 1959)
Weyl, H.: Über die Gleichverteilung von Zahlen modulo 1. Math. Ann. 77, 313–352 (1916)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Tim Goodman.
Research partially supported by grant SGST 09DZ2272900 from Fudan University, Shanghai, China.
Rights and permissions
About this article
Cite this article
Wu, Z., Sun, X. & Ma, L. Sampling scattered data with Bernstein polynomials: stochastic and deterministic error estimates. Adv Comput Math 38, 187–205 (2013). https://doi.org/10.1007/s10444-011-9233-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-011-9233-0
Keywords
- Bernstein polynomials
- Chebyshev inequality
- Discrepancy
- Erdős and Turán inequality
- Sampling
- Scattered data
- Uniform distribution
- Weyl’s criterion