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Abstract and Applied Analysis
Volume 2014, Article ID 454375, 10 pages
http://dx.doi.org/10.1155/2014/454375
Research Article

Analysis of Approximation by Linear Operators on Variable Spaces and Applications in Learning Theory

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received 7 May 2014; Accepted 30 June 2014; Published 16 July 2014

Academic Editor: Uno Hämarik

Copyright © 2014 Bing-Zheng Li and Ding-Xuan Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. N. Bernstein, “Démonstration du téoréme de Weirerstrass, fondée sur le calcul des probabilités,” Communications of the Kharkov Mathematical Society, vol. 13, pp. 1–2, 1913. View at Google Scholar
  2. L. V. Kantorovich, “Sur certaines developments suivant les polynômes de la forme de S. Bernstein I-II,” Comptes Rendus de l'Académie des Sciences de L'URSS A, vol. 563–568, pp. 595–600, 1930. View at Google Scholar
  3. J. L. Durrmeyer, Une formule d'inversion de la transformée Laplace: applications á la théorie des moments [Thése de 3e cycle: Sciences], Faculté des Sciences, l'Université Paris, Paris, France, 1967.
  4. H. Berens and G. G. Lorentz, “Inverse theorems for Bernstein polynomials,” Indiana University Mathematics Journal, vol. 21, pp. 693–708, 1972. View at Google Scholar · View at MathSciNet
  5. H. Berens and R. A. DeVore, “Quantitative Korovkin theorems for positive linear operators on LP-spaces,” Transactions of the American Mathematical Society, vol. 245, pp. 349–361, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Z. Ditzian and V. Totik, Moduli of Smoothness, vol. 9 of Springer Series in Computational Mathematics, Springer, New York, NY, USA, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  7. L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  8. W. Orlicz, “Über konjugierte Exponentenfolgen,” Studia Mathematica, vol. 3, pp. 200–211, 1931. View at Google Scholar
  9. E. Acerbi and G. Mingione, “Regularity results for a class of functionals with non-standard growth,” Archive for Rational Mechanics and Analysis, vol. 156, no. 2, pp. 121–140, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. O. Kovácik and J. Rákosnk, “On spaces Lpx and W1, px,” Czechoslovak Mathematical Journal, vol. 41, no. 116, pp. 592–618, 1991. View at Google Scholar · View at MathSciNet
  11. D. X. Zhou, “Approximation by positive linear operators on variables Lp(x) spaces,” Journal of Applied Functional Analysis, vol. 9, no. 3-4, pp. 379–391, 2014. View at Google Scholar · View at MathSciNet
  12. D. X. Zhou and K. Jetter, “Approximation with polynomial kernels and {SVM} classifiers,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 323–344, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. E. E. Berdysheva and K. Jetter, “Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions,” Journal of Approximation Theory, vol. 162, no. 3, pp. 576–598, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. A. B. Tsybakov, “Optimal aggregation of classifiers in statistical learning,” The Annals of Statistics, vol. 32, no. 1, pp. 135–166, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. S. Smale and D. X. Zhou, “Learning theory estimates via integral operators and their approximations,” Constructive Approximation, vol. 26, no. 2, pp. 153–172, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. S. Smale and D. X. Zhou, “Shannon sampling and function reconstruction from point values,” The American Mathematical Society: Bulletin, vol. 41, no. 3, pp. 279–305, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. T. Hu, J. Fan, Q. Wu, and D. X. Zhou, “Regularization schemes for minimum error entropy principle,” Analysis and Applications, 2014. View at Publisher · View at Google Scholar
  18. I. Steinwart and A. Christmann, “Estimating conditional quantiles with the help of the pinball loss,” Bernoulli, vol. 17, no. 1, pp. 211–225, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. D. H. Xiang, “A new comparison theorem on conditional quantiles,” Applied Mathematics Letters, vol. 25, no. 1, pp. 58–62, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. J. Lei, R. Jia, and E. W. Cheney, “Approximation from shift-invariant spaces by integral operators,” SIAM Journal on Mathematical Analysis, vol. 28, no. 2, pp. 481–498, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. K. Jetter and D. X. Zhou, “Order of linear approximation from shift-invariant spaces,” Constructive Approximation, vol. 11, no. 4, pp. 423–438, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. Derriennic, “On multivariate approximation by Bernstein-type polynomials,” Journal of Approximation Theory, vol. 45, no. 2, pp. 155–166, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. H. Berens and Y. Xu, “On Bernstein-Durrmeyer polynomials with Jacobi weights,” in Approximation Theory and Functional Analysis, C. K. Chui, Ed., pp. 25–46, Academic Press, Boston, Mass, USA, 1991. View at Google Scholar · View at MathSciNet
  24. B.-Z. Li, “Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels,” Journal of Approximation Theory, vol. 173, pp. 33–55, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. E. E. Berdysheva, “Uniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure,” Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 324–336, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. E. E. Berdysheva, “Bernstein–Durrmeyer operators with respect to arbitrary measure, II: pointwise convergence,” Journal of Mathematical Analysis and Applications, vol. 418, no. 2, pp. 734–752, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  27. D. X. Zhou, “Converse theorems for multidimensional Kantorovich operators,” Analysis Mathematica, vol. 19, no. 1, pp. 85–100, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. C. de Boor, R. A. DeVore, and A. Ron, “Approximation from shift-invariant subspaces of L2(Rd),” Transactions of the American Mathematical Society, vol. 341, no. 2, pp. 787–806, 1994. View at Publisher · View at Google Scholar · View at MathSciNet