Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Approximation by Generalized Kantorovich Sampling Type Series

  • Received : 2017.11.10
  • Accepted : 2019.01.28
  • Published : 2019.09.23
Download PDF
⟨ Previous Next ⟩

Abstract

In the present article, we analyse the behaviour of a new family of Kantorovich type sampling operators $(K^{\varphi}_wf)_{w>0}$. First, we give a Voronovskaya type theorem for these Kantorovich generalized sampling series and a corresponding quantitative version in terms of the first order of modulus of continuity. Further, we study the order of approximation in $C({\mathbb{R}})$, the set of all uniformly continuous and bounded functions on ${\mathbb{R}}$ for the family $(K^{\varphi}_wf)_{w>0}$. Finally, we give some examples of kernels such as B-spline kernels and the Blackman-Harris kernel to which the theory can be applied.

Keywords

References

  1. T. Acar, A. Ara and S. A. Mohiuddine, On Kantorovich modification of (p; q)-Bernstein operators, Iran. J. Sci. Technol. Trans. A Sci., 42(2018), 1459-1464. https://doi.org/10.1007/s40995-017-0154-8
  2. A. M. Acu, Stancu-Schurer-Kantorovich operators based on q-integers, Appl. Math. Comput., 259(2015), 896-907. https://doi.org/10.1016/j.amc.2015.03.032
  3. P. N. Agrawal and G. Prasad, Degree of approximation to integrable functions by Kantorovich polynomials, Boll. Un. Mat. Ital. A, 6(1985), 323-326.
  4. F. Altomare, M. M. Cappelletti and V. Leonessa, On a Generalization of Szasz-Mirakjan-Kantorovich operators, Results Math., 63(2013), 837-863. https://doi.org/10.1007/s00025-012-0236-z
  5. F. Altomare and V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel measures, Mediterr. J. Math., 3(2006), 363-382. https://doi.org/10.1007/s00009-006-0084-8
  6. G. A. Anastassiou and S. G. Gal, Approximation theory. Moduli of continuity and global smoothness preservation, Birkhauser, Boston, 2000.
  7. C. Bardaro and I. Mantellini, A Voronovskaya type theorem for a general class of discrete operators, Rocky Mountain J. Math., 39(2009), 1411-1442. https://doi.org/10.1216/RMJ-2009-39-5-1411
  8. C. Bardaro and I. Mantellini, Voronovskaya formulae for Kantorovich type generalized sampling series, Int. J. Pure. Appl. Math., 62(3)(2010), 247-262.
  9. C. Bardaro and I. Mantellini, A quantitative Voronovskaja formula for generalized sampling operators, East. J. Approx., 15(4)(2009), 459-471.
  10. C. Bardaro and I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim., 33(4)(2012), 374-396. https://doi.org/10.1080/01630563.2011.652270
  11. C. Bardaro and G. Vinti, An abstract approach to sampling type operators inspired by the work of P. L. Butzer, I. Linear operators, Sampl. Theory Signal Image Process, 2(2003), 271-295. https://doi.org/10.1007/BF03549399
  12. C. Bardaro, G. Vinti, P. L. Butzer and R. L. Stens, Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process, 6(2007), 29-52. https://doi.org/10.1007/BF03549462
  13. P. L. Butzer and R. L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev., 34(1992), 40-53. https://doi.org/10.1137/1034002
  14. P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, 157-183, Springer Texts Electrical Engrg., Springer, New York, 1993.
  15. M. M. Cappelletti and V. Leonessa, Approximation of some Feller semigroups associated with a modification of Szasz-Mirakjan-Kantorovich operators, Acta Math. Hungar., 139(3)(2013), 255-275. https://doi.org/10.1007/s10474-012-0267-7
  16. D. Costarelli and G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. U. M. I., 4(2011), 445-468.
  17. D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34(2013), 819-844. https://doi.org/10.1080/01630563.2013.767833
  18. D. Costarelli and G. Vinti, Order of approximation for sampling Kantorovich operators, J. Integral Equations Appl., 26(3)(2014), 345-368. https://doi.org/10.1216/JIE-2014-26-3-345
  19. D. Costarelli and G. Vinti, Rate of approximation for multivariate sampling Kantorovich operators on some function spaces, J. Integral Equations Appl., 26 (4)(2014), 455-481. https://doi.org/10.1216/JIE-2014-26-4-455
  20. D. Costarelli and G. Vinti, Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some function spaces, Numer. Funct. Anal. Optim., 36(8)(2015), 964-990. https://doi.org/10.1080/01630563.2015.1040888
  21. H. H. Gonska, P. Pitul and I. Rasa, On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators, Proc. of the Int.Conf. on Numerical Analysis and Approximation Theory, 55-80, Cluj-Napoca, 2006.
  22. V. Gupta and R. P. Agarwal, Convergence estimates in approximation theory, Springer, Cham, 2014.
  23. A. Kivinukk and G. Tamberg, Shannon sampling series with averaged kernels, Proc. SAMPTA 2007, 76-85, Thessaloniki, Greece, June 1-5, 2007.
  24. A. Kivinukk and G. Tamberg, Interpolating generalized Shannon sampling operators, their norms and approximation properties, Sampl. Theory Signal Image Process, 8(2009), 77-95. https://doi.org/10.1007/BF03549509
  25. I. Mantellini, Generalized sampling operators in modular spaces, Comment. Math., 38(1998), 77-92.
  26. S. Ries and R . L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions (Ed-s: Bl. Sendov, P. Petrushev, R. Maalev, S. Tashev), 746-756, Publ. House Bulgarian Academy of Sciences, Sofia, 1984.
  27. H. J. Schmeisser and W. Sickel, Sampling theory and function spaces, Applied Mathematics Reviews 1, 205-284, Word Scientific, Singapore, 2000.
  28. I. J. Schoenberg, Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 12, SIAM, Philadelphia, 1973.
  29. G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theory., 161(2009), 511-528 https://doi.org/10.1016/j.jat.2008.11.011