Skip to main content
SpringerLink
Log in

Approximation Results for Hadamard-Type Exponential Sampling Kantorovich Series

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

The present paper deals with construction of a new family of exponential sampling Kantorovich operators based on a suitable fractional-type integral operators. We study convergence properties of newly constructed operators and give a quantitative form of the rate of convergence thanks to logarithmic modulus of continuity. To obtain an asymptotic formula in the sense of Voronovskaja, we consider locally regular functions. The rest of the paper devoted to approximations of newly constructed operators in logarithmic weighted space of functions. By utilizing a suitable weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem via remainder of Mellin–Taylor’s formula. Furthermore, some examples of kernels which satisfy certain assumptions are presented and the results are examined by illustrative numerical tables and graphical representations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

All data generated or analyzed during this study are included in this published article.

References

  1. Acar, T., Alagöz, O., Aral, A., Costarelli, D., Turgay, M., Vinti, G.: Convergence of generalized sampling series in weighted spaces. Demonstr. Math. 55, 153–162 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Acar, T., Alagöz, O., Aral, A., Costarelli, D., Turgay, M., Vinti, G.: Approximation by sampling Kantorovich series in weighted spaces of functions. Turk. J. Math. 46(7), 2663–2676 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  3. Acar, T., Costarelli, D., Vinti, G.: Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series. Banach J. Math. Anal. 14, 1481–1508 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Acar, T., Draganov, B.R.: A stronge converse inequality for generalized sampling operators. Ann. Funct. Anal. 13(36), 1–16 (2022)

    MATH  Google Scholar 

  5. Acar, T., Eke, A., Kursun, S.: Bivariate generalized Kantorovich-type exponential sampling series. (Submitted) (2023)

  6. Acar, T., Kursun, S.: Pointwise convergence of generalized Kantorovich exponential sampling series. Dolomites Res. Notes Approx. 16, 1–10 (2023)

    MathSciNet  MATH  Google Scholar 

  7. Acar, T., Kursun, S.: Approximation properties of exponential sampling series in logarithmic weighted spaces, (Submitted) (2023)

  8. Acar, T., Kursun, S.: Convergence of bivariate exponential sampling series in logarithmic weighted space of functions, (Submitted) (2023)

  9. Acar, T., Kursun, S., Turgay, M.: Multidimensional Kantorovich modifications of exponential sampling series. Quaest. Math. 46(1), 57–72 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  10. Acar, T., Turgay, M.: Approximation by bivariate generalized sampling series in weighted spaces of functions. Dolomites Res. Notes Approx. 16, 11–22 (2023)

    MathSciNet  Google Scholar 

  11. Alagöz, O., Turgay, M., Acar, T., Parlak, M.: Approximation by sampling Durrmeyer operators in weighted space of functions. Numer. Funct. Anal. Optim. 43(10), 1223–1239 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  12. Angeloni, L., Cetin, N., Costarelli, D., Sambucini, A.R., Vinti, G.: Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces. Constr. Math. Anal. 4(2), 229–241 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Angeloni, L., Costarelli, D., Vinti, G.: Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing. Ann. Acad. Sci. Fenn. Math. 45, 751–770 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  14. Aral, A., Acar, T., Kursun, S.: Generalized Kantorovich forms of exponential sampling series, Anal. Math. Phys., 12(2), 1–19 (2022)

  15. Bajpeyi, S., Angamuthu, S.K.: On approximation by Kantorovich exponential sampling operators. Numer. Funct. Anal. Optim. 42(9), 1096–1113 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bajpeyi, S., Angamuthu, S.K., Mantellini, I.: Approximation by Durrmeyer type exponential sampling operators. Numer. Funct. Anal. Optim. 43(1), 16–34 (2022)

    Article  MathSciNet  Google Scholar 

  17. Balsamo, S., Mantellini, I.: On linear combinations of general exponential sampling series, Results Math., 74(4), (2019)

  18. Bardaro, C., Bevignani, G., Mantellini, I., Seracini, M.: Bivariate generelized exponential sampling series and applications to seismic waves. Const. Math. Anal. 2(4), 153–167 (2019)

    MATH  Google Scholar 

  19. Bardaro, C., Butzer, P.L., Mantellini, I.: The foundations of fractional calculus in the Mellin transform setting with applications. J. Fourier Anal. Appl. 21, 961–1017 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bardaro, C., Faina, L., Mantellini, I.: Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nach., 289, 1702–1720 (2016)

  21. Bardaro, C., Faina, L., Mantellini, I.: A generalization of the exponential sampling series and its approximation properties. Math. Slovaca 67(6), 1481–1496 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bardaro, C., Mantellini, I.: A note on the Voronovskaja theorem for Mellin-Fejer convolution operators, Appl. Math. Lett., 24, 2064–2067 (2011)

  23. Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling type operators. Zeitschrift für Anal. und ihre Anwendung 32, 279–298 (2013)

    Article  MATH  Google Scholar 

  24. Bardaro, C., Mantellini, I.: On Mellin convolution operators: a direct approach to the asymptotic formulae. Integral Transforms Spec. Funct. 25, 182–195 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bardaro, C., Mantellini, I.: On a Durrmeyer-type modification of the exponential sampling series. Rend. Circ. Mat. Palermo 70(3), 1289–1304 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bardaro, C., Mantellini, I., Schmeisser, G.: Exponential sampling series: convergence in Mellin- Lebesgue spaces. Results Math., 74, 1–20 (2019)

  27. Bardaro, C., Mantellini, I., Stens, R., Vautz, J., Vinti, G.: Generalized sampling approximation for multivariate discontinuous signals and application to image processing, New Perspectives on Approximation and Sampling Theory-Festschrift in honor of Paul Butzer’s 85th birthday, Birkhauser, 87-114 (2014)

  28. Bertero, M., Pike, E.R.: Exponential sampling method for Laplace and other dilationally invariant transforms I. Singularsystem analysis. II. Examples in photon correction spectroscopy and Frauenhofer diffraction, Inverse Probl., 7, 1-20; 21-41 (1991)

  29. Butzer, P.L., Engels, W., Ries, S., Stens, R.L.: The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines. SIAM J. Appl. Math. 46(2), 299–323 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Butzer, P.L., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3(4), 325–375 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Butzer, P.L., Jansche, S.: The exponential sampling theorem of signal analysis, Atti Sem. Mat. Fis. Univ. Modena Suppl., 46, (special issue dedicated to Professor Calogero Vinti) 99-122 (1998)

  32. Butzer, P.L., Jansche, S.: The finite Mellin Transform, Mellin-Fourier series and the Mellin-Poisson summation formula. Rend. Circ. Mat. Palermo 52, 55–81 (1998)

    MathSciNet  MATH  Google Scholar 

  33. Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integral. J. Math. Anal. Appl. 269, 1–27 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Butzer, P.L., Kilbas, A.A., Trujillo, J.J.: Stirling functions of the second kind in the setting of difference and fractional calculus. Numer. Funct. Anal. Optimiz. 4(7–8), 673–711 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Butzer, P.L., Splettstosser, W.: A sampling theorem for duration-limited functions with error estimates. Inf. Control. 34(1), 55–65 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Butzer, P.L., Stens, R.L.: Linear prediction by samples from the past. In: Advanced topics in Shannon sampling and interpolation theory, New York, Springer, 157-183 (1993)

  37. Butzer, P.L., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Dtsch. Math. Ver. 90(1), 1–70 (1998)

    MathSciNet  MATH  Google Scholar 

  38. Butzer, P.L., Westphal, U.: An introduction to fractional calculus. In: Hifler, H. (eds) Applications of Fractional Calculus in Physics, pp. 1–85. Wordl Scientific Publ, Singapore (2000)

  39. Casasent, D. (ed.).: Optical data processing, pp. 241–282. Springer, Berlin (1978)

  40. Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. 62(1), 265–280 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. Costarelli, D., Vinti, G.: Inverse results of approximation and the saturation order for the sampling Kantorovich series. J. Approx. Theory 24, 64–82 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Draganov, B.R.: A fast converging sampling operator. Constr. Math. Anal. 5(4), 190–201 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Gori, F.: Sampling in optics. In: Marks, R.J., II. (ed.) Advances Topics in Shannon Sampling and Interpolation Theory, pp. 37–83. Springer, New York (1993)

  44. Hadamard, J.: Essai sur l’etude des fonctions donnees par leur developpement de Taylor. J. Math. Pures et Appl. 4(8), 101–186 (1892)

    MATH  Google Scholar 

  45. Higgins, J.R.: Sampling theory in Fourier and signal analysis. Foundations. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  46. Kadak, U.: Fractional type multivariate sampling operators. RACSAM 115, 153 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kadak, U.: Fractional sampling operators of multivariate fuzzy functions and applications to image processing. Appl. Soft Comput. 132, 109901 (2023)

    Article  Google Scholar 

  48. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  49. Kotel’nikov, V.A.: On the carrying capacity of “ether” and wire in electrocommunications, Material for the First All-Union Conference on the Questions of Communications, Moscow, (1933)

  50. Kumar, S.A., Bajpeyi, S.: Direct and inverse results for Kantorovich type exponential sampling series, Results Math., 75(3), 1–17 (2020)

  51. Kursun, S., Turgay, M., Alagoz, O., Acar, T.: Approximation properties of multivariate exponential sampling series. Carpathian Math. Publ. 13(3), 666–675 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  52. Kursun, S., Aral, A., Acar, T.: Riemann-Liouville fractional integral type exponential sampling Kantorovich series, (Submitted) (2023)

  53. Mahmudov, N., Kara, M.: Approximation properties of the Riemann-Liouville fractional integral type Szász-Mirakyan-Kantorovich operators. J. Math. Inequalities 16(4), 1285–1308 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  54. Mamedov, R.G.: The Mellin transform and approximation theory (in Russian). “Elm”, Baku (1991)

  55. Ostrowsky, N., Sornette, D., Parker, P., Pike, E.R.: Exponential sampling method for light scattering polydispersity analysis. Opt. Acta 28, 1059–1070 (1994)

    Article  Google Scholar 

  56. Shannon, C.E.: Communications in the presence of noise. Proc. IRE. 37, 10–21 (1949)

    Article  MathSciNet  Google Scholar 

  57. Turgay, M., Aral, A., Acar, T.: Approximation properties of the Riemann-Liouville fractional integral type generalized sampling Kantorovich series, (Submitted) (2023)

  58. Whittaker, E.T.: On the functions, which are represented by expansions of the interpolation theory. Proc. R. Soc. Edinb. 35, 181–194 (1915)

    Article  MATH  Google Scholar 

  59. Zayed, A.I.: Advances in Shannon’s sampling theory. CRC Press, Boca Raton (1993)

Download references

Author information

Authors and Affiliations

  1. Faculty of Science, Department of Mathematics, Selcuk University, Selcuklu, 42003, Konya, Turkey

    Sadettin Kursun & Tuncer Acar

  2. Faculty of Science and Arts, Department of Mathematics, Kirikkale University, Yahsihan, Kirikkale, 71450, Turkey

    Ali Aral

Authors
  1. Sadettin Kursun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Aral
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tuncer Acar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

SK wrote the main manuscript text and prepared the figures/tables. AA and TA analyzed the theorems and proofs in the paper. All authors reviewed the manuscript.

Corresponding author

Correspondence to Sadettin Kursun.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kursun, S., Aral, A. & Acar, T. Approximation Results for Hadamard-Type Exponential Sampling Kantorovich Series. Mediterr. J. Math. 20, 263 (2023). https://doi.org/10.1007/s00009-023-02459-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02459-2

Keywords

Mathematics Subject Classification

Search

Navigation