Skip to main content
SpringerLink
Log in

Bivariate generalized Kantorovich-type exponential sampling series

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we introduce a family generalized Kantorovich-type exponential sampling operators of bivariate functions by using the bivariate Mellin-Gauss-Weierstrass operator. Approximation behaviour of the series is established at continuity points of log-uniformly continuous functions. A rate of convergence of the family of operators is presented by means of logarithmic modulus of continuity and a Voronovskaja-type theorem is proved in order to determine rate of pointwise convergence. Convergence of the family of operators is also investigated for functions belonging to weighted space. Furthermore, some examples of the kernels which support our results are given.

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

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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)

    MathSciNet  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)

    MathSciNet  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)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  5. Acar, T., Kursun, S.: On some results of generalized Kantorovich forms of exponential sampling series. Dolomites Res. Notes Approx. 16, 1–10 (2023)

    MathSciNet  Google Scholar 

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

  7. Acar, T., Kursun, S., Turgay, M.: Multidimensional Kantorovich modifications of exponential sampling series. Quaest. Math. 46, 57–72 (2022). https://doi.org/10.2989/16073606.2021.1992033

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. 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)

    MathSciNet  Google Scholar 

  10. Angamuthu, S.K., Bajpeyi, S.: Inverse approximation and GBS of bivariate Kantorovich type sampling series. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 114(2), 82 (2020)

    MathSciNet  Google Scholar 

  11. Angamuthu, S.K., Bajpeyi, S.: Direct and inverse result for Kantorovich type exponential sampling series. Results Math. 75(3), 119 (2020)

    MathSciNet  Google Scholar 

  12. Angamuthu, S.K., Ponnanian, D.: Approximation by generalized bivariate Kantorovich sampling type series. J. Anal. 27, 429–449 (2019)

    MathSciNet  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)

    MathSciNet  Google Scholar 

  14. 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  Google Scholar 

  15. Angeloni, L., Vinti, G.: A characterization of absolute continuity by means of Mellin integral operators. Z. Anal. Anwend. 34, 343–356 (2015)

    MathSciNet  Google Scholar 

  16. Angeloni, L., Vinti, G.: Convergence in variation and a characterization of the absolute continuity. Integral Transforms Spec. Funct. 26, 829–844 (2015)

    MathSciNet  Google Scholar 

  17. Aral, A., Acar, T., Kursun, S.: Generalized Kantorovich forms of exponential sampling series. Anal. Math. Phys. 12(2), 50 (2022)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  19. Balsamo, S., Mantellini, I.: On linear combinations of general exponential sampling series. Results Math. 74, 180 (2019)

    MathSciNet  Google Scholar 

  20. 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)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  23. Bardaro, C., Mantellini, I.: Voronovskaya-type estimate for Mellin convolution operators. Results Math. 50(1–2), 1–16 (2007)

    MathSciNet  Google Scholar 

  24. Bardaro, C., Mantellini, I.: Asymptotic formulae for bivariate Mellin convolution operators. Anal. Theory Appl. 24(4), 377–394 (2008)

    MathSciNet  Google Scholar 

  25. Bardaro, C., Mantellini, I.: Generalized sampling approximation of bivariate signals: rate of pointwise convergence. Numer. Funct. Anal. Optim. 31(2), 131–154 (2010)

    MathSciNet  Google Scholar 

  26. Bardaro, C., Mantellini, I.: A quantitative Voronovskaya formula for Mellin convolution operators. Mediterr. J. Math. 7(4), 483–501 (2010)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  28. Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling type operators. Z. Anal. Anwend. 32(3), 279–298 (2013)

    MathSciNet  Google Scholar 

  29. Bardaro, C., Mantellini, I.: Asymptotic expansion of generalized Durrmeyer sampling type series. Jaen J. Approx. 6(2), 143–165 (2014)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  32. Bardaro, C., Mantellini, I., Schmeisser, G.: Exponential sampling series: convergence in Mellin-Lebesgue spaces. Results Math. 74, 119 (2019)

    MathSciNet  Google Scholar 

  33. Bardaro, C., Mantellini, I., Tittarelli, I.: Convergence of semi-discrete exponential sampling operators in Mellin-Lebesgue spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117(1), 30 (2023)

    MathSciNet  Google Scholar 

  34. Bardaro, C., Mantellini, I., Stens, R., Vautz, J., Vinti, G.: Generalized sampling approximation for multivariate discontinuous signals and application to image processing. In: Zayed, A.I., Schmeisser, G. (eds.) New Perspectives on Approximation and Sampling Theory-Festschrift in Honor of Paul Butzer’s 85th Birthday, pp. 87–114. Birkhauser, Basel (2014)

    Google Scholar 

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

    Google Scholar 

  36. Butzer, P.L., Fischer, A., Stens, R.L.: Generalized sampling approximation of multivariate signals; theory and some applications. Note Mat. 10(1), 173–191 (1990)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  39. Butzer, P.L., Jansche, S.: A self-contained approach to Mellin transform analysis for square integrable functions; applications. Integral Transforms Spec. Funct. 8, 175–198 (1999)

    MathSciNet  Google Scholar 

  40. Butzer, P.L., Splettstösser, W.: A sampling theorem for duration limited functions with error estimates. Inform. Contr. 34, 55–65 (1977)

    MathSciNet  Google Scholar 

  41. Casasent, D.: Optical signal processing. In: Casasent, D. (ed.) Optical Data Processing, pp. 241–282. Springer, Berlin (1978)

    Google Scholar 

  42. Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083–1103 (2017)

    MathSciNet  Google Scholar 

  43. Costarelli, D., Piconi, M., Vinti, G.: Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces. Sampl. Theory Signal Process. Data Anal. 21, 3 (2023)

    MathSciNet  Google Scholar 

  44. Costarelli, D., Vinti, G.: Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz Spaces. Boll. Unione Mat. Ital. 4(3), 445–468 (2011)

    MathSciNet  Google Scholar 

  45. Costarelli, D., Vinti, G.: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integral Equ. Appl. 26(4), 455–481 (2014)

    MathSciNet  Google Scholar 

  46. Costarelli, D., Vinti, G.: Order of approximation for sampling Kantorovich operators. J. Integral Equ. Appl. 26(3), 345–368 (2014)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  51. Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press, Oxford (1996)

    Google Scholar 

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

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

  55. Prashant, K., Angamuthu, S. K., Bajpeyi, S.: On bivariate Kantorovich exponential sampling series, (2020) arXiv:2007.09373v1

  56. Ries, S., Stens, R. L.: Approximation by generalized sampling series. In: B. Sendov, P. Petrushev, R. Maleev, S. Tashev (eds.), Proc. Internat. Conf. "Constructive Theory of Functions", Varna, Bulgaria, June 1984, pp. 746–756, Bulgarian Acad. Sci., Sofia (1984)

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

    MathSciNet  Google Scholar 

  58. Splettstösser, W.: On generalized sampling sums based on convolution integrals. Arch. Elek. Ubertr. 32, 267–275 (1978)

    Google Scholar 

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

    Google Scholar 

  60. Zayed, A.I.: Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton (1993)

Download references

Author information

Authors and Affiliations

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

    Tuncer Acar & Abdulkadir Eke

  2. Department of Basic Sciences, National Defence University, Turkish Army Academy, 06420, Çankaya, Ankara, Turkey

    Sadettin Kursun

Authors
  1. Tuncer Acar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdulkadir Eke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sadettin Kursun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tuncer Acar.

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

Acar, T., Eke, A. & Kursun, S. Bivariate generalized Kantorovich-type exponential sampling series. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 35 (2024). https://doi.org/10.1007/s13398-023-01535-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-023-01535-2

Keywords

Mathematics Subject Classification

Search

Navigation