Bivariate Sampling Kantorovich series in Weighted Spaces of Functions

Abstract

This paper introduces and analyzes the approximation properties of bivariate sampling Kantorovich series in weighted spaces of functions. We define the operators using a kernel function that satisfies specific conditions, establishing their linearity and well-definiteness within the weighted spaces of functions. The main results include proving both pointwise convergence for weighted spaces of continuous functions and uniformly continuous functions. Furthermore, we provide a quantitative analysis of the approximation error, deriving an estimate for the rate of convergence in terms of the bivariate weighted modulus of continuity, which demonstrates the dependency of the error on the moments of the kernel function and the smoothness of the approximated function. The theoretical findings are further illustrated through numerical tables and graphical examples in the final section, providing a visual confirmation of the convergence properties and the accuracy of the approximation.

Keywords

• Sampling Kantorovich operators
• Bivariate weighted modulus of continuity
• Weighted approximation

References

1. [1] Higgins, J. R., "Sampling theory in Fourier and signal analysis: Foundations", Oxford, Oxford University Press (1996). DOI: https://doi.org/10.1093/oso/9780198596998.001.0001
2. [2] Kivinukk, A. and Tamberg, G., "Interpolating generalized Shannon sampling operators", Their Norms and Approximation Properties. Sampling Theory in Signal and Image Processing, 8: 77-95, (2009). DOI: https://doi.org/10.1007/BF03549509
3. [3] Butzer, P.L. and Nessel, R.J., "Fourier analysis and approximation I", Academic press, New York-London, (1971).
4. [4] Ries S. and Stens R.L., "Approximation by generalized sampling series" Proceedings of the International Conference on Constructive Theory of Functions (Varna, 1984), Bulgarian Academy of Science, Sofia, (1984), 746-756.
5. [5] Angeloni, L., Costarelli, D. and Vinti, G., "A characterization of the convergence in variation for the generalized sampling series", Annales Fennici Mathematici, 43(2): 755-767, (2018). DOI: https://doi.org/10.5186/aasfm.2018.4343
6. [6] Bardaro, C., Butzer, P.L., Stens R.L. and Vinti, G., "Prediction by samples from the past with error estimates covering discontinuous signals", in IEEE Transactions on Information Theory, 56(1): 614-633, (2010). DOI: https://doi.org/10.1109/TIT.2009.2034793
7. [7] Butzer, P.L., Fischer, A. and Stens, R.L., "Generalized sampling approximation of multivariate signals: theory and applications", Note di Matematica, 10(1): 173-191, (1990). DOI: https://doi.org/10.1285/i15900932v10supn1p173
8. [8] Butzer, P.L. and 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). DOI: https://doi.org/10.1007/978-1-4613-9757-1_5

9. [9] Butzer, P.L., Fischer, A., and Stens, R.L., "Generalized sampling approximation of multivariate signals; general theory", Atti del Seminario Matematico e Fisico dell'Universita di Modena, 41(1): 17-37, (1993).
10. [10] Tamberg, G., "Approximation by generalized Shannon sampling operators generated by band-limited kernels", Proceedings in Applied Mathematics and Mechanics, 8(1): 10937-10940, (2008). DOI: https://doi.org/10.1002/pamm.200810937
11. [11] Tamberg, G., "On truncation errors of some generalized Shannon sampling operators" Numerical Algorithms, 55(2): 367-382, (2010). DOI: https://doi.org/10.1007/s11075-010-9418-5
12. [12] Bardaro, C., Butzer, P.L., Stens R.L. and Vinti, G., "Kantorovich-type generalized sampling series in the setting of Orlicz spaces", Sampling Theory in Signal and Image Processing, 6(1): 29-52, (2007). DOI: https://doi.org/10.1007/BF03549462
13. [13] Acar, T., Costarelli, D. and Vinti, G., "Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series", Banach Journal of Mathematical Analysis, 14(4): 1481-1508, (2020). DOI: https://doi.org/10.1007/s43037-020-00071-0
14. [14] Bardaro, C. and Mantellini, I., "Asymptotic formulae for multivariate Kantorovich type generalized sampling series", Acta Mathematica Sinica, English Series, 27(7): 1247-1258, (2011). DOI: https://doi.org/10.1007/s10114-011-0227-0
15. [15] Bardaro, C. and Mantellini, I., "On convergence properties for a class of Kantorovich discrete operators", Numerical Functional Analysis and Optimization, 33(4): 374-396, (2012). DOI: https://doi.org/10.1080/01630563.2011.652270
16. [16] Costarelli, D. and Vinti, G., "Inverse results of approximation and the saturation order for the sampling Kantorovich series", Journal of Approximation Theory, 242: 64-82, (2019). DOI: https://doi.org/10.1016/j.jat.2019.03.001
17. [17] Orlova, O. and Tamberg, G., "On approximation properties of generalized Kantorovich-type sampling operators", Journal of Approximation Theory, 201: 73-86, (2016). DOI: https://doi.org/10.1016/j.jat.2015.10.001
18. [18] Gadjiev, A.D., "The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P. P. Korovkin", Doklady Akademii Nauk, 218(5): 1001-1004, (1974).
19. [19] Gadjiev, A.D., "On P. P. Korovkin type theorems", Matematicheskie Zametki, 20(5): 781-786, (1976).
20. [20] Acar, T., Alagoz, O., Aral, A., Costarelli, D., Turgay, M. and Vinti, G., "Convergence of generalized sampling series in weighted spaces", Demonstratio Mathematica, 55: 153-162, (2022). DOI: https://doi.org/10.1515/dema-2022-0014
21. [21] Acar, T., Alagoz, O., Aral, A., Costarelli, D., Turgay, M. and Vinti, G., "Approximation by sampling Kantorovich series in weighted spaces of functions", Turkish Journal of Mathematics, 46(7): 2663-2676, (2022). DOI: https://doi.org/10.55730/1300-0098.3293
22. [22] Acar, T., Kursun, S. and Turgay, M. "Multidimensional Kantorovich modifications of exponential sampling series", Quaestiones Mathematicae, 46(1): 57-72, (2023). DOI: https://doi.org/10.2989/16073606.2021.1992033
23. [23] Acar, T. and Kursun, S., "Pointwise convergence of generalized Kantorovich exponential sampling series", Dolomites Research Notes on Approximation, 16: 1-10, (2023). DOI: https://doi.org/10.14658/pupj-/pupj-drna-2023-2-1
24. [24] Acar, T. and Turgay, M., "Approximation by bivariate generalized sampling series in weighted spaces of functions", Dolomites Research Notes on Approximation, 16: 11-22, (2023). DOI: https://doi.org/10.14658/pupj-/pupj-drna-2023-2-2
25. [25] Alagoz, O., Turgay, M., Acar, T. and Parlak, M., "Approximation by sampling Durrmeyer operators in weighted space of functions", Numerical Functional Analysis and Optimization, 43(10): 1223-1239, (2022). DOI: https://doi.org/10.1080/01630563.2022.2096630
26. [26] Alagoz, O., "Weighted approximations by sampling type operators: recent and new results", Constructive Mathematical Analysis, 7(3): 114-125, (2024). DOI: https://doi.org/10.33205/cma.1528004
27. [27] Aral, A., Acar, T. and Kursun, S., "Generalized Kantorovich forms of exponential sampling series", Analysis and Mathematical Physics, 12: 50, (2022). DOI: https://doi.org/10.1007/s13324-022-00667-9
28. [28] Draganov, B.R., "A fast converging sampling operator", Constructive Mathematical Analysis, 5(4): 190-201, (2022). DOI: https://doi.org/10.33205/cma.1172005
29. [29] Kursun, S., Turgay, M., Alagoz, O. and Acar., T., "Approximation properties of multivariate exponential sampling series", Carpathian Mathematical Publications, 13(3): 666-675, (2021). DOI: https://doi.org/10.15330/cmp.13.3.666-675
30. [30] Kursun, S., Aral, A., and Acar, T., "Approximation results for Hadamard-type exponential sampling Kantorovich series", Mediterranean Journal of Mathematics, 20: 263, (2023). DOI: https://doi.org/10.1007/s00009-023-02459-2
31. [31] Kursun, S., Aral, A., and Acar, T., "Riemann–Liouville fractional integral type exponential sampling Kantorovich series", Expert Systems with Applications, 238(F): 122350, (2024). DOI: https://doi.org/10.1016/j.eswa.2023.122350
32. [32] Turgay, M. and Acar, T., "Approximation by modified generalized sampling series", Mediterranean Journal of Mathematics, 21: 107, (2024). DOI: https://doi.org/10.1007/s00009-024-02653-w
33. [33] Turgay, M. and Acar, T., "Modified sampling Kantorovich operators in weighted spaces of functions", FILOMAT, 39(24): 8477-8491 , (2025). DOI: https://doi.org/10.2298/FIL2524477T
34. [34] Costarelli, D. and Vinti, G., "Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz Spaces", Bollettino dell'Unione Matematica Italiana, 4(3): 445-468, (2011).
35. [35] Acar, T., Aral, A. and I. Rasa, "The new forms of Voronovskaya's theorem in weighted spaces", Positivity, 20(2): 25-40, (2016). DOI: https://doi.org/10.1007/s11117-015-0338-4
36. [36] Acar, T., Montano, M.C., Garrancho P. and Leonessa, V., "On Bernstein-Chlodovsky operators preserving e^(-2x)", Bulletin of the Belgian Mathematical Society - Simon Stevin, 26(5): 681-698, (2019). DOI: https://doi.org/10.36045/bbms/1579402817
37. [37] Aral, A., “Weighted approximation: Korovkin and quantitative type theorems”, Modern Mathematical Methods, 1(1): 1-21, (2023). DOI: https://doi.org/10.64700/mmm.25
38. [38] Ispir N. and Atakut, C., "Approximation by modified Szász-Mirakjan operators on weighted spaces", Proceedings of the Indian Academy of Sciences: Mathematical Sciences, 112(4): 571-578, (2002). DOI: https://doi.org/10.1007/BF02829690
39. [39] Ispir, N., "On modified Baskakov operators on weighted spaces", Turkish Journal of Mathematics, 26(3): 355-365, (2001).