Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces

Abstract

In this paper, we establish a quantitative estimate for multivariate sampling Kantorovich operators by means of the modulus of continuity in the general setting of Orlicz spaces. As a consequence, the qualitative order of convergence can be obtained, in case of functions belonging to suitable Lipschitz classes. In the particular instance of L^p-spaces, using a direct approach, we obtain a sharper estimate than that one that can be deduced from the general case.

Keywords

• Multivariate sampling Kantorovich operators
• Orlicz spaces
• modulus of smoothness
• quantitative estimates
• Lipschitz classes

References

1. T. Acar, D. Costarelli and G. Vinti: Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series, Banach J. Math. Anal., 14 (4) (2020), 1481-1508.
2. F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter studies in Mathematics, (2011).
3. F. Altomare, M. Cappelletti Montano and V. Leonessa: On a Generalization of Szász-Mirakjan-Kantorovich Operators, Results Math., 63 (2013), 837-863.
4. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa: A generalization of Kantorovich operators for convex compact subsets, Banach J. Math. Anal., 11 (3) (2017), 591-614.
5. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa: Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (1) (2018), 153-173.
6. L. Angeloni, D. Costarelli, M. Seracini, G. Vinti and L. Zampogni: Variation diminishing-type properties for multivariate sampling Kantorovich operators, Bollettino U.M.I., Special issue dedicated to Prof. Domenico Candeloro, 13 (4) (2020), 595-605.
7. L. Angeloni, D. Costarelli and G. Vinti: A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn. Math., 43 (2018), 755-767.
8. L. Angeloni, D. Costarelli and G. Vinti: Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing, Ann. Acad. Sci. Fenn. Math., 45 (2020), 751-770.

9. F. Asdrubali, G. Baldinelli, F. Bianchi, D. Costarelli, A. Rotili, M. Seracini and G. Vinti: Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160-171.
10. C. Bardaro, I. Mantellini: Voronovskaya formulae for Kantorovich type generalized sampling series, Int. J. Pure Appl. Math., 62 (3) (2010), 247-262.
11. C. Bardaro, I. Mantellini: Asymptotic formulae for multivariate Kantorovich type generalized sampling series, Acta Math. Sinica (ES), 27 (7) (2011), 1247-1258.
12. C. Bardaro, J. Musielak and G. Vinti: Nonlinear Integral Operators and Applications, in: de Gruyter Series in Nonlinear Analysis and Applications, vol. 9, Walter de Gruyter & Co., Berlin, (2003).
13. C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti: Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process., 6 (1) (2007), 29-52.
14. M. Cantarini, D. Costarelli and G. Vinti: A solution of the problem of inverse approximation for the sampling Kantorovich operators in case of Lipschitz functions, Dolomites Res. Notes Approx. DRNA, 13 (2020), 30-35.
15. N. Çetin, D. Costarelli and G. Vinti: Quantitative estimates for nonlinear sampling Kantorovich operators, arXiv 2102.08651 (2021).
16. F. Cluni, D. Costarelli, V. Gusella and G. Vinti: Reliability increase of masonry characteristics estimation by sampling algorithm applied to thermographic digital images, Probabilist Eng. Mech., 60 (2020), 103022.
17. L. Coroianu, S. G. Gal: Lp- approximation by truncated max-product sampling operators of Kantorovich-type based on Fejer kernel, J. Integral Equations Applications, 29 (2) (2017), 349-364.
18. L. Coroianu, S. G. Gal: Approximation by truncated max-product operators of Kantorovich-type based on generalized (Φ, Ψ)-kernels, Math. Methods Appl. Sci., 41 (17) (2018), 7971-7984.
19. L. Coroianu, S. G. Gal: Approximation by max-product operators of Kantorovich type, Stud. Univ. Babes-Bolyai Math., 64 (2) (2019), 207-223.
20. D. Costarelli, M. Seracini and G. Vinti: A segmentation procedure of the pervious area of the aorta artery from CT images without contrast medium, Math. Methods Appl. Sci., 43 (2020), 114-133.
21. D. Costarelli, M. Seracini and G. Vinti: A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods, Appl. Math. Comput., 374 (2020), 125046.
22. D. Costarelli, A. R. Sambucini and G. Vinti: Convergence in Orlicz spaces by means of the multivariate max-product neural network operators of the Kantorovich type, Neural Comput. & Applic., 31 (9) (2019), 5069-5078.
23. D. Costarelli, R. Spigler: How sharp is the Jensen inequality ?, J. Inequal. Appl., 2015:69 (2015) 1-10.
24. D. Costarelli, G. Vinti: Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special issue dedicated to Prof. Giovanni Prodi, 9 (4) (2011), 445-468.
25. D. Costarelli, G. Vinti: A quantitative estimate for the sampling Kantorovich series in terms of the modulus of continuity in Orlicz spaces, Constr. Math. Anal., 2 (1) (2019), 8-14.
26. D. Costarelli, G. Vinti: An inverse result of approximation by sampling Kantorovich series, Proc. Edinburgh Math. Soc., 62 (1) (2019), 265-280.
27. D. Costarelli, G. Vinti: Inverse results of approximation and the saturation order for the sampling Kantorovich series, J. Approx. Theor., 242 (2019), 64-82.
28. D. Costarelli, G. Vinti: Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels, Anal. Math. Phys., 9 (2019), 2263-2280.
29. E. D’Aniello, M. Maiuriello: A survey on composition operators on some function spaces, Aequat. Math., (2020).
30. A. Krivoshein, M. A. Skopina: Multivariate sampling-type approximation, Anal. Appl., 15 (4) (2017), 521-542.
31. J. Musielak, W. Orlicz: On modular spaces, Studia Math., 28 (1959), 49-65.
32. J. Musielak: Orlicz Spaces and Modular Spaces, in: Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, (1983).
33. M. M. Rao, Z.D. Ren: Theory of Orlicz Spaces, Marcel Dekker Inc., Pure and Appl. Math., New York-Basel-Hong Kong, (1991).
34. M. M. Rao, Z. D. Ren: Applications of Orlicz Spaces, Marcel Dekker Inc., Monographs and Textbooks in Pure and applied Mathematics, vol. 250, New York, (2002).
35. G. Vinti, L. Zampogni: Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theor., 161 (2009), 511-528.