A Quantitative Estimate for the Sampling Kantorovich Series in Terms of the Modulus of Continuity in Orlicz Spaces

Year 2019, Volume: 2 Issue: 1, 8 - 14, 01.03.2019

Danilo Costarellı , Gianluca Vıntı

https://doi.org/10.33205/cma.484500

Cited By: 10

Abstract

In the present paper we establish a quantitative estimate for the sampling Kantorovich operators with respect to the modulus of continuity in Orlicz spaces defined in terms of the modular functional. At the end of the paper, concrete examples are discussed, both for what concerns the kernels of the above operators, as well as for some concrete instances of Orlicz spaces.

Keywords

Sampling Kantorovich series, Orlicz spaces, Modulus of continuity, Quantitative estimates, Kernels

References

• [1] A. Abdurexit and T. N. Bekjan: Noncommutative Orlicz modular spaces associated with growth functions. Banach J. Math. Anal. 9 (4) (2015), 115–125.
• [2] T. Acar, A. Alotaibi and S. A. Mohiuddine: Construction of new family of Bernstein-Kantorovich operators. Math. Methods Appl. Sci. 40 (18) (2017), 7749–7759.
• [3] G. Allasia, R. Cavoretto and A. De Rossi: A class of spline functions for landmark-based image registration, Math. Methods Appl. Sci. 35 (8) (2012), 923–934.
• [4] G. Allasia, R. Cavoretto and A. De Rossi: Lobachevsky spline functions and interpolation to scattered data, Comput. Appl. Math. 32 (1) (2013), 71–87.
• [5] 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.
• [6] 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. Comp. 317 (2018), 160–171.
• [7] F. Asdrubali, G. Baldinelli, F. Bianchi, D. Costarelli, L. Evangelisti, A. Rotili, M. Seracini and G. Vinti: A model for the improvement of thermal bridges quantitative assessment by infrared thermography. Applied Energy 211 (2018), 854–864.
• [8] 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.
• [9] C. Bardaro and I. Mantellini: On convergence properties for a class of Kantorovich discrete operators. Num. Funct. Anal. Optim. 33 (4) (2012), 374–396.
• [10] C. Bardaro, J. Musielak and G. Vinti: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, 9 New York, Berlin, 2003.
• [11] B. Bartoccini, D. Costarelli and G. Vinti: Extension of saturation theorems for the sampling Kantorovich operators. In print in: Complex Analysis and Operator Theory (2018), DOI: 10.1007/s11785-018-0852-z.
• [12] P. L. Butzer: A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1) (1983), 185–212.
• [13] P. L. Butzer and R. J. Nessel: Fourier Analysis and Approximation, Vol. I: One-dimensional theory, Pure and Applied Mathematics, 40, Academic Press, New York-London, 1971.
• [14] P. L. Butzer, S. Ries and R. L. Stens: Approximation of Continuous and Discontinuous Functions by Generalized Sampling Series. J. Approx. Theory 50 (1) (1987), 25–39.
• [15] L. Coroianu and S. G. Gal: $L^p$- approximation by truncated max-product sampling operators of Kantorovich-type based on Fejér kernel. J. Integral Equations Appl. 29 (2) (2017), 349–364.
• [16] L. Coroianu and S. G. Gal: Approximation by truncated max-product operators of Kantorovich-type based on generalized $(\Phi,\Psi)$-kernels. Math. Methods Appl. Sci. 41 (2018), 7971-7984.
• [17] D. Costarelli, A.M. Minotti and G. Vinti: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450 (2) (2017), 1083–1103.
• [18] D. Costarelli and A.R. Sambucini: Approximation results in Orlicz spaces for sequences of Kantorovich max-product neural network operators. Results Math. 73 (1) (2018), Art. 15, 15 pp. DOI: 10.1007/s00025-018-0799-4.
• [19] D. Costarelli and R. Spigler: How sharp is the Jensen inequality ?, J. Inequal. Appl. 2015:69 (2015) 1–10.
• [20] D. Costarelli and G. Vinti: Approximation by Nonlinear Multivariate Sampling-Kantorovich Type Operators and Applications to Image Processing. Numer. Funct. Anal. Optim. 34 (8) (2013), 819–844.
• [21] D. Costarelli and G. Vinti: Order of approximation for sampling Kantorovich operators, J. Integral Equations Appl. 26 (3) (2014), 345–368.
• [22] D. Costarelli and G. Vinti: Convergence for a family of neural network operators in Orlicz spaces. Math. Nachr. 290 (2-3) (2017), 226–235.
• [23] D. Costarelli and G. Vinti: An inverse theorem of approximation by sampling Kantorovich series. In print in: Proc. Edinb. Math. Soc. (2018), DOI:10.1017/S0013091518000342.
• [24] D. Cruz-Uribe and P. Hasto: Extrapolation and interpolation in generalized Orlicz spaces. Trans. Amer. Math. Soc. 370 (6) (2018), 4323–4349.
• [25] P. A. Hasto: The maximal operator on generalized Orlicz spaces. J. Funct. Anal. 269 (12) (2015), 4038–4048.
• [26] Y. S. Kolomoitsev and M. A. Skopina: Approximation by multivariate Kantorovich-Kotelnikov operators. J. Math. Anal. Appl. 456 (1) (2017), 195–213.
• [27] A. Krivoshein and M. A. Skopina: Multivariate sampling-type approximation, Anal. Appl. 15 (4) (2017), 521–542.
• [28] K. Kuaket and P. Kumam: Fixed points of asymptotic pointwise contractions in modular spaces. Appl. Math. Lett. 24 (11) (2011), 1795–1798.
• [29] J. Musielak: Orlicz spaces and Modular Spaces. Lecture Notes in Math. 1034 Springer-Verlag, Berlin, 1983.
• [30] J. Musielak and W. Orlicz: On modular spaces. Studia Math. 18 (1959), 49–65.
• [31] O. Orlova and G. Tamberg: On approximation properties of generalized Kantorovich-type sampling operators. J. Approx. Theory 201 (2016), 73–86.
• [32] S. Ries and R. L. Stens: Approximation by generalized sampling series. In: Proc. Internat. Conf. Constructive Theory of Functions, Varna, Bulgaria, June 1984, pp. 746–756, Bulgarian Acad. Sci. Sofia, 1984.
• [33] M. Unser: Ten good reasons for using spline wavelets. Proc. SPIE Vol. 3169,Wavelets Applications in Signal and Image Processing V (1997), 422–431.