Convergence of a class of generalized sampling Kantorovich operators perturbed by multiplicative noise. (English) Zbl 07910308
Candela, Anna Maria (ed.) et al., Recent advances in mathematical analysis. Celebrating the 70th anniversary of Francesco Altomare. Based on the presentations at the international conference, ReDiMA 2021, Bari, Italy, September 23–24, 2021. Cham: Birkhäuser. Trends Math., 249-267 (2023).
In this article a family of sampling type series is introduced. From the mathematical point of view, the present definition generalizes the notion of the well-known sampling Kantorovich operators, in fact providing a weighted version of the original family of operators by noice functions. From the application point of view, this situation represents the reconstruction problem of signals perturbed by linear or nonlinear multiplicative noise sources. First, pointwise and uniform approximation theorems have been proved. Then, convergence theorems have been derived in the general setting of Orlicz spaces. Also, an \(L_p\)-convergence theorem is established. Finally, the concept of delta convergent sequence is introduced and also used in order to prove that the above family of sampling type operators extend the well-known generalized sampling series of P. L. Butzer.
For the entire collection see [Zbl 1515.46001].
For the entire collection see [Zbl 1515.46001].
Reviewer: Vijay Gupta (New Delhi)
Keywords:
uniform convergence; modular convergence; generalized sampling Kantorovich series; noise functions; Orlicz spaces; delta convergenceSoftware:
DLMFReferences:
| [1] | Acar, T., Costarelli, D., Vinti, G.: Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling serie. Banach J. Math. Anal. 14(4), 1481-1508 (2020) · Zbl 1450.41005 |
| [2] | Altomare, F.: On some convergence criteria for nets of positive operators on continuous function spaces. J. Math. Anal. Appl. 398(2), 542-552 (2013) · Zbl 1275.41028 |
| [3] | Altomare, F.: On the convergence of sequences of positive linear operators and functionals on bounded function spaces. Proc. Am. Math. Soc. 149, 3837-3848 (2021) · Zbl 1511.41011 |
| [4] | Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. De Gruyter Studies in Mathematics, vol. 17. De Gruyter, Berlin (1994) · Zbl 0924.41001 |
| [5] | Altomare, F., Leonessa, V.: On a sequence of positive linear operators associated with a continuous selection of Borel measures. Mediter. J. Math. 3, 363-382 (2006) · Zbl 1121.41019 |
| [6] | Altomare, F., Cappelletti Montano, M., Leonessa, V.: On a generalization of Szász-Mirakjan-Kantorovich operators. Res. Math. 63, 837-863 (2013) · Zbl 1272.41003 |
| [7] | Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the convergence in variation for the generalized sampling series. Ann. Acad. Sci. Fennicae Math. 43, 755-767 (2018) · Zbl 1405.41010 |
| [8] | Bardaro, C., Mantellini, I.: Asymptotic expansion of generalized Durrmeyer sampling type series. Jean J. Approx. 6(2), 143-165 (2014) · Zbl 1379.41017 |
| [9] | Bardaro, C., Musielak, J., Vinti, G.: Nonlinear Integral Operators and Applications. De Gruyter Series in Nonlinear Analysis and Applications, vol. 9. De Gruyter, Berlin (2003) · Zbl 1030.47003 |
| [10] | Bardaro, C., Butzer, P.L., Stens R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Samp. Theory Sign. Image Proc. 6, 29-52 (2007) · Zbl 1156.41307 |
| [11] | Bardaro, C., Faina, L., Mantellini, I.: Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series. Math. Nachr. 289(14-15), 1702-1720 (2016) · Zbl 1354.41008 |
| [12] | Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic Press, New York (1971) · Zbl 1515.42001 |
| [13] | Butzer, P.L., Stens, R.L.: Linear prediction by samples from past. In: Advanced Topics in Shannon Sampling and Interpolation Theory R. J. Marks II. Springer Texts Electrical Engineering, pp. 157-183. Springer, New York (1993) |
| [14] | Butzer, P.L., Fisher, A., Stens, R.L.: Approximation of continuous and discontinuous functions by generalized sampling series. J. Approx. Theory 50, 25-39 (1987) · Zbl 0654.41004 |
| [15] | Butzer, P.L., Fisher, A., Stens, R.L.: Generalized sampling aproximation of multivariate signals. Atti Sem. Mat. Fis. Univ. Modena 41, 17-37 (1993) · Zbl 0789.42026 |
| [16] | Cantarini, M., Costarelli, D., Vinti, G.: A solution of the problem of inverse approximation for the sampling Kantorovich operators in case of Lipschitz functions. Dolomites Res. Notes Approx. 13, 30-35 (2020) · Zbl 1540.41034 |
| [17] | Cantarini, M., Costarelli, D., Vinti, G.: Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators. J. Math. Anal. Appl. 509, 125913 (2022) · Zbl 1487.41033 |
| [18] | Costarelli, D., Spigler, R.: How sharp is the Jensen inequality? J. Inequalities Appl. 2015, 1-10 (2015) · Zbl 1309.26014 |
| [19] | Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinburgh Math. Soc. 62(1), 265-280 (2019) · Zbl 1428.41019 |
| [20] | Costarelli, D., Vinti, G.: Inverse results of approximation and the saturation order for the sampling Kantorovich series. J. Approx. Theory 242, 64-82 (2019) · Zbl 1416.41019 |
| [21] | Costarelli, D., Vinti, G.: Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels. Anal. Math. Phys. 9, 2263-2280 (2019) · Zbl 1478.42005 |
| [22] | Costarelli, D., Piconi, M., Vinti, G.: On the convergence properties of Durrmeyer-Sampling type operators in Orlicz spaces. Math. Nachrichten (2021). https://doi.org/10.1002/mana.202100117 |
| [23] | Kanwal, R.P.: Generalized Functions: Theory and Applications, 3rd edn. Springer, New York (2004) · Zbl 1069.46001 |
| [24] | Karsli, H.: On Urysohn type generalized sampling operators. Dolomites Res. Notes Approx. 14(2), 58-67 (2021) · Zbl 1540.41043 |
| [25] | Karsli, H.: On multidimensional Urysohn type generalized sampling operators. Math. Found. Comput. 4(4), 271-280 (2021). Special issue on: Approximation by linear and nonlinear operators with applications · Zbl 1493.41011 |
| [26] | Kivinukk, A., Tamberg, G.: On window methods in generalized Shannon sampling operators. In: New Perspectives on Approximation and Sampling Theory. Birkhäuser, Cham, pp. 63-85 (2014) · Zbl 1318.42002 |
| [27] | Musielak, J., Orlicz, W.: Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983) · Zbl 0557.46020 |
| [28] | Olver, F.W.J., Lozier, D.W., Boisfert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) · Zbl 1198.00002 |
| [29] | Orlova, O., Tamberg, G.: On approximation properties of generalized Kantorovich-type sampling operators. J. Approx. Theory 201, 73-86 (2016) · Zbl 1329.41030 |
| [30] | Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146. Marcel Dekker, New York (1991) · Zbl 0724.46032 |
| [31] | Vinti, G.: A general approximation result for nonlinear integral operators and applications to signal processing. Appl. Anal. 79, 217-238 (2001) · Zbl 1020.41009 |
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