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Acar, Tuncer; Alagöz, Osman; Aral, Ali; Costarelli, Danilo; Turgay, Metin; Vinti, Gianluca

Approximation by sampling Kantorovich series in weighted spaces of functions. (English) Zbl 1524.41028

Turk. J. Math. 46, No. 7, 2663-2676 (2022).
Summary: This paper studies the convergence of the so-called sampling Kantorovich operators for functions belonging to weighted spaces of continuous functions. This setting allows us to establish uniform convergence results for functions that are not necessarily uniformly continuous and bounded on \(\mathbb{R}\). In that context we also prove quantitative estimates for the rate of convergence of the family of the above operators in terms of weighted modulus of continuity. Finally, pointwise convergence results in quantitative form by means of Voronovskaja type theorems have been also established.

Cited in 10 Documents

MSC:

41A25 Rate of convergence, degree of approximation
41A36 Approximation by positive operators

Keywords:

generalized sampling series; sampling Kantorovich operators; pointwise and uniform convergence; weighted approximation; asymptotic formulas
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References:

[1] Acar T, Alagoz O, Aral A, Costarelli D, Turgay M et al. Convergence of generalized sampling series in weighted spaces. Demonstratio Mathematica 2022; 55: 153-162. doi: 10.1515/dema-2022-0014. · Zbl 1509.41010 · doi:10.1515/dema-2022-0014
[2] Acar T, Aral A, Raşa I. The new forms of Voronovskaya’s theorem in weighted spaces. Positivity 2016; 20 (1): 25-40. doi: 10.1007/s11117-015-0338-4 · Zbl 1334.41015 · doi:10.1007/s11117-015-0338-4
[3] Acar T, Montano MC, Garrancho P, Leonessa V. On Bernstein-Chlodovsky operators preserving e −2x . Bulletin of the Belgian Mathematical Society-Simon Stevin 2019; 26 (5): 681-698. doi: 10.36045/bbms/1579402817 · Zbl 1434.41009 · doi:10.36045/bbms/1579402817
[4] Acar T, Costarelli D, Vinti G. Linear prediction and simultaneous approximation by m -th order Kantorovich type sampling series. Banach Journal of Mathematical Analysis 2020; 14 (4): 1481-1508. doi: 10.1007/s43037-020-00071-0 · Zbl 1450.41005 · doi:10.1007/s43037-020-00071-0
[5] Acar T, Montano MC, Garrancho P, Leonessa V. Voronovskaya type results for Bernstein-Chlodovsky op-erators preserving e −2x . Journal of Mathematical Analysis and Applications 2020; 491 (1): 124307. doi: 10.1016/j.jmaa.2020.124307 · Zbl 1447.41008 · doi:10.1016/j.jmaa.2020.124307
[6] Angeloni L, Costarelli D, Vinti G. A characterization of the convergence in variation for the generalized sampling series. Annales Academiae Scientiarum Fennicae Mathematica 2018; 43: 755-767. doi: 10.5186/aasfm.2018.4343 · Zbl 1405.41010 · doi:10.5186/aasfm.2018.4343
[7] Bardaro C, Butzer PL, Stens RL, Vinti G. Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampling Theory in Signal and Image Processing 2007; 6 (1): 29-52. doi: 10.1007/BF03549462 · Zbl 1156.41307 · doi:10.1007/BF03549462
[8] Bardaro C, Butzer PL, Stens RL, Vinti G. Prediction by samples from the past with error estimates covering discon-tinuous signals. In: IEEE Transactions on Information Theory 2010; 56 (1): 614-633. doi: 10.1109/TIT.2009.2034793 · Zbl 1366.94216 · doi:10.1109/TIT.2009.2034793
[9] Bardaro C, Mantellini I. Asymptotic formulae for multivariate Kantorovich type generalized sampling series. Acta Mathematica Sinica, English Series 2011; 27 (7): 1247-1258. doi: 10.1007/s10114-011-0227-0 · Zbl 1303.41008 · doi:10.1007/s10114-011-0227-0
[10] Bardaro C, Mantellini I. On convergence properties for a class of Kantorovich discrete operators. Numerical Functional Analysis and Optimization 2012; 33 (4): 374-396. doi: 10.1080/01630563.2011.652270 · Zbl 1267.41021 · doi:10.1080/01630563.2011.652270
[11] Bardaro C, Musielak J, Vinti G. Nonlinear integral operators and applications. De Gruyter Series in Nonlinear Analysis and Applications 9, New York-Berlin, De Gruyter, 2003. doi: 10.1515/9783110199277 · Zbl 1030.47003 · doi:10.1515/9783110199277
[12] Butzer PL, Stens RL. Linear prediction by samples from the past. In: Advanced topics in Shannon sampling and interpolation theory. New York, Springer, 1993, pp. 157-183. doi: 10.1007/978-1-4613-9757-1_5 · doi:10.1007/978-1-4613-9757-1_5
[13] Cantarini M, Costarelli D, Vinti G. Approximation of differentiable and not differentiable signals by the first derivative of sampling Kantorovich operators. Journal of Mathematical Analysis and Applications 2022; 509: 125913. doi: 10.1016/j.jmaa.2021.125913 · Zbl 1487.41033 · doi:10.1016/j.jmaa.2021.125913
[14] Coroianu L, Gal SG. Approximation by nonlinear generalized sampling operators of max-product kind. Sampling Theory in Signal and Image Processing 2010; 9 (1-3): 59-75. doi: 10.1007/BF03549524 · Zbl 1228.41015 · doi:10.1007/BF03549524
[15] Coroianu L, Gal SG. Approximation by max-product sampling operators based on sinc -type kernels. Sampling Theory in Signal and Image Processing 2011; 10 (3): 211-230. doi: 10.1007/BF03549542 · Zbl 1346.94074 · doi:10.1007/BF03549542
[16] Coroianu L, Gal SG. Saturation results for the truncated max-product sampling operators based on sinc and Fejer-type kernels. Sampling Theory in Signal and Image Processing 2012; 11 (1): 113-132. doi: 10.1007/BF03549552 · Zbl 1346.41004 · doi:10.1007/BF03549552
[17] Costarelli D, Minotti AM, Vinti G. Approximation of discontinuous signals by sampling Kantorovich series. Journal of Mathematical Analysis and Applications 2017; 450 (2): 1083-1103. doi: 10.1016/j.jmaa.2017.01.066 · Zbl 1373.41018 · doi:10.1016/j.jmaa.2017.01.066
[18] Costarelli D, Vinti G. Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. Journal of Integral Equations and Applications 2014; 26 (4): 455-481. doi: 10.1216/JIE-2014-26-4-455 · Zbl 1308.41017 · doi:10.1216/JIE-2014-26-4-455
[19] Costarelli D, Vinti G. Inverse results of approximation and the saturation order for the sampling Kantorovich series. Journal of Approximation Theory 2019; 242: 64-82. doi: 10.1016/j.jat.2019.03.001 · Zbl 1416.41019 · doi:10.1016/j.jat.2019.03.001
[20] Costarelli D, G. Vinti G. Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels. Analysis and Mathematical Physics 2019; 9: 2263-2280. doi: 10.1007/s13324-019-00334-6 · Zbl 1478.42005 · doi:10.1007/s13324-019-00334-6
[21] Gadziev AD. 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 SSSR 1974; 218 (5): 1001-1004.
[22] Gadjiev AD. Theorems of Korovkin type. Mathematical Notes of the Academy of Sciences of the USSR 1976; 20: 995-998. doi: 10.1007/BF01146928 · Zbl 0398.41014 · doi:10.1007/BF01146928
[23] Higgins JR. Sampling theory in Fourier and signal analysis: Foundations. Oxford, Oxford Univ. Press, 1996. · Zbl 0872.94010
[24] Ispir N. On modified Baskakov operators on weighted spaces. Turkish Journal of Mathematics 2001; 26 (3): 355-365. · Zbl 0999.41013
[25] Kivinukk A, Tamberg G. Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampling Theory in Signal and Image Processing 2009; 8: 77-95. doi: 10.1007/bf03549509 · Zbl 1182.41012 · doi:10.1007/bf03549509
[26] Kivinukk A, Tamberg G. On approximation properties of sampling operators by dilated kernels. In: 8th International Conference on Sampling Theory and Applications (SAMPTA’09);
[27] Marseille, France; 2009.
[28] Orlova O, Tamberg G. On approximation properties of generalized Kantorovich-type sampling operators. Journal of Approximation Theory 2016; 201: 73-86. doi: 10.1016/j.jat.2015.10.001 · Zbl 1329.41030 · doi:10.1016/j.jat.2015.10.001
[29] Ries S, Stens RL. 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. · Zbl 0588.41013
[30] Tamberg G. Approximation by generalized Shannon sampling operators generated by band-limited kernels. Pro-ceedings in Applied Mathematics and Mechanics 2008; 8 (1): 10937-10940. doi: 10.1002/pamm.200810937 · Zbl 1395.94228 · doi:10.1002/pamm.200810937
[31] Tamberg G. On truncation errors of some generalized Shannon sampling operators. Numerical Algorithms 2010; 55 (2): 367-382. doi: 10.1007/s11075-010-9418-5 · Zbl 1205.41015 · doi:10.1007/s11075-010-9418-5
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