Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. (English) Zbl 1308.41017
The problem of the order of approximation for the multivariate sampling Kantorovich operators is studied. The cases of uniform approximation for uniformly continuous and bounded functions belonging to Lipschitz classes and the case of the modular approximation for functions in Orlicz spaces are considered. In the latter context, Lipschitz classes of Zygmund-type which take into account of the modular functional involved are introduced. Applications to \(L^p(R^n),\) interpolation and exponential spaces can be deduced from the general theory formulated in the setting of Orlicz spaces. Special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fejér’s and \(B\)-spline kernels have been discussed.
Reviewer: Vijay Gupta (New Delhi)
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A30 | Approximation by other special function classes |
Keywords:
multivariate sampling Kantorovich operators; Orlicz spaces; order of approximation; Lipschitz classes; irregular samplingReferences:
| [1] | L. Angeloni and G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing , Diff. Int. Eq. 18 (2005), 855-890. · Zbl 1212.41045 |
| [2] | C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Approximation of the Whittaker sampling series in terms of an average modulus of smoothness covering discontinuous signals , J. Math. Anal. Appl. 316 (2006), 269-306. · Zbl 1087.94013 · doi:10.1016/j.jmaa.2005.04.042 |
| [3] | C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Kantorovich-type generalized sampling series in the setting of Orlicz spaces , Sampl. Theor. Signal Image Process. 6 (2007), 29-52. · Zbl 1156.41307 |
| [4] | —-, Prediction by samples from the past with error estimates covering discontinuous signals , IEEE Trans. Inform. Theor. 56 (2010), 614-633. · Zbl 1366.94216 · doi:10.1109/TIT.2009.2034793 |
| [5] | C. Bardaro and I. Mantellini, Modular approximation by sequences of nonlinear integral operators in Musielak-Orlicz spaces , Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 403-425. · Zbl 0918.46030 |
| [6] | —-, On convergence properties for a class of Kantorovich discrete operators , Numer. Funct. Anal. Optim. 33 (2012), 374-396. · Zbl 1267.41021 · doi:10.1080/01630563.2012.652270 |
| [7] | C. Bardaro, J. Musielak and G. Vinti, Nonlinear integral operators and applications , De Gruyter Ser. Nonlin. Anal. Appl. 9 (2003), New York. · Zbl 1030.47003 |
| [8] | C. Bardaro and G. Vinti, Some inclusion theorems for Orlicz and Musielak-Orlicz type spaces , Ann. Mat. Pura Appl. 168 (1995), 189-203. · Zbl 0877.47017 · doi:10.1007/BF01759260 |
| [9] | —-, A general approach to the convergence theorems of generalized sampling series , Appl. Anal. 64 (1997), 203-217. · Zbl 0878.47016 · doi:10.1080/00036819708840531 |
| [10] | —-, An abstract approach to sampling type operators inspired by the work of P.L. Butzer -Part I- Linear operators , Sampl. Theor. Sig. Image Process. 2 (2003), 271-296. |
| [11] | L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend. 12 (1993), 511-534. · Zbl 0786.30019 |
| [12] | P.L. Butzer, A. Fisher and R.L. Stens, Generalized sampling approximation of multivariate signals: theory and applications , Note Matem. 10 (1990), 173-191. · Zbl 0768.42013 |
| [13] | P.L. Butzer and R.J. Nessel, Fourier analysis and approximation I, Academic Press, New York, 1971. |
| [14] | P.L. Butzer, S. Ries and R.L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series , J. Approx. Theor. 50 (1987), 25-39. · Zbl 0654.41004 · doi:10.1016/0021-9045(87)90063-3 |
| [15] | P.L. Butzer and R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview , SIAM Rev. 34 (1992), 40-53. · Zbl 0746.94002 · doi:10.1137/1034002 |
| [16] | —-, Linear prediction by samples from the past , in Advanced topics in Shannon sampling and interpolation theory , R.J. Marks II, eds., Springer-Verlag, New York, 1993. · doi:10.1007/978-1-4613-9757-1_5 |
| [17] | F. Cluni, D. Costarelli, A.M. Minotti and G. Vinti, Multivariate sampling Kantorovich operators: approximation and applications to civil engineering , EURASIP Open Library, Proc. SampTA 2013, 10th International Conference on Sampling Theory and Applications, July 1-5, 2013, Jacobs University, Bremen, 2013, 400-403. |
| [18] | —-, Enhancement of thermographic images as tool for structural analysis in earthquake engineering , NDT&E Inter. Indep. Nondestr. Test. Eval. (2014) DOI: 10.1016/j.ndteint.2014.10.001. |
| [19] | —-, Applications of sampling Kantorovich operators to thermographic images for seismic engineering , J. Comp. Anal. Appl. 19 (2015), 602-617. · Zbl 1369.94021 |
| [20] | D. Costarelli, Interpolation by neural network operators activated by ramp functions , J. Math. Anal. Appl. 419 (2014), 574-582. · Zbl 1298.41037 · doi:10.1016/j.jmaa.2014.05.013 |
| [21] | —-, Neural network operators: constructive interpolation of multivariate functions , submitted, 2014. |
| [22] | D. Costarelli and R. Spigler, Approximation by series of sigmoidal functions with applications to neural networks , Ann. Matem. Pura Appl., 2013, doi: 10.1007/s10231-013-0378-y. · Zbl 1321.41021 · doi:10.1007/s10231-013-0378-y |
| [23] | —-, Approximation results for neural network operators activated by sigmoidal functions , Neural Networks 44 (2013), 101-106. · Zbl 1296.41017 · doi:10.1016/j.neunet.2013.03.015 |
| [24] | —-, Multivariate neural network operators with sigmoidal activation functions , Neural Networks 48 (2013), 72-77. · Zbl 1297.41006 · doi:10.1016/j.neunet.2013.07.009 |
| [25] | —-, Convergence of a family of neural network operators of the Kantorovich type , J. Approx. Theor. 185 (2014), 80-90. · Zbl 1297.41003 · doi:10.1016/j.jat.2014.06.004 |
| [26] | —-, How sharp is Jensen’s inequality? , submitted. |
| [27] | D. Costarelli and G. Vinti, Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces , Boll. U.M.I. 4 (2011), 445-468. · Zbl 1234.41018 |
| [28] | —-, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing , Num. Funct. Anal. Opt. 34 (2013), 819-844. · Zbl 1277.41007 · doi:10.1080/01630563.2013.767833 |
| [29] | —-, Order of approximation for sampling Kantorovich operators , J. Int. Equat. Appl. 26 (2014), 345-368. · Zbl 1308.41016 · doi:10.1216/JIE-2014-26-3-345 |
| [30] | —-, Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces , Comment. Math. 5 (2013), 171-192. |
| [31] | M.M. Dodson and A.M. Silva, Fourier analysis and the sampling theorem , Proc. Ir. Acad. 86 (1985), 81-108. · Zbl 0583.42003 |
| [32] | C. Donnini and G. Vinti, Approximation by means of Kantorovich generalized sampling operators in Musielak-Orlicz spaces , PanAmer. Math. J. 18 (2008), 1-18. · Zbl 1144.41304 |
| [33] | J.R. Higgins, Sampling theory in Fourier and signal analysis : Foundations , Oxford Univ. Press, Oxford, 1996. · Zbl 0872.94010 |
| [34] | J.R. Higgins and R.L. Stens, Sampling theory in Fourier and signal analysis : Advanced topics , Oxford Science Publications, Oxford University Press, Oxford, 1999. · Zbl 0938.94001 |
| [35] | A.J. Jerry, The Shannon sampling-its various extensions and applications: a tutorial review , Proc. IEEE 65 (1977), 1565-1596. · Zbl 0442.94002 · doi:10.1109/PROC.1977.10771 |
| [36] | W.M. Kozlowski, Modular function spaces , Pure Appl. Math., Marcel Dekker, New York, 1988. |
| [37] | M.A. Krasnosel’ski\? and Ya.B. Ruticki\?, Convex functions and Orlicz spaces , P. Noordhoff Ltd., Groningen, The Netherlands, 1961. |
| [38] | L. Maligranda, Orlicz spaces and interpolation , Sem. Mat. IMECC, Campinas, 1989. · Zbl 0874.46022 |
| [39] | J. Musielak, Orlicz spaces and modular spaces , Lect. Notes Math. 1034 , Springer-Verlag, New York, 1983. · Zbl 0557.46020 · doi:10.1007/BFb0072210 |
| [40] | J. Musielak and W. Orlicz, On modular spaces , Stud. Math. 28 (1959), 49-65. · Zbl 0086.08901 |
| [41] | M.M. Rao and Z.D. Ren, Theory of Orlicz spaces , Pure Appl. Math., Marcel Dekker, Inc., New York, 1991. · Zbl 0724.46032 |
| [42] | —-, Applications of Orlicz spaces , Mono. Text. Pure Appl. Math. 250 , Marcel Dekker, Inc., New York, 2002. · Zbl 0997.46027 |
| [43] | S. Ries and R.L. Stens, Approximation by generalized sampling series , in Constructive theory of functions ’84, Sofia, 1984, 746-756. · Zbl 0588.41013 |
| [44] | F. Ventriglia and G. Vinti, A unified approach for the convergence of nonlinear Kantorovich type operators , Comm. Appl. Nonlin. Anal. 21 (2014), 45-74. · Zbl 1300.41017 |
| [45] | C. Vinti, A survey on recent results of the mathematical seminar in Perugia , inspired by the Work of Professor P.L. Butzer, Result. Math. 34 (1998), 32-55. · Zbl 1010.01021 · doi:10.1007/BF03322036 |
| [46] | G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing , Appl. Anal. 79 (2001), 217-238. · Zbl 1020.41009 · doi:10.1080/00036810108840958 |
| [47] | —-, Approximation in Orlicz spaces for linear integral operators and applications , Rend. Circ. Mat. Palermo 76 (2005), 103-127. · Zbl 1136.41306 |
| [48] | G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces , J. Approx. Theor. 161 (2009), 511-528. · Zbl 1183.41016 · doi:10.1016/j.jat.2008.11.011 |
| [49] | —-, A Unifying approach to convergence of linear sampling type operators in Orlicz spaces , Adv. Diff. Equat. 16 (2011), 573-600. · Zbl 1223.41014 |
| [50] | —-, Approximation results for a general class of Kantorovich type operators , Adv. Nonlin. Stud. 14 (2014), 991-1011. · Zbl 1305.41027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
