Journal of Integral Equations and Applications

Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces

Danilo Costarelli and Gianluca Vinti

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Abstract

In this paper, 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/signals 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. The special cases of multivariate sampling Kantorovich operators based on kernels of the product type and constructed by means of Fej\'er's and B-spline kernels have been studied in details.

Article information

Source
J. Integral Equations Applications, Volume 26, Number 4 (2014), 455-481.

Dates
First available in Project Euclid: 9 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1420812881

Digital Object Identifier
doi:10.1216/JIE-2014-26-4-455

Mathematical Reviews number (MathSciNet)
MR3299827

Zentralblatt MATH identifier
1308.41017

Subjects
Primary: 41A25: Rate of convergence, degree of approximation 41A30: Approximation by other special function classes 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47A58: Operator approximation theory 47B38: Operators on function spaces (general) 94A12: Signal theory (characterization, reconstruction, filtering, etc.)

Keywords
Multivariate sampling Kantorovich operators Orlicz spaces order of approximation Lipschitz classes irregular sampling

Citation

Costarelli, Danilo; Vinti, Gianluca. Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integral Equations Applications 26 (2014), no. 4, 455--481. doi:10.1216/JIE-2014-26-4-455. https://projecteuclid.org/euclid.jiea/1420812881


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References

  • L. Angeloni and G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing, Diff. Int. Eq. 18 (2005), 855–890.
  • 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.
  • 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.
  • ––––, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Inform. Theor. 56 (2010), 614–633.
  • 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.
  • ––––, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim. 33 (2012), 374–396.
  • C. Bardaro, J. Musielak and G. Vinti, Nonlinear integral operators and applications, De Gruyter Ser. Nonlin. Anal. Appl. 9 (2003), New York.
  • C. Bardaro and G. Vinti, Some inclusion theorems for Orlicz and Musielak-Orlicz type spaces, Ann. Mat. Pura Appl. 168 (1995), 189–203.
  • ––––, A general approach to the convergence theorems of generalized sampling series, Appl. Anal. 64 (1997), 203–217.
  • ––––, 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.
  • L. Bezuglaya and V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Z. Anal. Anwend. 12 (1993), 511–534.
  • P.L. Butzer, A. Fisher and R.L. Stens, Generalized sampling approximation of multivariate signals: theory and applications, Note Matem. 10 (1990), 173–191.
  • P.L. Butzer and R.J. Nessel, Fourier analysis and approximation I, Academic Press, New York, 1971.
  • 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.
  • P.L. Butzer and R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Rev. 34 (1992), 40–53.
  • ––––, 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.
  • 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.
  • ––––, 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.
  • ––––, Applications of sampling Kantorovich operators to thermographic images for seismic engineering, J. Comp. Anal. Appl. 19 (2015), 602–617.
  • D. Costarelli, Interpolation by neural network operators activated by ramp functions, J. Math. Anal. Appl. 419 (2014), 574–582.
  • ––––, Neural network operators: constructive interpolation of multivariate functions, submitted, 2014.
  • 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.
  • ––––, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks 44 (2013), 101–106.
  • ––––, Multivariate neural network operators with sigmoidal activation functions, Neural Networks 48 (2013), 72–77.
  • ––––, Convergence of a family of neural network operators of the Kantorovich type, J. Approx. Theor. 185 (2014), 80–90.
  • ––––, How sharp is Jensen's inequality?, submitted.
  • 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.
  • ––––, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Num. Funct. Anal. Opt. 34 (2013), 819–844.
  • ––––, Order of approximation for sampling Kantorovich operators, J. Int. Equat. Appl. 26 (2014), 345–368.
  • ––––, Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces, Comment. Math. 5 (2013), 171–192.
  • M.M. Dodson and A.M. Silva, Fourier analysis and the sampling theorem, Proc. Ir. Acad. 86 (1985), 81–108.
  • C. Donnini and G. Vinti, Approximation by means of Kantorovich generalized sampling operators in Musielak-Orlicz spaces, PanAmer. Math. J. 18 (2008), 1–18.
  • J.R. Higgins, Sampling theory in Fourier and signal analysis: Foundations, Oxford Univ. Press, Oxford, 1996.
  • J.R. Higgins and R.L. Stens, Sampling theory in Fourier and signal analysis: Advanced topics, Oxford Science Publications, Oxford University Press, Oxford, 1999.
  • A.J. Jerry, The Shannon sampling-its various extensions and applications: a tutorial review, Proc. IEEE 65 (1977), 1565–1596.
  • W.M. Kozlowski, Modular function spaces, Pure Appl. Math., Marcel Dekker, New York, 1988.
  • M.A. Krasnosel'skiǐ and Ya.B. Rutickiǐ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, The Netherlands, 1961.
  • L. Maligranda, Orlicz spaces and interpolation, Sem. Mat. IMECC, Campinas, 1989.
  • J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math. 1034, Springer-Verlag, New York, 1983.
  • J. Musielak and W. Orlicz, On modular spaces, Stud. Math. 28 (1959), 49–65.
  • M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Pure Appl. Math., Marcel Dekker, Inc., New York, 1991.
  • ––––, Applications of Orlicz spaces, Mono. Text. Pure Appl. Math. 250, Marcel Dekker, Inc., New York, 2002.
  • S. Ries and R.L. Stens, Approximation by generalized sampling series, in Constructive theory of functions'84, Sofia, 1984, 746–756.
  • F. Ventriglia and G. Vinti, A unified approach for the convergence of nonlinear Kantorovich type operators, Comm. Appl. Nonlin. Anal. 21 (2014), 45–74.
  • 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.
  • G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Appl. Anal. 79 (2001), 217–238.
  • ––––, Approximation in Orlicz spaces for linear integral operators and applications, Rend. Circ. Mat. Palermo 76 (2005), 103–127.
  • G. Vinti and L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, J. Approx. Theor. 161 (2009), 511–528.
  • ––––, A Unifying approach to convergence of linear sampling type operators in Orlicz spaces, Adv. Diff. Equat. 16 (2011), 573–600.
  • ––––, Approximation results for a general class of Kantorovich type operators, Adv. Nonlin. Stud. 14 (2014), 991–1011.