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Extension of Saturation Theorems for the Sampling Kantorovich Operators

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Abstract

In this paper, we extend the saturation results for the sampling Kantorovich operators proved in a previous paper, to more general settings. In particular, exploiting certain Voronovskaja-formulas for the well-known generalized sampling series, we are able to extend a previous result from the space of \(C^2\)-functions to the space of \(C^1\)-functions. Further, requiring an additional assumption, we are able to reach a saturation result even in the space of the uniformly continuous and bounded functions. In both the above cases, the assumptions required on the kernels, which define the sampling Kantorovich operators, have been weakened with respect to those assumed previously. On this respect, some examples have been discussed at the end of the paper.

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References

  1. Agrawal, P.N., Baxhaku, B.: Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein–Stancu–Kantorovich operators. Appl. Math. Comput. 306, 56–72 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Aleskeev, V.G.: Jackson and Jackson–Vallee Poussin-type kernels and their probability applications. Theory Probab. Appl. 41(1), 137–195 (1996)

    MathSciNet  Google Scholar 

  3. Allasia, G., Cavoretto, R., De Rossi, A.: A class of spline functions for landmark-based image registration. Math. Methods Appl. Sci. 35, 923–934 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comput. Appl. Math. 32, 71–87 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the absolute continuity in terms of convergence in variation for the sampling Kantorovich operators (2017) (submitted)

  7. Angeloni, L., Vinti, G.: Approximation with respect to Goffman–Serrin variation by means of non-convolution type integral operators. Numer. Funct. Anal. Optim. 31, 519–548 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Angeloni, L., Vinti, G.: Approximation in variation by homothetic operators in multidimensional setting. Differ. Integr. Equ. 26(5–6), 655–674 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Asdrubali, F., Baldinelli, G., Bianchi, F., Costarelli, D., Rotili, A., Seracini, M., Vinti, G.: Detection of thermal bridges from thermographic images by means of image processing approximation algorithms. Appl. Math. Comput. 317, 160–171 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Asdrubali, F., Baldinelli, G., Bianchi, F., Costarelli, D., Evangelisti, L., Rotili, A., Seracini, M., Vinti, G.: A model for the improvement of thermal bridges quantitative assessment by infrared thermography. Appl. Energy 211, 854–864 (2018)

    Article  MATH  Google Scholar 

  11. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 6(1), 29–52 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Bardaro, C., Karsli, H., Vinti, G.: Nonlinear integral operators with homogeneous kernels: Pointwise approximation theorems. Appl. Anal. 90(3–4), 463–474 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling operators. J. Anal. Appl. 32, 279–298 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Bezuglaya, L., Katsnelson, V.: The sampling theorem for functions with limited multi-band spectrum I. Zeitschrift für Analysis und ihre Anwendungen 12, 511–534 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boccuto, A., Bukhvalov, A.V., Sambucini, A.R.: Inequalities in classical spaces with mixed norms. Positivity 6, 393–411 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boccuto, A., Candeloro, D., Sambucini, A.R.: Vitali-type theorems for filter convergence related to Riesz space-valued modulars and applications to stochastic processes. J. Math. Anal. Appl. 419(2), 818–838 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, New York (2010)

    Book  Google Scholar 

  18. Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation, Vol.1. Pure and Applied Mathematics, 40, Academic Press (1971)

  19. Coroianu, L., Gal, S.G.: \(L^p\)- approximation by truncated max-product sampling operators of Kantorovich-type based on Fejer kernel. J. Integr. Equ. Appl. 29(2), 349–364 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  20. Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083–1103 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Costarelli, D., Vinti, G.: Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces. Numer. Funct. Anal. Optim. 36(8), 964–990 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Costarelli, D., Vinti, G.: Approximation by max-product neural network operators of Kantorovich type. Results Math. 69(3), 505–519 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Costarelli, D., Vinti, G.: Max-product neural network and quasi-interpolation operators activated by sigmoidal functions. J. Approx. Theory 209, 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Costarelli, D., Vinti, G.: Pointwise and uniform approximation by multivariate neural network operators of the max-product type. Neural Netw. 81, 81–90 (2016)

    Article  MATH  Google Scholar 

  25. Costarelli, D., Vinti, G.: Convergence for a family of neural network operators in Orlicz spaces. Math. Nachr. 290(2–3), 226–235 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Costarelli, D., Vinti, G.: Convergence results for a family of Kantorovich max-product neural network operators in a multivariate setting. Mathematica Slovaca 67(6), 1469–1480 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Costarelli, D., Vinti, G.: Saturation classes for max-product neural network operators activated by sigmoidal functions. Results Math. 72(3), 1555–1569 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. In: Proceedings of the Edinburgh Mathematical Society (2018). arXiv:1801.08695

  29. Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon sampling operators, their norms and approximation properties. Samp. Theory Signal Image Process. 8, 77–95 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Kivinukk, A., Tamberg, G.: On approximation properties of sampling operators by dilated kernels. In: 8th Internatuional in Conference on Sampling Theory and Applications, SampTA 2009, pp. 18-22 (2009)

  31. Kolomoitsev, Y.S., Krivoshein, A., Skopina, M.A.: Differential and falsified sampling expansions. J. Fourier Anal. Appl. (2017) https://doi.org/10.1007/s00041-017-9559-1

  32. Menekse Yilmaz, M., Uysal, G.: Convergence of singular integral operators in weighted Lebesgue spaces. Eur. J. Pure Appl. Math. 10(2), 335–347 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Nishishiraho, T.: Saturation of bounded linear operators. Tohoku Math. J. 30, 69–81 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  34. Orlova, O., Tamberg, G.: On approximation properties of generalized Kantorovich-type sampling operators. J. Approxim. Theory 201, 73–86 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ravier, R.J., Stichartz, R.S.: Sampling theory with average values on the Sierpinski gasket. Construct. Approx. 44(2), 159–194 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ries, S., Stens, R.L.: Approximation by generalized sampling series. In: Constructive Theory of Functions’84, Sofia, pp. 746–756 (1984)

  37. Runovski, K., Schmeisser, H.J.: On approximation methods generated by Bochner–Riesz kernels. J. Fourier Anal. Appl. 14(1), 16–38 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Runovski, K., Schmeisser, H.J.: On families of linear polynomial operators generated by Riesz kernels. Eurasian Math. J. 1(4), 24–139 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Szabados, J.: Convergence and saturation problems of discrete linear operators. Linear Oper. Approx. I I, 405–419 (1974)

    MathSciNet  MATH  Google Scholar 

  40. Tamberg, G.: Approximation by generalized Shannon sampling operators generated by band-limited kernels. Proc. Appl. Math. Mech. 8(1), 10937–10940 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vinti, G., Zampogni, L.: A General approximation approach for the simultaneous treatment of integral and discrete operators. Adv. Nonlinear Stud. (2017). https://doi.org/10.1515/ans-2017-6038

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors are partially supported by the “Department of Mathematics and Computer Science” of the University of Perugia (Italy). Moreover, the second author of the paper has been partially supported within the 2017 GNAMPA-INdAM Project “Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all’imaging”.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Perugia, 1, Via Vanvitelli, 06123, Perugia, Italy

    Benedetta Bartoccini, Danilo Costarelli & Gianluca Vinti

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  1. Benedetta Bartoccini
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  2. Danilo Costarelli
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  3. Gianluca Vinti
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Correspondence to Danilo Costarelli.

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Communicated by Uwe Kaehler.

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Bartoccini, B., Costarelli, D. & Vinti, G. Extension of Saturation Theorems for the Sampling Kantorovich Operators. Complex Anal. Oper. Theory 13, 1161–1175 (2019). https://doi.org/10.1007/s11785-018-0852-z

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  • DOI: https://doi.org/10.1007/s11785-018-0852-z

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