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Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels

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Abstract

In the present paper, we study the saturation order in the space \(L^1({\mathbb {R}})\) for the sampling Kantorovich series based upon bandlimited kernels. The above study is based on the so-called Fourier transform method, introduced in 1960 by P. L. Butzer. As a first result, the saturation order is derived in a Bernstein class; here, it is crucial to derive the Fourier transform of the above sampling-type series, which can be expressed in a suitable closed form. Subsequently, the saturation is reached in the whole space \(L^1({\mathbb {R}})\). At the end of the paper, several examples of bandlimited kernels, such as the Fejér’s and Bochner–Riesz’s kernel, have been recalled and the saturation order of the corresponding sampling Kantorovich series has been stated.

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Notes

  1. Obviously, such property for the space \(B^1_{\sigma }({\mathbb {R}})\), holds also for the spaces \(B^p_{\sigma }({\mathbb {R}})\), \(1 \le p \le 2\), where in this case, the definition of the Fourier transform of \(f \in L^p({\mathbb {R}})\) is used, see [12]; moreover, if \(p>2\) the same result holds in the distributional sense.

References

  1. Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the convergence in variation for the generalized sampling series. Annales Academiae Scientiarum Fennicae Mathematica 43, 755–767 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the absolute continuity in terms of convergence in variation for the sampling Kantorovich operators. Mediterr. J. Math. (2019). https://doi.org/10.1007/s00009-019-1315-0

  3. 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 

  4. Bardaro, C., Butzer, P.L., Mantellini, I.: The foundations of fractional calculus in the Mellin transform setting with applications. J. Fourier Anal. Appl. 21(5), 961–1017 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. Bardaro, C., Karsli, H., Vinti, G.: On pointwise convergence of linear integral operators with homogeneous kernels. Integral Transforms Spec. Funct. 19(6), 429–439 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boas Jr., R.P.: Entire Functions. Academic Press, New York (1954)

    MATH  Google Scholar 

  8. 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 

  9. 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 

  10. Butzer, P.L.: Fourier-transform methods in the theory of approximation. Arch. Ration. Mech. Anal. 5(1), 390–415 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  11. Butzer, P.L., Feichtinger, H.G., Gröchenig, K.: Error analysis in regular and irregular sampling theory. Appl. Anal. 50(3–4), 167–189 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic Press, New York (1971)

    Book  MATH  Google Scholar 

  13. Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein space \(B^p_{\sigma }\) and their extensions to functions from larger spaces with error estimates in term of their distances from \(B^p_{\sigma }\). J. Fourier Anal. Appl. 19, 333–375 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Butzer, P.L., Splettstösser, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math. Verein. 90, 1–70 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Butzer, P.L., Stens, R.L.: Linear prediction by samples from the past. In: Marks, R.J. (ed.) Advanced Topics in Shannon Sampling and Interpolation Theory, pp. 157–183. Springer, New York (1993)

    Chapter  Google Scholar 

  16. Constales, D., De Bie, H., Lian, P.: A new construction of the Clifford–Fourier kernel. J. Fourier Anal. Appl. 23(2), 462–483 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Coroianu, L., Costarelli, D., Gal, S.G., Vinti, G.: The max-product generalized sampling operators: convergence and quantitative estimates. Appl. Math. Comput. 355, 173–183 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Coroianu, L., Gal, S.G.: Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels. Sampl. Theory Signal Image Process. 11(1), 113–132 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  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.: Order of approximation for sampling Kantorovich operators. J. Integral Equ. Appl. 26(3), 345–368 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Costarelli, D., Vinti, G.: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces. J. Integral Equ. Appl. 26(4), 455–481 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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 

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

    MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. 62(1), 265–280 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. Do, M.N., Lu, Y.M.: A theory for sampling signals from a union of subspaces. IEEE Trans. Signal Process. 56(6), 2334–2345 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Feichtinger, H.G., Gröchenig, K.: Irregular sampling theorems and series expansions of band-limited functions. J. Math. Anal. Appl. 167, 530–556 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press, Oxford (1996)

    MATH  Google Scholar 

  30. 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

  31. Kolomoitsev, Y.S., Skopina, M.A.: Approximation by multivariate Kantorovich–Kotelnikov operators. J. Math. Anal. Appl. 456(1), 195–213 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krivoshein, A., Skopina, M.A.: Multivariate sampling-type approximation. Anal. Appl. 15(4), 521–542 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lind, M., Petrushev, P.: Nonlinear nonnested spline approximation. Constr. Approx. 45(2), 143–191 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lyubarskii, Y., Ortega-Cerdá, J.: Bandlimited Lipschitz functions. Appl. Comput. Harmonic Anal. 37(2), 307–324 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Marzo, J., Pridhnani, B.: Sufficient conditions for sampling and interpolation on the sphere. Constr. Approx. 40(2), 241–257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. 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 

  37. Nguyen, H.Q., Unser, M.: A sampling theory for non-decaying signals. Appl. Comput. Harmonic Anal. 43(1), 76–93 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

  41. Rosenthal, M., Schmeisser, H.J.: On the boundedness of singular integrals in Morrey spaces and its preduals. J. Fourier Anal. Appl. 22(2), 462–490 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Skopina, M.A.: Band-limited scaling and wavelet expansions. Appl. Comput. Harmonic Anal. 36(1), 143–157 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Stens, R.L.: Error estimates for sampling sums based on convolution integrals. Inf. Control 45, 37–47 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  44. Tamberg, G.: On truncation errors of some generalized Shannon sampling operators. Numerical Algorithms 55(2), 367–382 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Timan, A.F.: Theory of approximation of functions of a real variable. MacMillan, New York (1965)

    Google Scholar 

  46. Urbach, H.P.: Generalized sampling theorem for band-limited functions. Math. Comput. Modell. 38, 133–140 (2003)

    Article  MATH  Google Scholar 

  47. Vinti, G., Zampogni, L.: Approximation results for a general class of Kantorovich type operators. Adv. Nonlinear Stud. 14, 991–1011 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author has been partially supported within the 2019 GNAMPA-INdAM Project “Metodi di analisi reale per l’approssimazione attraverso operatori discreti e applicazioni”, while the second author within the projects: (1) Ricerca di Base 2017 dell’Università degli Studi di Perugia - “Metodi di teoria degli operatori e di Analisi Reale per problemi di approssimazione ed applicazioni” , (2) Ricerca di Base 2018 dell’Università degli Studi di Perugia - “Metodi di Teoria dell’Approssimazione, Analisi Reale, Analisi Nonlineare e loro Applicazioni” , (3) “Metodi e processi innovativi per lo sviluppo di una banca di immagini mediche per fini diagnostici” funded by the Fondazione Cassa di Risparmio di Perugia, 2018. This research has been accomplished within RITA (Research ITalian network on Approximation).

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

    Danilo Costarelli & Gianluca Vinti

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  1. Danilo Costarelli
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Correspondence to Danilo Costarelli.

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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). https://doi.org/10.1007/s13324-019-00334-6

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