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A characterization of the rate of approximation of Kantorovich sampling operators in variable exponent Lebesgue spaces

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Abstract

We establish a direct and a matching two-term converse estimate by a K-functional and a modulus of smoothness for the rate of approximation by generalized Kantorovich sampling operators in variable exponent Lebesgue spaces. They yield the saturation property and class of these operators. We also prove a Voronovskaya-type estimate.

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References

  1. Acar, T., Costarelli, D., Vinti, G.: Linear prediction and simultaneous approximation by \(m\)-th order Kantorovich type sampling series. Banach J. Math. Anal. 14, 1481–1508 (2020)

    Article  MathSciNet  Google Scholar 

  2. Acar, T., Draganov, B.R.: A characterization of the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification. J. Math. Anal. Appl. 530(2), 127740 (2024)

    Article  MathSciNet  Google Scholar 

  3. Akgün, R.: Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent. Ukrain. Math. J. 63(1), 3–23 (2011)

    Article  MathSciNet  Google Scholar 

  4. Akgün, R.: Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth. Georg. Math. J. 18(2), 203–235 (2011)

    Article  MathSciNet  Google Scholar 

  5. Angeloni, L., Çetin, N., Costarelli, D., Sambucini, A.R., Vinti, G.: Multivariate sampling Kantorovich operators: quantitative estimates in Orlicz spaces. Constr. Math. Anal. 4(2), 229–241 (2021)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Bardaro, C., Mantellini, I.: Voronovskaya formulae for Kantorovich type generalized sampling series. Int. J. Pure Appl. Math. 62(3), 247–262 (2010)

    MathSciNet  Google Scholar 

  8. Bartoccini, B., Costarelli, D., Vinti, G.: Extension of saturation theorems for the sampling Kantorovich operators. Complex Anal. Oper. Theory 13, 1161–1175 (2019)

    Article  MathSciNet  Google Scholar 

  9. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation. Birkhäser, Basel (1971)

    Book  Google Scholar 

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

    Chapter  Google Scholar 

  11. Costarelli, D., Vinti, G.: Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces. Boll. Unione Mat. Ital. Serie (9) 4(3), 445–468 (2011)

    MathSciNet  Google Scholar 

  12. Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. (2) 62(1), 265–280 (2019)

    Article  MathSciNet  Google Scholar 

  13. Costarelli, D., Vinti, G.: Inverse results of approximation and the saturation order for the sampling Kantorovich series. J. Approx. Theory 242, 64–82 (2019)

    Article  MathSciNet  Google Scholar 

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. Costarelli, D., Vinti, G.: Approximation results by multivariate sampling Kantorovich series in Musielak–Orlicz spaces. Dolomites Res. Notes Approx. 12, 7–16 (2019)

    MathSciNet  Google Scholar 

  16. Costarelli, D., Vinti, G.: Approximation properties of the sampling Kantorovich operators: regularization, saturation, inverse results and Favard classes in \(L^p\)-Spaces. J. Fourier Anal. Appl. 28, 49 (2022)

    Article  Google Scholar 

  17. Costarelli, D., Vinti, G.: Convergence of sampling Kantorovich operators in modular spaces with applications. Rend. Circ. Mat. Palermo 2(70), 1115–1136 (2021)

    Article  MathSciNet  Google Scholar 

  18. Costarelli, D., Vinti, G.: A quantitative estimate for the sampling Kantorovich series in terms of the modulus of continuity in Orlicz spaces. Constr. Math. Anal. 2(1), 8–14 (2019)

    MathSciNet  Google Scholar 

  19. Cruz-Uribe, D.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. CRM Preprints, Barcelona (2012)

    Google Scholar 

  20. Cruz-Uribe, D., Diening, L., Fiorenza, A.: A new proof of the boundedness of maximal operators on variable Lebesgue spaces. Boll. Unione Mat. Ital. (9) 2(1), 151–173 (2009)

    MathSciNet  Google Scholar 

  21. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 28(1), 223–238 (2003)

    MathSciNet  Google Scholar 

  22. Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: Corrections to: “The maximal function on variable \(L^p\) spaces” [Ann. Acad. Sci. Fenn. Math. 28(1), 223–238 (2003)]. Ann. Acad. Sci. Fenn. Math. 29(1), 247–249 (2004)

  23. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Book  Google Scholar 

  24. Ditzian, Z., Ivanov, K.G.: Strong converse inequalities. J. Anal. Math. 61, 61–111 (1993)

    Article  MathSciNet  Google Scholar 

  25. Donnini, C., Vinti, G.: Approximation by means of Kantorovich generalized sampling operators in Musielak–Orlicz spaces. PanAmerican Math. J. 18(2), 1–18 (2008)

    MathSciNet  Google Scholar 

  26. Guven, A., Israfilov, D.M.: Trigonometric approximation in generalized Lebesgue spaces \(L^{p(x)}\). J. Math. Inequal. 4(2), 285–299 (2010)

    Article  MathSciNet  Google Scholar 

  27. Israfilov, D., Kokilashvili, V., Samko, S.: Approximation in weighted Lebesgue and Smirnov spaces with variable exponents. Proc. A. Razmadze Math. Inst. 143, 25–35 (2007)

    MathSciNet  Google Scholar 

  28. Israfilov, D.M., Testici, A.: Approximation problems in the Lebesgue spaces with variable exponent. J. Math. Anal. Appl. 459(1), 112–123 (2018)

    Article  MathSciNet  Google Scholar 

  29. Jia, R.-Q.: Approximation with scaled shift-invariant spaces by means of quasi-projection operators. J. Approx. Theory 131(1), 30–46 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Kolomoitsev, Yu., Skopina, M.: Approximation by sampling-type operators in \(L_{p}\)-spaces. Math. Meth. Appl. Sci. 43(16), 9358–9374 (2020)

    Article  Google Scholar 

  32. Kolomoitsev, Yu., Skopina, M.: Approximation by multivariate quasi-projection operators and Fourier multipliers. Appl. Math. Comput. 400, 125955 (2021)

    MathSciNet  Google Scholar 

  33. Kolomoitsev, Yu., Skopina, M.: Uniform approximation by multivariate quasi-projection operators. Anal. Math. Phys. 12, 68 (2022)

    Article  MathSciNet  Google Scholar 

  34. Nekvinda, A.: Hardy-Littlewood maximal operator on \(L^{p(x)}(^n)\). Math. Inequal. Appl. 7(2), 255–265 (2004)

    MathSciNet  Google Scholar 

  35. Ries, S., Stens, R.L.: Approximation by generalized sampling series. In: Sendov, B., Petrushev, P., Maleev, R., Tashev, S. (eds.) Proc. Internat. Conf. “Constructive Theory of Functions”, Varna, Bulgaria, June 1984, pp. 746–756. Bulgarian Acad. Sci., Sofia (1984)

  36. Samko, S.: Convolution type operators in \(L^{p(x)}\). Integr. Transforms Spec. Funct. 7(1–2), 123–144 (1998)

    Article  Google Scholar 

  37. Samko, S.: Differentiation and integration of variable order and the spaces \(L^{p(x)}\). In: “Operator theory for complex and hypercomplex analysis’’ (Mexico City, 1994), Contemp. Math, vol. 212, pp. 203–219. American Mathematical Society, Providence (1998)

    Chapter  Google Scholar 

  38. Sharapudinov, I.I.: Some aspects of approximation theory in the spaces \(L^{p(x)}\). Anal. Math. 33(2), 135–153 (2007)

    MathSciNet  Google Scholar 

  39. Sharapudinov, I.I.: Approximation of functions in \(L^{p(x)}_{2\pi }\) by trigonometric polynomials. Izv. RAN Ser. Mat. 77(2), 197–224 (2013) (in Russian); English transl. Izv. Math. 77(2), 407–434 (2013)

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Funding

This study is financed by the European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria, project No BG-RRP-2.004-0008.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

    Borislav R. Draganov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113, Sofia, Bulgaria

    Borislav R. Draganov

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  1. Borislav R. Draganov
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Correspondence to Borislav R. Draganov.

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Draganov, B.R. A characterization of the rate of approximation of Kantorovich sampling operators in variable exponent Lebesgue spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 71 (2024). https://doi.org/10.1007/s13398-024-01571-6

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  • DOI: https://doi.org/10.1007/s13398-024-01571-6

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