Weighted approximations by sampling type operators: recent and new results

Year 2024, , 114 - 125, 15.09.2024

Osman Alagoz

https://doi.org/10.33205/cma.1528004

Abstract

In this paper, we collect some recent results on the approximation properties of generalized sampling operators and Kantorovich operators, focusing on pointwise and uniform convergence, rate of convergence, and Voronovskaya-type theorems in weighted spaces of functions. In the second part of the paper, we introduce a new generalization of sampling Durrmeyer operators including a special function $\rho$ which satisfies certain assumptions. For the family of newly constructed operators, we obtain pointwise convergence, uniform convergence and rate of convergence for functions belonging to weighted spaces of functions.

Keywords

Sampling series, generalized sampling operator, Kantorovich operator, Durrmeyer type sampling operator, weighted approximation

Supporting Institution

TUBITAK

Thanks

This study was supported by Scientific and Technological Research Council of Turkey (TUBITAK) under the Grant Number 119F263. The author thank to TUBITAK for their supports.

References

• T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay and G. Vinti: Convergence of generalized sampling series in weighted spaces, Demonstr. Math., 55 (2022), 153–162.
• T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay and G. Vinti: Approximation by sampling Kantorovich series in weighted spaces of functions, Turkish J. Math., 46 (7) (2022), 2663–2676.
• T. Acar, M. C. Montano, P. Garrancho and V. Leonessa: Voronovskaya type results for Bernstein-Chlodovsky operators preserving e−2x, J. Math. Anal. Appl., 491 (1) (2020), 124307.
• T. Acar, M. C. Montano, P. Garrancho and V. Leonessa: On Bernstein-Chlodovsky operators preserving e−2x, Bull. Belg. Math. Soc. Simon Stevin, 26 (5) (2019), 681–698.
• A. Aral: Weighted approximation: Korovkin and quantitative type theorems, Modern Math. Methods, 1 (1) (2023), 1–21.
• F. Asdrubali, G. Baldinelli, F. Bianchi, D. Costarelli, A. Rotili, M. Seracini and G. Vinti: Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput., 317 (2018), 160–171.
• C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti: Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process., 6 (1) (2007), 29–52.
• B. R. Dragonov: A fast converging sampling operator, Constr. Math. Anal., 5 (4) (2022), 190–201.
• P. L. Butzer, R. L. Stens: The sampling theorem and linear prediction in signal analysis, Jahresber. Dtsch. Math. Ver., 90 (1) (1998), 1–70.
• P. L. Butzer, R. L. Stens: Linear prediction by samples from the past, Advanced topics in Shannon sampling and interpolation theory, Springer, New York, 1993, 157–183.
• P. L. Butzer, W. Engels, S. Ries and R. L. Stens: The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM J. Appl. Math., 46 (2) (1986), 299–323.
• P. L. Butzer, W. Splettstosser, A sampling theorem for duration-limited functions with error estimates, Inf. Control, 34 (1) (1977), 55–65.
• G. Baldinelli, F. Bianchi, A. Rotili, D. Costarelli, M. Seracini, G. Vinti and L. Evangelisti: A model for the improvement of thermal bridges quantitative assessment by infrared thermography, Appl. Energy, 211 (2018), 854–864.
• C. Bardaro, I. Mantellini: Asymptotic expansion of generalized Durrmeyer sampling type series, Jaen J. Approx., 6 (2) (2014), 143–165.
• D. Costarelli, G. Vinti: Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integral Equ. Appl., 26 (4) (2014), 455–481.
• D. Costarelli, A. R. Sambucini: A comparison among a fuzzy algorithm for image rescaling with other methods of digital image processing, Constr. Math. Anal., 7 (2) (2024), 45–68.
• L. Boccali, D. Costarelli, G. Vinti: A Jackson-type estimate in terms of the τ-modulus for neural network operators in Lp-spaces, Modern Math. Methods, 2 (2) (2024), 90–102.
• F. Cluni, D. Costarelli, A. M. Minotti and G. Vinti: Enhancement of thermographic images as tool for structural analysis in earthquake engineering, NDT E Int., 70 (2015), 60–72.
• A. D. Gadjiev: The convergence problem for a sequence of positive linear operators on unbounded sets, and Theorems analogous to that of P. P. Korovkin, Dokl. Akad. Nauk SSSR, 218 (5) (1974), 1001–1004.
• A. D. Gadjiev: Theorems of Korovkin type, Math. Notes Acad. Sci. USSR, 20 (1976), 995–998.
• A. Holhos: Quantitative estimates for positive linear operators in weighted space, Gen. Math., 16 (4) (2008), 99–110.
• N. Ispir: On modified Baskakov operators on weighted spaces, Turk. J. Math., 25 (3) (2001), 355–365.
• M. Turgay, T. Acar: Approximation by Modified Generalized Sampling Series, Mediterr. J. Math., 21 (2024), 107.