Weighted Approximations by Sampling-type Series

In this talk, we consider the approximation properties of generalized sampling type series and their exponential forms in weighted spaces of continuous functions and logarithmic weighted spaces of continuous functions, respectively. This type of problem (approximation in weighted spaces of functions) was considered for the first time in 1974 by Gadjiev for positive linear operators. Approximation by generalized sampling type series and their exponential forms in weighted spaces of functions were first considered by our research group in 2022. Published in 2022, our studies "T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, G. Vinti, Convergence of generalized sampling series in weighted spaces, Demonstratio Mathematica 55, 2022, 153-162" and "A. Aral, T. Acar, S. Kursun, Generalized Kantorovich forms of exponential sampling series, Analysis and Mathematical Physics 12:50, 2022" have been quite remarkable in this field. In this way, we have extended the class of approximated functions.
In this talk, I will present these all recent results.
References:
[1] P. L. Butzer, W. Splettstosser, A sampling theorem for duration limited functions with error estimates, Inform. Contr., 34, 1977, 55–65.
[2] C. Bardaro, L. Faina, I. Mantellini, A generalization of the exponential sampling series and its approximation properties, Math. Slovaca, 67 (6), 2017, 1481–1496.
[3] A. D. Gadziev, 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, no. 5; English Translated Soviet Math. Dokl., 15 (5), 1974.
[4] T. Acar, O. Alagoz, A. Aral, D. Costarelli, M. Turgay, G. Vinti, Approximation by sampling Kantorovich series, in weighted spaces of functions, Turkish Journal of Mathematics, 46 (7), 2022, 2663–2676.
[5] T. Acar, M. Turgay, Approximation by bivariate generalized sampling series in weighted spaces of functions, Dolomites Research Notes on Approximation Theory, 16, 2023, 11–22.
[6] T. Acar, S. Kursun, Pointwise convergence of generalized Kantorovich exponential sampling series, Dolomites Research Notes on Approximation Theory, 16, 2023, 1–10.
[7] T. Acar, S. Kursun, O. Acar, Approximation properties of exponential sampling series in logarithmic weighted spaces, Bulletin of the Iranian Mathematical Society, 50:36, 2024.