Furthermore, we also present a $P$-statistical approximation result in the space of periodic continuous functions of period $2\pi$ for short $C^*$.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In this study, our main goal is to obtain approximation properties of convolution operators for multivariables via a special method which is not included in any other methods given before, also known as $P$-statistical convergence.
📊
Key Findings & Data:We present the $P$-statistical rate of this approximation and provide examples of convolution operators.
💡
Conclusion & Impact:Furthermore, we also present a $P$-statistical approximation result in the space of periodic continuous functions of period $2\pi$ for short $C^*$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Power series method · statistical convergence · convolution operators
Abstract
In this study, our main goal is to obtain approximation properties of convolution operators for multivariables via a special method which is not included in any other methods given before, also known as $P$-statistical convergence. We present the $P$-statistical rate of this approximation and provide examples of convolution operators. It is noteworthy to express that one can not approximate $f$ by earlier results for our examples. Therefore, our results fill an important gap in the existing literature. Furthermore, we also present a $P$-statistical approximation result in the space of periodic continuous functions of period $2\pi$ for short $C^*$.
Dinar, R., Yurdakadim, T. (2025). Approximation properties of convolution operators via statistical convergence based on a power series. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 74 Issue: 1. https://doi.org/10.31801/cfsuasmas.1566336
Atlıhan, Ö. G., Yurdakadim, T., Taş, E., A new approach to approximation by positive linear operators in weighted spaces, Ukrainian Mathematical Journal, 74(11) (2023), 1447-1453. https://doi.org/10.1007/s11253-02302161-2
Atlıhan, Ö. G., Yurdakadim, T., Taş, E., Summation process of convolution operators for multivariables, Sarajevo Journal of Mathematics, 26(2) (2017), 207-216.
Atlıhan, Ö. G., Yurdakadim, T., Taş, E., Korovkin type approximation theorems in weighted spaces via power series method, Oper. Matrices, 12(2) (2018), 529-535. https://doi.org/10.7153/oam-2018-12-32
Atlıhan, Ö. G., Yurdakadim, T., Taş, E., Statistical approximation properties of convolution operators for multivariables, AIP Conference Proceedings, (2013), 1156-1159. https://doi.org/10.1063/1.4825713
Bernstein, S. N., Demonstration du theoreme de Weierstrass fondee sur le calcul des probablites, Communications of the Kharkov Mathematical Society, 13 (1912), 1-2.
Çınar, S., Yıldız, S., P-statistical summation process of sequences of convolution operators, Indian Journal of Pure and Applied Mathematics, 53(3) (2022), 648-659. https://doi.org/10.1007/s13226-021-00156-y
Demirkale, S., Taş, E., Statistical convergence of spliced sequences in terms of power series on topological spaces, Mathematical Sciences and Applications E-Notes, 11(2) (2023), 104-111. https://doi.org/10.36753/mathenot.1212331
DeVore, R. A., Lorentz, G. G., Constructive Approximation, Springer-Verlag, 1993.
Duman, O., Khan, M. K., Orhan, C., A-statistical convergence of approximating operators,Math. Inequal. Appl., 6 (2003), 689-699. https://doi.org/10.7153/mia-06-62
Duman, O., A-statistical convergence of sequences of convolution operators, Taiwanese Journal of Mathematics, 12(2) (2008), 523-536. https://doi.org/10.11650/twjm/1500574172
Erkus, E., Duman, O., A Korovkin type approximation theorem in statistical sense, Studia Sci. Math. Hungar., 43(3) (2006), 285-294. https://doi.org/10.1556/sscmath.43.2006.3.2
Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
Fridy, J. A., Miller, H. I., Orhan, C., Statistical rates of convergence, Acta Sci. Math., 69 (2003), 147-157. (2003).
Gadjiev, A. D., Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain Journal of Math., 32 (2002), 129-137. https://doi.org/10.1216/rmjm/1030539612
Kratz, W., Stadtmüller, U., Tauberian theorems for $J_p$-summability, J. Math. Anal. Appl., 139 (1989), 362-371.
Korovkin, P. P., On convergence of linear positive operators in the space of continuous functions, Doklady Akad. Nauk SSR., 90 (1953), 961-964.
Popoviciu, T., Asupra demonstratiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare, Lucrarile Ses. Gen. Şt. Acad. Romane din., (1950), 1664-1657.
Salat, T., On statistically convergent sequences of real numbers, Mat. Slovaca., 30 (1980), 139-150.
Soylemez, D., Ünver, M., Rates of power series statistical convergence of positive linear operators and power series statistical convergence of q-Meyer–Konig and Zeller operators, Lobachevskii Journal of Mathematics, 42(2) (2021), 426-434. https://doi.org/10.1134/S1995080221020189
Srivastava, H. M., Gupta, V., A certain family of summation-integral type operators, Math Comput. Modelling, 37 (2003), 1307-1315.
Tasdelen, F., Olgun, A., Tunca Başcanbaz, G., Approximation of functions of two variables by certain linear positive operators, Proc. Indian Acad. Sci., 117(3) (2007), 387-399. https://doi.org/10.1007/s12044-007-0033-x
Ünver, M., Orhan, C., Statistical convergence with respect to power series methods and applications to approximation theory, Numer. Func. Anal. Opt., 40 (2019), 535-547. https://doi.org/10.1080/01630563.2018.1561467
Yurdakadim, T., Taş, E., Effects of fuzzy settings in Korovkin theory via $P_{p}$-statistical convergence, Romanian Journal of Mathematics and Computer Science, 2(12) (2022), 1-8.
Weierstrass, K. G., Über die Analytische Dorstellbarkeit Sogenannter Willkürlicher Funktionen Einer Reellen, Veranderlichen. Sitzungsber, Akad. Berlin, 1885.