Abstract
In this paper, we deal with an approximation problem for matrix-valued positive linear operators via statistical convergence with respect to the power series method which is a new statistical type convergence. Then, we present an application that shows our theorem is more applicable than the classical one. We also compute the rates of P-statistical convergence of these operators.
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References
Anastassiou, G.A., and O. Duman. 2011. Towards intelligent modeling: statistical approximation theory. Intelligent systems reference library, vol. 14. Berlin: Springer.
Bardaro, C., A. Boccuto, K. Demirci, I. Mantellini, and S. Orhan. 2015. Korovkin-type theorems for modular \(\Psi -A-\)statistical convergence. J. Funct. Sp. 2015: 1–11. https://doi.org/10.1155/2015/160401.
Şahin Bayram, N. 2021. Criteria for statistical convergence with respect to power series methods. Positivity 25 (3): 1097–1105. https://doi.org/10.1007/s11117-020-00801-6.
Belen, C., M. Yıldırım, and C. Sümbül. 2020. On statistical and strong convergence with respect to a modulus function and a power series method. Filomat 34 (12): 3981–3993. https://doi.org/10.2298/FIL2012981B.
Boos, J. 2000. Classical and modern methods in summability. Oxford: Oxford University Press.
Çınar, S., and S. Yıldız. 2021. \(P\)-statistical summation process of sequences of convolution operators. Indian J. Pure Appl. Math.https://doi.org/10.1007/s13226-021-00156-y.
Dirik, F., and K. Demirci. 2010. Four-dimensional matrix transformation and the rate of \(A-\)statistical convergence of continuous functions. Comput. Math. Appl. 59 (8): 2976–2981. https://doi.org/10.1016/j.camwa.2010.02.015.
Duman, O. 2007. Regular matrix transformations and rates of convergence of positive linear operators. Calcolo 44 (3): 159–164. https://doi.org/10.1007/s10092-007-0134-z.
Duman, O., and E. Erkuş-Duman. 2011. Statistical Korovkin-type theory for matrix-valued functions. Stud. Sci. Math. Hung. 48: 489–508. https://doi.org/10.1556/sscmath.2011.1179.
Duman, O., and C. Orhan. 2005. Rates of \(A-\)statistical convergence of positive linear operators. Appl. Math. Lett. 18 (12): 1339–1344. https://doi.org/10.1016/j.aml.2005.02.029.
Demirci, K., and F. Dirik. 2010. A Korovkin type approximation theorem for double sequences of positive linear operators of two variables in \(A-\)statistical sense. Bull. Korean Math. Soc. 47 (4): 825–837. https://doi.org/10.4134/BKMS.2010.47.4.825.
Demirci, K., and S. Orhan. 2017. Statistical relative approximation on modular spaces. RM 71 (3): 1167–1184. https://doi.org/10.1007/s00025-016-0548-5.
Fast, H. 1951. Sur la convergence statistique. Colloquium Mathematicae 2: 241–244.
Kratz, W., and U. Stadtmüller. 1989. Tauberian theorems for \(J_{p}-\)summability. J. Math. Anal. Appl. 139: 362–371. https://doi.org/10.1016/0022-247X(89)90113-3.
Gadjiev, A.D., Orhan, C. 2002. Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics 32: 129-138. https://www.jstor.org/stable/44238888
Korovkin, P.P. 1960. Linear operators and approximation theory. Delhi: Hindustan Publishing Corporation.
Niven, I., and H.S. Zuckerman. 1980. An introduction to the theory of numbers. New York: John Wiley and Sons.
Serra-Capizzano, S. 1999. A Korovkin based approximation of multilevel Toeplitz matrices (with rectangular unstructured blocks) via multilevel trigonometric matrix spaces. SIAM J. Numer. Anal. 36 (6): 1831–1857. https://doi.org/10.1137/S0036142997322497.
Orhan, S., and K. Demirci. 2015. Statistical approximation by double sequences of positive linear operators on modular spaces. Positivity 19 (1): 23–36. https://doi.org/10.1007/s11117-014-0280-x.
Söylemez, D., and M. Ünver. 2021. Rates of power series statistical convergence of positive linear operators and power series statistical convergence of q-Meyer-König and Zeller operators. Lobachevskii J. Math. 42 (2): 426–434. https://doi.org/10.1134/S1995080221020189.
Stadtmüller, U., and A. Tali. 1999. On certain families of generalized Nörlund methods and power series methods. J. Math. Anal. Appl. 238: 44–66. https://doi.org/10.1006/jmaa.1999.6503.
Steinhaus, H. 1951. Sur la convergence ordinaire et la convergence asymtotique. Colloq. Math. 2: 73–74.
Ünver, M., and C. Orhan. 2019. Statistical convergence with respect to power series methods and applications to approximation theory. Numer. Funct. Anal. Optim. 40 (5): 535–547. https://doi.org/10.1080/01630563.2018.1561467.
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Demirci, K., Yıldız, S. & Çınar, S. Approximation of matrix-valued functions via statistical convergence with respect to power series methods. J Anal 30, 1179–1192 (2022). https://doi.org/10.1007/s41478-022-00400-6
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DOI: https://doi.org/10.1007/s41478-022-00400-6