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Approximation theorems via Pp-statistical convergence on weighted spaces

  • Sevda Yıldız EMAIL logo and Nilay Şahin Bayram
Published/Copyright: June 24, 2024
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Abstract

In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp-statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces.

  1. Communicated by Gregor Dolinar

References

[1] Akdağ, S.: Summation process of positive linear operators in two-dimensional weighted spaces, Math. Slovaca 65(6) (2015), 1475–1490.10.1515/ms-2015-0100Search in Google Scholar

[2] Atlihan, O. G.—Ünver, M.—Duman, O.: Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar. 75 (2017), 201–209.10.1007/s10998-017-0187-ySearch in Google Scholar

[3] Bardaro, C.—Boccuto, A.—Demirci, K.—Mantellini, I.—Orhan, S.: Korovkin-type theorems for modular Ψ -A-statistical convergence, J. Funct. Spaces 2015 (2015), Art. ID 160401.Search in Google Scholar

[4] Baron, S.—Stadtmüller, U.: Tauberian theorems for power series methods applied to double sequences, J. Math. Anal. Appl. 211(2) (1997), 574–589.10.1006/jmaa.1997.5473Search in Google Scholar

[5] Belen, C.—Yildirim, M.—Sümbül, C.: On statistical and strong convergence with respect to a modulus function and a power series method, Filomat 34(12) (2020), 3981–3993.10.2298/FIL2012981BSearch in Google Scholar

[6] Cao, F.—Liu, Y.: Approximation theorems by positive linear operators in weighted spaces, Positivity 15 (2011), 87–103.10.1007/s11117-009-0043-2Search in Google Scholar

[7] Çinar, S.—Yildiz, S.: P-statistical summation process of sequences of convolution operators, Indian J. Pure Appl. Math. 53 (2021), 648–659.10.1007/s13226-021-00156-ySearch in Google Scholar

[8] Demirci, K.: On lacunary statistical limit points, Demonstr. Math. 35(1) (2002), 93–102.10.1515/dema-2002-0111Search in Google Scholar

[9] Demirci, K.—Dirik, F.: Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions, Math. Comput. Model. 52(9–10) (2010), 1858–1866.10.1016/j.mcm.2010.07.015Search in Google Scholar

[10] Demirci, K.—Dirik, F.: Approximation for periodic functions via statistical σ -convergence, Math. Commun. 16(1) (2011), 77–84.Search in Google Scholar

[11] Demirci, K.—Dirik, F.—Yildiz, S.: Approximation via equi-statistical convergence in the sense of power series method, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116(2) (2022), Art. No. 65.10.1007/s13398-021-01191-4Search in Google Scholar

[12] Demirci, K.—ĐJ určić, D.—Kočinac, L.—Yildiz, S.: A theory of variations via P–statistical convergence, Publ. Inst. Math. (Beograd) (N.S.) 110(124) (2021), 11-27.10.2298/PIM2124011DSearch in Google Scholar

[13] Demirci, K.—Orhan, S.: Statistical relative approximation on modular spaces, Results Math. 71(3) (2017), 1167–1184.10.1007/s00025-016-0548-5Search in Google Scholar

[14] Demirci, K.—Yildiz, S.—Dirik, F.: Approximation via power series method in two-dimensional weighted spaces, Bull. Malays. Math. Sci. Soc. 43(6) (2020), 3871–3883.10.1007/s40840-020-00902-1Search in Google Scholar

[15] Dirik, F.—Demirci, K.: Korovkin-type approximation theorem for functions of two variables in statistical sense, Turkish J. Math. 34(1) (2010), 73–83.10.3906/mat-0802-21Search in Google Scholar

[16] Dirik, F.—Duman, O.—Demirci, K.: Approximation in statistical sense to B-continuous functions by positive linear operators, Stud. Sci. Math. Hung. 47(3) (2010), 289–298.10.1556/sscmath.2009.1129Search in Google Scholar

[17] Fast, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 24–244.10.4064/cm-2-3-4-241-244Search in Google Scholar

[18] Fridy, J.—Orhan, C.: Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125(12) (1997), 3625–3631.10.1090/S0002-9939-97-04000-8Search in Google Scholar

[19] Gadjiev, A. D.: Theorems of Korovkin-type, Mat. Zametki 20 (1976), 781–786.10.1007/BF01146928Search in Google Scholar

[20] Gadjiev, A. D.—Orhan, C.: Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129–138.10.1216/rmjm/1030539612Search in Google Scholar

[21] Gönül Bilgin, N.—Özgür, N.: Approximation by two dimensional Gadjiev-Ibragimov type operators, Ikonion Journal of Mathematics 1(1) (2019), 1–10.Search in Google Scholar

[22] Karakuş, S.—Demirci, K.—Duman, O.: Statistical approximation by positive linear operators on modular spaces, Positivity 14(2) (2010), 321–334.10.1007/s11117-009-0020-9Search in Google Scholar

[23] Korovkin, P. P.: Linear Operators and Approximation Theory, Hindustan Publ. Co., Delhi, 1960.Search in Google Scholar

[24] Miller, H. I.: A measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347(5) (1995), 1811–1819.10.1090/S0002-9947-1995-1260176-6Search in Google Scholar

[25] Moricz, F.: Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2004), 82–89.10.1007/s00013-003-0506-9Search in Google Scholar

[26] Pringsheim, A.: Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289–321.10.1007/BF01448977Search in Google Scholar

[27] Söylemez, D.: A Korovkin-type approximation theorem for Balázs type Bleimann, Butzer and Hahn operators via power series statistical convergence, Math. Slovaca 72(1) (2022), 153–164.10.1515/ms-2022-0011Search in Google Scholar

[28] Söylemez, D.—Ünver, M.: 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) (2021), 426–434.10.1134/S1995080221020189Search in Google Scholar

[29] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymtotique, Colloq. Math. 2 (1951), 73–74.10.4064/cm-2-2-98-108Search in Google Scholar

[30] Şahin Bayram, N.: Criteria for statistical convergence with respect to power series methods, Positivity 25(3) (2021), 1097–1105.10.1007/s11117-020-00801-6Search in Google Scholar

[31] Şahin Bayram, N.—Yildiz, S.: Approximation by statistical convergence with respect to power series methods, Hacet. J. Math. Stat. 51(4) (2022), 1108–1120.10.15672/hujms.1022072Search in Google Scholar

[32] Taşer, H.—Yurdakadim, T.: Approximation for q-Chlodowsky operators via statistical convergence with respect to power series method, Mathematical Sciences and Applications E-Notes 10(2) (2022), 72–81.10.36753/mathenot.992220Search in Google Scholar

[33] Ünver, M.—Orhan, C.: Statistical convergence with respect to power series methods and applications to approximation theory, Numer. Funct. Anal. Optim. 40(5) (2019), 535–547.10.1080/01630563.2018.1561467Search in Google Scholar

[34] Ünver, M.: Abel transforms of positive linear operators on weighted spaces, Bull. Belg. Math. Soc. 21(5) (2014), 813–822.10.36045/bbms/1420071855Search in Google Scholar

[35] Ünver, M.—Şahin Bayram, N.: On statistical convergence with respect to power series methods, Positivity 26(3) (2022), Art. No. 55.10.1007/s11117-022-00921-1Search in Google Scholar

[36] Yildiz, S.: Korovkin-type approximation via statistical e-convergence on two dimensional weighted spaces, Math. Slovaca 71(5) (2021), 1167–1178.10.1515/ms-2021-0046Search in Google Scholar

[37] Yildiz, S.: Approximation via statistical Ka2 -convergence on two-dimensional weighted spaces, Rev. Un. Mat. Argentina 63(1) (2022), 21–39.10.33044/revuma.2010Search in Google Scholar

[38] Yildiz, S.—Demirci, K.—Dirik, F.: Korovkin theory via Pp-statistical relative modular convergence for double sequences, Rend. Circ. Mat. Palermo (2) 72 (2023), 1125–1141.10.1007/s12215-021-00681-zSearch in Google Scholar

Received: 2022-10-20
Accepted: 2023-11-10
Published Online: 2024-06-24
Published in Print: 2024-06-25

© 2024 Mathematical Institute Slovak Academy of Sciences

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