Approximation via statistical $K^2_a$-convergence on two-dimensional weighted spaces
DOI:
https://doi.org/10.33044/revuma.2010Abstract
We give a non-regular statistical summability method named statistical $K^2_a$-convergence and prove a Korovkin type approximation theorem for this new and interesting convergence method on two-dimensional weighted spaces. We also study the rate of statistical $K^2_a$-convergence by using the weighted modulus of continuity and afterwards we present a non-trivial application.
Downloads
References
S. Akdağ, Summation process of positive linear operators in two-dimensional weighted spaces, Math. Slovaca 65 (2015), no. 6, 1475–1490. MR 3458994.
G. Atlihan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56 (2008), no. 5, 1188–1195. MR 2437286.
G. Atlihan, M. Ünver and O. Duman, Korovkin theorems on weighted spaces: revisited, Period. Math. Hungar. 75 (2017), no. 2, 201–209. MR 3718515.
C. Belen, M. Mursaleen and M. Yildirim, Statistical $A$-summability of double sequences and a Korovkin type approximation theorem, Bull. Korean Math. Soc. 49 (2012), no. 4, 851–861. MR 2978422.
F. Cao and Y. Liu, Approximation theorems by positive linear operators in weighted spaces, Positivity 15 (2011), no. 1, 87–103. MR 2782749.
K. Demirci and F. Dirik, A Korovkin type approximation theorem for double sequences of positive linear operators of two variables in $A$-statistical sense, Bull. Korean Math. Soc. 47 (2010), no. 4, 825–837. MR 2681469.
F. Dirik and K. Demirci, Korovkin type approximation theorem for functions of two variables in statistical sense, Turkish J. Math. 34 (2010), no. 1, 73–83. MR 2654417.
O. Duman and C. Orhan, Statistical approximation by positive linear operators, Studia Math. 161 (2004), no. 2, 187–197. MR 2033235.
H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244 (1952). MR 0048548.
A. D. Gadžiev, Theorems of the type of P. P. Korovkin's theorems, Mat. Zametki 20 (1976), no. 5, 781–786. MR 0493081.
N. Gönül Bilgin and N. Özgür, Approximation by two dimensional Gadjiev-Ibragimov type operators. Ikonion J. Math., 1 (2019), no. 1, 1–10.
M. Lazić and V. Jovović, Cauchy's operators and convergence methods, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 4 (1993), 81–87. MR 1295605.
F. Móricz, Tauberian theorems for double sequences that are statistically summable $(C,1,1)$, J. Math. Anal. Appl. 286 (2003), no. 1, 340–350. MR 2009641.
F. Móricz, Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2003), no. 1, 82–89. MR 2002719.
F. Móricz and B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 2, 283–294. MR 0948914.
P. Okçu Şahin and F. Dirik, Statistical relative uniform convergence of double sequences of positive linear operators, Appl. Math. E-Notes 17 (2017), 207–220. MR 3743903.
S. Orhan and K. Demirci, $K_a$-convergence and Korovkin type approximation, Period. Math. Hungar. 77 (2018), no. 1, 108–118. MR 3842974.
A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), no. 3, 289–321. MR 1511092.
Steinhaus, H. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951), 73–74.
E. Taş, T. Yurdakadim and O. G. Atlihan, Korovkin type approximation theorems in weighted spaces via power series method, Oper. Matrices 12 (2018), no. 2, 529–535. MR 3812188.
S. Yıldız, $K_a$-convergence for double sequences and Korovkin type approximation, Appl. Math. E-Notes 21 (2021), 62–71. MR 4242237.
S. Yıldız, F. Dirik and K. Demirci, Korovkin type approximation theorem via $K_a$-convergence on weighted spaces, Math. Methods Appl. Sci. 42 (2019), no. 16, 5371–5382. MR 4033321.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Sevda Yıldız
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal. The Journal may retract the paper after publication if clear evidence is found that the findings are unreliable as a result of misconduct or honest error.