Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

Sevda Yıldız

doi:10.17776/csj.831339

EN

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

Abstract

In the present paper, we introduce a new kind of convergence, called the statistical relative uniform convergence, for a double sequence of functions at a point, where the relative uniform convergence of the set of the neighborhoods of the given point is considered. By the use of the statistical relative uniform convergence, we investigate a Korovkin type approximation theorem which makes the proposed method stronger than the ones studied before. After that, we give an example using this new type of convergence. We also study the rate of convergence of the proposed convergence.

Keywords

References

[1] Fast H., Sur la Convergence Statistique, Colloq. Math., 2 (1951) 241-244.
[2] Steinhaus H., Sur la Convergence Ordinaire et la Convergence Asymtotique, Colloq. Math., 2 (1951) 73-74.
[3] Gadjiev A. D., Orhan C., Some Approximation Theorems via Statistical Convergence, Rocky Mountain J. Math., 32 (2002) 129-138.
[4] Korovkin P. P., Linear Operators and Approximation Theory. Delhi: Hindustan Publ. Co., (1960).
[5] Belen C., Yıldırım M., Generalized A-Statistical Convergence and a Korovkin Type Approximation Theorem for Double Sequences, Miskolc Mathematical Notes, 14 (1) (2013) 31-39.
[6] Demirci K., Dirik F., Four-Dimensional Matrix Transformation and Rate of A-Statistical Convergence of Periodic Functions, Math. Comput. Modelling, 52 (9-10) (2010) 1858-1866.
[7] Dirik F., Demirci K., Korovkin Type Approximation Theorem for Functions of Two Variables in Statistical Sense, Turk J Math, 34 (2010) 73-83.
[8] Ünver M., Orhan C., Statistical Convergence with Respect to Power Series Methods and Applications to Approximation Theory, Numerical Functional Analysis and Optimization, 40 (5) (2019) 535-547.

[9] Moore E. H., An Introduction to a Form of General Analysis, The New Haven Mathematical Colloquium, Yale University Press, New Haven, (1910).
[10] Chittenden E. W., On the Limit Functions of Sequences of Continuous Functions Converging Relatively Uniformly, Transactions of the AMS, 20 (1919) 179-184.
[11] Demirci K., Orhan S., Statistically Relatively Uniform Convergence of Positive Linear Operators, Results Math., 69 (2016) 359-367.
[12] Demirci K., Kolay B., A-Statistical Relative Modular Convergence of Positive Linear Operators, Positivity, 21 (3) (2017) 847-863.
[13] Demirci K., Orhan S., Kolay B., Relative Hemen Hemen Yakınsaklık ve Yaklaşım Teoremleri, Sinop Üniversitesi Fen Bilimleri Dergisi, 1 (2) (2016) 114-122.
[14] Demirci K., Yıldız S., Statistical Relative Uniform Convergence in Dually Residuated Lattice Totally Ordered Semigroups, Sarajevo J. Math., 15 (2) (2019) 227-237.
[15] Sahin P. O., Dirik F., Statistical Relative Uniform Convergence of Double Sequences of Positive Linear Operators, Appl. Math. E-Notes, 17 (2017) 207-220.
[16] Sahin P. O., Dirik F., Statistical Relative Equal Convergence of Double Function Sequences and Korovkin-Type Approximation Theorem, Applied Mathematics E-Notes, 19 (2019) 101-115.
[17] Yılmaz B., Demirci K., Orhan S., Relative Modular Convergence of Positive Linear Operators, Positivity, 20 (3) (2016) 565-577.
[18] Klippert J., Williams G., Uniform Convergence of a Sequence of Functions at a Point, Internat. J. Math. Ed. Sci. Tech., 33 (1) (2002) 51-58.
[19] Demirci K., Boccuto A., Yıldız S., Dirik F., Relative Uniform Convergence of a Sequence of Functions at a Point and Korovkin-Type Approximation Theorems, Positivity, 24 (1) (2020) 1-11.
[20] Boccuto A., Demirci K., Yıldız S., Abstract Korovkin-type theorems in the filter setting with respect to relative uniform convergence, Turkish Journal of Mathematics, 44 (4) (2020) 1238-1249.
[21] Dirik F., Demirci K., Yıldız S., Acu A. M., The Uniform Convergence of a Double Sequence of Functions at a Point and Korovkin-Type Approximation Theorems, Georgian Mathematical Journal, 1 (ahead-of-print) (2020)
[22] Pringsheim A., Zur Theorie der Zweifach Unendlichen Zahlenfolgen, Math. Ann., 53 (1) (1900) 289-321.
[23] Moricz F., Statistical Convergence of Multiple Sequences, Arch. Math., (Basel) 81 (2004) 82-89.
[24] Stancu D. D., A Method for Obtaining Polynomials of Bernstein Type of Two Variables, The American Mathematical Monthly, 70 (3) (1963) 260-264.
[25] DeVore R.A., Lorentz G.G., Constructive Approximation (Grund. Math. Wiss. 303). Berlin: Springer Verlag, (1993).
[26] Altomare F., Campiti M., Korovkin-Type Approximation Theory and its Applications. New York: Walter de Gruyter, (1994).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Sevda Yıldız ^*
0000-0002-4730-2271
Türkiye

Publication Date

March 29, 2021

Submission Date

November 25, 2020

Acceptance Date

March 3, 2021

Published in Issue

Year 2021 Volume: 42 Number: 1

DOI

https://doi.org/10.17776/csj.831339

IZ

https://izlik.org/JA92HB77TN

Cite

RIS / Bibtex

APA

Yıldız, S. (2021). Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory. Cumhuriyet Science Journal, 42(1), 123-131. https://doi.org/10.17776/csj.831339

Cited By

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method

Afrika Matematika

https://doi.org/10.1007/s13370-023-01078-0

Öz

Statistical relative uniform convergence of a double sequence of functions at a point and applications to approximation theory

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite

Cited By

Approximation via statistical relative uniform convergence of sequences of functions at a point with respect to power series method