Loading [a11y]/accessibility-menu.js
Research Article
BibTex RIS Cite

Year 2021, Volume: 42 Issue: 1, 123 - 131, 29.03.2021

Sevda Yıldız

https://doi.org/10.17776/csj.831339
Cited By: 1

Abstract

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).

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

Year 2021, Volume: 42 Issue: 1, 123 - 131, 29.03.2021

Sevda Yıldız

https://doi.org/10.17776/csj.831339
Cited By: 1

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

Double sequence Korovkin theorem rate of convergence statistical relative uniform convergence

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).
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Sevda Yıldız 0000-0002-4730-2271

Publication Date March 29, 2021
Submission Date November 25, 2020
Acceptance Date March 3, 2021
Published in Issue Year 2021Volume: 42 Issue: 1

Cite

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