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Year 2022, , 72 - 81, 01.06.2022
https://doi.org/10.36753/mathenot.992220

Abstract

References

  • [1] Acar Is ̧ler, N.: Bernstein operator approach for solving linear differential equations. Mathematical Sciences and Applications E-Notes. 9(1), 28-35 (2021).
  • [2] Altomare, F., Campiti, M.: Korovkin type approximation theory and its applications. De Gruyter Studies in Mathematics. 17, Walter de Gruyter&Co. Berlin (1994).
  • [3] Atlıhan, Ö.G., Orhan, C.: Summation process of positive linear operators. Computers and Mathematics with Applications. 56, 1188-1195 (2008).
  • [4] Atlıhan, Ö.G., Ünver, M., Duman, O.: Korovkin theorems on weighted spaces: revisited. Periodica Mathematica Hungarica. 75, 201-209 (2017).
  • [5] Bernstein, S.N.: Demonstration du theoreme de Weierstrass fondee. Communications of the Kharkov Mathematical Society. 13, 1-2 (1912).
  • [6] Boos, J.: Classical and Modern Methods in Summability. Oxford University Press. Oxford (2000).
  • [7] Chlodowsky, I.: Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M. S. Bernstein. Compositio Mathematica. 4, 380-393 (1937).
  • [8] Duman, O., Orhan, C.: Statistical approximation by positive linear operators. Studia Mathematica. 161, 187-197 (2004).
  • [9] Fast, H.: Sur la convergence statistique. Colloquium Mathematicum. 2(3/4), 241–244 (1951).
  • [10] Fridy, J. A.: On statistical convergence. Analysis 5, 301–313 (1985).
  • [11] Gadjiev,A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics. 32, 129-137 (2002).
  • [12] Karsli, H., Gupta, V.: Some approximation properties of q-Chlodowsky operators. Applied Mathematics and Compu- tation. 195, 220-229 (2008).
  • [13] Korovkin,P.P.: Onconvergenceoflinearpositiveoperatorsinthespaceofcontinuousfunctions.DokladyAkademii nauk SSSR. 90, 961-964 (1953).
  • [14] Kratz, W., Stadtmüller, U.: Tauberian theorems for Jp-summability. Journal of Mathematical Analysis and Applica- tions. 139, 362-371 (1989).
  • [15] Lorentz, G. G.: Bernstein Polynomials. University of Toronto Press. Toronto (1953).
  • [16] Lupas ̧, A.: A q-analogue of the Bernstein operator, seminar on numerical and statistical calculus. University of Cluj-Napoc. 9, 85-92 (1987).
  • [17] Oruc, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. Journal of Approximation Theory. 117, 301-313 (2002).
  • [18] Ostrovska,S:,Ontheimprovementanalyticpropertiesunderthelimitq-Bernsteinoperator.JournalofApproximation Theory. 138, 37-53 (2006).
  • [19] Ostrovska, S.: q-Bernstein polynomials and their iterates. Journal of Approximation Theory. 123, 232-255 (2003).
  • [20] Özgüç, I., Tas ̧, E.: A Korovkin-type approximation theorem and power series method. Results in Mathematics. 69(3) (2016).
  • [21] Phillips,G.M.:Bernstein polynomials basedon the q-integers. Annals of Numerical Mathematics. 4, 511-518 (1997).
  • [22] Salat, T.: On statistically convergent sequences of real numbers. Mathematica Slovaca. 30(2), 139-150 (1980).
  • [23] Tas ̧, E., Orhan, C., Yurdakadim, T.: The Stancu-Chlodowsky operators based on q-Calculus. In: Proceedings of the 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), Sept 21–27 2013, Rhodes Island, GREECE. AIP Conference Proceeding. 1558, 1152-1155 (2013).
  • [24] Tas ̧, E., Yurdakadim, T., Atlihan, Ö.G..: Korovkin type approximation theorems in weighted spaces via power series method. Operators and Matrices. 12, 529-535 (2018).
  • [25] Uysal, G., Yılmaz, B.: On convergence of partial derivatives of multidimensional convolution operators. Mathematical Sciences and Applications E-Notes. 9(1), 9-21 (2021).
  • [26] Ünver, M., Orhan, C.: Statistical convergence with respect to power series methods and applications to approximation theory. Numerical Functional Analysis and Optimization. 40(5), 535-547 (2019).
  • [27] Wang,H.:Voronovskaja-typeformulasandsaturationofconvergenceforq-Bernsteinpolynomialsfor0<q<1.Journal of Approximation Theory. 145, 182-195 (2007).

Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method

Year 2022, , 72 - 81, 01.06.2022
https://doi.org/10.36753/mathenot.992220

Abstract

Many results which are obtained or unable to obtained by classical calculus have also been studied by q-calculus. It is effective to use q-calculus since it acts as a bridge between mathematics and physics. The q-analog of Chlodowsky operators has been introduced and the approximation properties of these operators have been studied in [12]. Then in [23], the q-analog of Stancu-Chlodowsky operators has been introduced and some approximation results of these operators have been studied via A-statistical convergence which is a more general setting.In this paper, we present the approximation properties of q-Chlodowsky operators via statistical convergence with respect to power series method. It is noteworthy to mention that statistical convergence and statistical convergence with respect to power series method are incompatible.

References

  • [1] Acar Is ̧ler, N.: Bernstein operator approach for solving linear differential equations. Mathematical Sciences and Applications E-Notes. 9(1), 28-35 (2021).
  • [2] Altomare, F., Campiti, M.: Korovkin type approximation theory and its applications. De Gruyter Studies in Mathematics. 17, Walter de Gruyter&Co. Berlin (1994).
  • [3] Atlıhan, Ö.G., Orhan, C.: Summation process of positive linear operators. Computers and Mathematics with Applications. 56, 1188-1195 (2008).
  • [4] Atlıhan, Ö.G., Ünver, M., Duman, O.: Korovkin theorems on weighted spaces: revisited. Periodica Mathematica Hungarica. 75, 201-209 (2017).
  • [5] Bernstein, S.N.: Demonstration du theoreme de Weierstrass fondee. Communications of the Kharkov Mathematical Society. 13, 1-2 (1912).
  • [6] Boos, J.: Classical and Modern Methods in Summability. Oxford University Press. Oxford (2000).
  • [7] Chlodowsky, I.: Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M. S. Bernstein. Compositio Mathematica. 4, 380-393 (1937).
  • [8] Duman, O., Orhan, C.: Statistical approximation by positive linear operators. Studia Mathematica. 161, 187-197 (2004).
  • [9] Fast, H.: Sur la convergence statistique. Colloquium Mathematicum. 2(3/4), 241–244 (1951).
  • [10] Fridy, J. A.: On statistical convergence. Analysis 5, 301–313 (1985).
  • [11] Gadjiev,A.D., Orhan, C.: Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics. 32, 129-137 (2002).
  • [12] Karsli, H., Gupta, V.: Some approximation properties of q-Chlodowsky operators. Applied Mathematics and Compu- tation. 195, 220-229 (2008).
  • [13] Korovkin,P.P.: Onconvergenceoflinearpositiveoperatorsinthespaceofcontinuousfunctions.DokladyAkademii nauk SSSR. 90, 961-964 (1953).
  • [14] Kratz, W., Stadtmüller, U.: Tauberian theorems for Jp-summability. Journal of Mathematical Analysis and Applica- tions. 139, 362-371 (1989).
  • [15] Lorentz, G. G.: Bernstein Polynomials. University of Toronto Press. Toronto (1953).
  • [16] Lupas ̧, A.: A q-analogue of the Bernstein operator, seminar on numerical and statistical calculus. University of Cluj-Napoc. 9, 85-92 (1987).
  • [17] Oruc, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. Journal of Approximation Theory. 117, 301-313 (2002).
  • [18] Ostrovska,S:,Ontheimprovementanalyticpropertiesunderthelimitq-Bernsteinoperator.JournalofApproximation Theory. 138, 37-53 (2006).
  • [19] Ostrovska, S.: q-Bernstein polynomials and their iterates. Journal of Approximation Theory. 123, 232-255 (2003).
  • [20] Özgüç, I., Tas ̧, E.: A Korovkin-type approximation theorem and power series method. Results in Mathematics. 69(3) (2016).
  • [21] Phillips,G.M.:Bernstein polynomials basedon the q-integers. Annals of Numerical Mathematics. 4, 511-518 (1997).
  • [22] Salat, T.: On statistically convergent sequences of real numbers. Mathematica Slovaca. 30(2), 139-150 (1980).
  • [23] Tas ̧, E., Orhan, C., Yurdakadim, T.: The Stancu-Chlodowsky operators based on q-Calculus. In: Proceedings of the 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), Sept 21–27 2013, Rhodes Island, GREECE. AIP Conference Proceeding. 1558, 1152-1155 (2013).
  • [24] Tas ̧, E., Yurdakadim, T., Atlihan, Ö.G..: Korovkin type approximation theorems in weighted spaces via power series method. Operators and Matrices. 12, 529-535 (2018).
  • [25] Uysal, G., Yılmaz, B.: On convergence of partial derivatives of multidimensional convolution operators. Mathematical Sciences and Applications E-Notes. 9(1), 9-21 (2021).
  • [26] Ünver, M., Orhan, C.: Statistical convergence with respect to power series methods and applications to approximation theory. Numerical Functional Analysis and Optimization. 40(5), 535-547 (2019).
  • [27] Wang,H.:Voronovskaja-typeformulasandsaturationofconvergenceforq-Bernsteinpolynomialsfor0<q<1.Journal of Approximation Theory. 145, 182-195 (2007).
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Halime Taşer 0000-0003-2338-9242

Tuğba Yurdakadim 0000-0003-2522-6092

Publication Date June 1, 2022
Submission Date September 7, 2021
Acceptance Date November 29, 2021
Published in Issue Year 2022

Cite

APA Taşer, H., & Yurdakadim, T. (2022). Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method. Mathematical Sciences and Applications E-Notes, 10(2), 72-81. https://doi.org/10.36753/mathenot.992220
AMA Taşer H, Yurdakadim T. Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method. Math. Sci. Appl. E-Notes. June 2022;10(2):72-81. doi:10.36753/mathenot.992220
Chicago Taşer, Halime, and Tuğba Yurdakadim. “Approximation for $q$-Chlodowsky Operators via Statistical Convergence With Respect to Power Series Method”. Mathematical Sciences and Applications E-Notes 10, no. 2 (June 2022): 72-81. https://doi.org/10.36753/mathenot.992220.
EndNote Taşer H, Yurdakadim T (June 1, 2022) Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method. Mathematical Sciences and Applications E-Notes 10 2 72–81.
IEEE H. Taşer and T. Yurdakadim, “Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method”, Math. Sci. Appl. E-Notes, vol. 10, no. 2, pp. 72–81, 2022, doi: 10.36753/mathenot.992220.
ISNAD Taşer, Halime - Yurdakadim, Tuğba. “Approximation for $q$-Chlodowsky Operators via Statistical Convergence With Respect to Power Series Method”. Mathematical Sciences and Applications E-Notes 10/2 (June 2022), 72-81. https://doi.org/10.36753/mathenot.992220.
JAMA Taşer H, Yurdakadim T. Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method. Math. Sci. Appl. E-Notes. 2022;10:72–81.
MLA Taşer, Halime and Tuğba Yurdakadim. “Approximation for $q$-Chlodowsky Operators via Statistical Convergence With Respect to Power Series Method”. Mathematical Sciences and Applications E-Notes, vol. 10, no. 2, 2022, pp. 72-81, doi:10.36753/mathenot.992220.
Vancouver Taşer H, Yurdakadim T. Approximation for $q$-Chlodowsky Operators via Statistical Convergence with Respect to Power Series Method. Math. Sci. Appl. E-Notes. 2022;10(2):72-81.

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