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STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS

Year 2017, Volume: 66 Issue: 2, 263 - 275, 01.08.2017

Vishnu Narayan Mıshra Prashantkumar Patel

https://doi.org/10.1501/Commua1_0000000817

Abstract

In this work, we investigate weighted properties of q-Durrmeyer-Stancu operators. We also give some corrections in limit of q-Durrmeyer-Stancu operators defined in [1] and discuss their convergence properties

Keywords

Durrmeyer operators Korovkin type theorems rate of the weighted-statistical convergent

References

  • V. N. Mishra, P. Patel, A short note on approximation properties of Stancu generalization of q-Durrmeyer operators, Fixed Point Theory and Application 2013 (1) (2013) 84.
  • H. Fast, Sur la convergence statistique, Colloquium Mathematicum 2 (1951) 241ï£ ¡244.
  • Ö. Dalmanoglu, O. Dogru, On statistical approximation properties of Kantorovich type q- Bernstein operators, Mathematical and Computer Modelling 52 (5) (2010) 760–771.
  • V. N. Mishra, K. Khatri, L. N. Mishra, Statistical approximation by Kantorovich-type discrete q-Beta operators, Advances in Diğerence Equations 2013:345, DOI: 10.1186/10.1186/1687- 1847-2013-345. (1) (2013) 1–15.
  • M. Örkcü, Approximation properties of bivariate extension of q-Szász-Mirakjan-Kantorovich operators, Journal of Inequalities and Applications 2013 (1) (2013) 1–10.
  • P. Patel, V. N. Mishra, Jain-Baskakov operators and its diğerent generalization, Acta Math- ematica Vietnamica 40 (4) (2014) 715–746.
  • Q. Lin, Statistical approximation of modi…ed Schurer-type q-Bernstein Kantorovich operators, Journal of Inequalities and Applications 2014 (1) (2014) 465.
  • H. Aktu¼glu, Korovkin type approximation theorems proved via statistical convergence, Journal of Computational and Applied Mathematics 259 (2014) 174–181.
  • V. Karakaya, A. Karaisa, Korovkin type approximation theorems for weighted statistical convergence, Bulletin of Mathematical Sciences 5 (2) (2015) 159–169.
  • V. Karakaya, Weighted statistical convergence, Iranian Journal of Science and Technology (Sciences) 33 (3) (2009) 219–223.
  • M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Applied Mathematics and Computation 218 (18) (2012) 9132–9137.
  • G. M. Phillips, Bernstein polynomials based on the q-integers, Annals of Numerical Mathe- matics 4 (1996) 511–518.
  • S. Ostrovska, The …rst decade of the q-Bernstein polynomials: results and perspectives, Journal on Mathematical Analysis Approximation Theory 2 (1) (2007) 35–51.
  • A. Lupa¸s, A q-analogue of the Bernstein operator, in: University of Cluj-Napoca, Seminar on numerical and statistical calculus, Vol. 9, 1987.
  • V. Gupta, W. Heping, The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Mathematical Methods in the Applied Sciences 31 (16) (2008) 1946–1955.
  • Z. Finta, V. Gupta, Approximation by q-Durrmeyer operators, Journal of Applied Mathe- matics and Computing 29 (1) (2009) 401–415.
  • V. N. Mishra, P. Patel, The Durrmeyer type modi…cation of the q-Baskakov type operators with two parameter and , Numerical Algorithms 67 (4) (2014) 753–769.
  • B. ·Ibrahim, E. Ibikli, The approximation properties of generalized Bernstein polynomials of two variables, Applied Mathematics and Computation 156 (2) (2004) 367–380.
  • B. ·Ibrahim, Ç. Atakut, On Stancu type generalization of q-Baskakov operators, Mathematical and Computer Modelling 52 (5) (2010) 752–759.
  • B. ·Ibrahim, Approximation by Stancu–Chlodowsky polynomials, Computers & Mathematics with Applications 59 (1) (2010) 274–282.
  • Ç. Atakut, B. ·Ibrahim, Stancu type generalization of the Favard–Szàsz operators, Applied Mathematics Letters 23 (12) (2010) 1479–1482.
  • B. ·Ibrahim, On the approximation properties of two-dimensional q-Bernstein-Chlodowsky polynomials, Mathematical Communications 14 (2) (2009) 255–269.
  • G. ·Içöz, R. Mohapatra, Weighted approximation properties of Stancu type modi…cation of q-Szász-Durrmeyer operators, Communications Series A1 Mathematics & Statistics 65 (1).
  • G. Icoz, B. Cekim, Durrmeyer-type generalization of Mittag-Le- er operators, Gazi University Journal of Science 28 (2) (2015) 259–263.
  • O. Do¼gru, G. Içöz, K. Kanat, On the rates of convergence of the q-Lupa¸s-Stancu operators, Filomat 30 (5).
  • O. Dalmanoglu, S. K. Serenbay, Approximation by Chlodowsky type q-Jakimovski-Leviatan operators.
  • A. Aral, A. Karaisa, Some approximation properties of Kontorovich variant of Chlodowsky operators based on q-ineteger, Commun. Fac. Sci. Univ. Ank. SÈr. A1 Math. Stat. 65 (2) (2016) 97–119.
  • V. N. Mishra, K. Khatri, L. N. Mishra, Deepmala, Inverse result in simultaneous approxi- mation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications 2013 (1) (2013) 586.
  • R. B. Gandhi, Deepmala, V. N. Mishra, Local and global results for modi…ed Szàsz-Mirakjan operators, Math. Method. Appl. Sci.DOI: 10.1002/mma.4171.
  • A. R. Gairola, Deepmala, L. N. Mishra, Rate of approximation by …nite iterates of q- Durrmeyer operators, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 86 (2) (2016) 229–234.
  • A. Wa…, N. Rao, R. Deepmala, Approximation properties by generalized Baskakov Kantorovich Stancu type operators, Applied Mathematics & Information Sciences Letters 4 (3) (2016) 111–118.
  • K. K. Singh, A. R. Gairola, Deepmala, Approximation theorems for q-analouge of a linear positive operator by A. Lupa¸s, International Journal of Analysis and Applications 12 (1) (2016) 30–37.
  • V. N. Mishra, P. Sharma, L. N. Mishra, On statistical approximation properties of q- Baskakov–Szász–Stancu operators, Journal of the Egyptian Mathematical Society 24 (3) (2016) 396–401.
  • V. N. Mishra, H. Khan, K. Khatri, L. N. Mishra, Hypergeometric representation for Baskakov- Durrmeyer-Stancu type operators, Bulletin of Mathematical Analysis and Applications 5 (3) (2013) 18–26.
  • V. N. Mishra, K. Khatri, L. N. Mishra, On simultaneous approximation for Baskakov- Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Physical Sciences 24 (3A) (2012) 567–577.
  • V. N. Mishra, P. Patel, On generalized integral Bernstein operators based on q-integers, Applied Mathematics and Computation 242 (2014) 931–944.
  • V. Kac, P. Cheung, Quantum calculus, Springer, 2002.
  • A. Il’inskii, S. Ostrovska, Convergence of generalized Bernstein polynomials, Journal of Ap- proximation Theory 116 (1) (2002) 100–112.
  • V. Gupta, Some approxmation properties of q-Durrmeyer operators, Applied Mathematics and Computation 197 (2008) 172–178.
  • G. G. Lorentz, Bernstein polynomials, American Mathematical Society, 1953.
  • X. M. Zeng, D. Lin, L. Li, A note on approximation properties of q-Durrmeyer operators, Applied Mathematics and Computation 216 (3) (2010) 819–821.
  • Current address : Vishnu Narayan Mishra (Corresponding author): Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Ichchhanath Mahadev Du- mas Road, Surat-395 007 (Gujarat), India, L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite - Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001, India
  • E-mail address : vishnu_narayanmishra@yahoo.co.in; vishnunarayanmishra@gmail.com
  • Current address : Prashantkumar Patel: Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat-395 007 (Gu- jarat), India, Department of Mathematics, St. Xavier’s College(Autonomous), Ahmedabad-380 009 (Gujarat), India
  • E-mail address : prashant225@gmail.com

Year 2017, Volume: 66 Issue: 2, 263 - 275, 01.08.2017

Vishnu Narayan Mıshra Prashantkumar Patel

https://doi.org/10.1501/Commua1_0000000817

Abstract

References

  • V. N. Mishra, P. Patel, A short note on approximation properties of Stancu generalization of q-Durrmeyer operators, Fixed Point Theory and Application 2013 (1) (2013) 84.
  • H. Fast, Sur la convergence statistique, Colloquium Mathematicum 2 (1951) 241ï£ ¡244.
  • Ö. Dalmanoglu, O. Dogru, On statistical approximation properties of Kantorovich type q- Bernstein operators, Mathematical and Computer Modelling 52 (5) (2010) 760–771.
  • V. N. Mishra, K. Khatri, L. N. Mishra, Statistical approximation by Kantorovich-type discrete q-Beta operators, Advances in Diğerence Equations 2013:345, DOI: 10.1186/10.1186/1687- 1847-2013-345. (1) (2013) 1–15.
  • M. Örkcü, Approximation properties of bivariate extension of q-Szász-Mirakjan-Kantorovich operators, Journal of Inequalities and Applications 2013 (1) (2013) 1–10.
  • P. Patel, V. N. Mishra, Jain-Baskakov operators and its diğerent generalization, Acta Math- ematica Vietnamica 40 (4) (2014) 715–746.
  • Q. Lin, Statistical approximation of modi…ed Schurer-type q-Bernstein Kantorovich operators, Journal of Inequalities and Applications 2014 (1) (2014) 465.
  • H. Aktu¼glu, Korovkin type approximation theorems proved via statistical convergence, Journal of Computational and Applied Mathematics 259 (2014) 174–181.
  • V. Karakaya, A. Karaisa, Korovkin type approximation theorems for weighted statistical convergence, Bulletin of Mathematical Sciences 5 (2) (2015) 159–169.
  • V. Karakaya, Weighted statistical convergence, Iranian Journal of Science and Technology (Sciences) 33 (3) (2009) 219–223.
  • M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Applied Mathematics and Computation 218 (18) (2012) 9132–9137.
  • G. M. Phillips, Bernstein polynomials based on the q-integers, Annals of Numerical Mathe- matics 4 (1996) 511–518.
  • S. Ostrovska, The …rst decade of the q-Bernstein polynomials: results and perspectives, Journal on Mathematical Analysis Approximation Theory 2 (1) (2007) 35–51.
  • A. Lupa¸s, A q-analogue of the Bernstein operator, in: University of Cluj-Napoca, Seminar on numerical and statistical calculus, Vol. 9, 1987.
  • V. Gupta, W. Heping, The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Mathematical Methods in the Applied Sciences 31 (16) (2008) 1946–1955.
  • Z. Finta, V. Gupta, Approximation by q-Durrmeyer operators, Journal of Applied Mathe- matics and Computing 29 (1) (2009) 401–415.
  • V. N. Mishra, P. Patel, The Durrmeyer type modi…cation of the q-Baskakov type operators with two parameter and , Numerical Algorithms 67 (4) (2014) 753–769.
  • B. ·Ibrahim, E. Ibikli, The approximation properties of generalized Bernstein polynomials of two variables, Applied Mathematics and Computation 156 (2) (2004) 367–380.
  • B. ·Ibrahim, Ç. Atakut, On Stancu type generalization of q-Baskakov operators, Mathematical and Computer Modelling 52 (5) (2010) 752–759.
  • B. ·Ibrahim, Approximation by Stancu–Chlodowsky polynomials, Computers & Mathematics with Applications 59 (1) (2010) 274–282.
  • Ç. Atakut, B. ·Ibrahim, Stancu type generalization of the Favard–Szàsz operators, Applied Mathematics Letters 23 (12) (2010) 1479–1482.
  • B. ·Ibrahim, On the approximation properties of two-dimensional q-Bernstein-Chlodowsky polynomials, Mathematical Communications 14 (2) (2009) 255–269.
  • G. ·Içöz, R. Mohapatra, Weighted approximation properties of Stancu type modi…cation of q-Szász-Durrmeyer operators, Communications Series A1 Mathematics & Statistics 65 (1).
  • G. Icoz, B. Cekim, Durrmeyer-type generalization of Mittag-Le- er operators, Gazi University Journal of Science 28 (2) (2015) 259–263.
  • O. Do¼gru, G. Içöz, K. Kanat, On the rates of convergence of the q-Lupa¸s-Stancu operators, Filomat 30 (5).
  • O. Dalmanoglu, S. K. Serenbay, Approximation by Chlodowsky type q-Jakimovski-Leviatan operators.
  • A. Aral, A. Karaisa, Some approximation properties of Kontorovich variant of Chlodowsky operators based on q-ineteger, Commun. Fac. Sci. Univ. Ank. SÈr. A1 Math. Stat. 65 (2) (2016) 97–119.
  • V. N. Mishra, K. Khatri, L. N. Mishra, Deepmala, Inverse result in simultaneous approxi- mation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications 2013 (1) (2013) 586.
  • R. B. Gandhi, Deepmala, V. N. Mishra, Local and global results for modi…ed Szàsz-Mirakjan operators, Math. Method. Appl. Sci.DOI: 10.1002/mma.4171.
  • A. R. Gairola, Deepmala, L. N. Mishra, Rate of approximation by …nite iterates of q- Durrmeyer operators, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 86 (2) (2016) 229–234.
  • A. Wa…, N. Rao, R. Deepmala, Approximation properties by generalized Baskakov Kantorovich Stancu type operators, Applied Mathematics & Information Sciences Letters 4 (3) (2016) 111–118.
  • K. K. Singh, A. R. Gairola, Deepmala, Approximation theorems for q-analouge of a linear positive operator by A. Lupa¸s, International Journal of Analysis and Applications 12 (1) (2016) 30–37.
  • V. N. Mishra, P. Sharma, L. N. Mishra, On statistical approximation properties of q- Baskakov–Szász–Stancu operators, Journal of the Egyptian Mathematical Society 24 (3) (2016) 396–401.
  • V. N. Mishra, H. Khan, K. Khatri, L. N. Mishra, Hypergeometric representation for Baskakov- Durrmeyer-Stancu type operators, Bulletin of Mathematical Analysis and Applications 5 (3) (2013) 18–26.
  • V. N. Mishra, K. Khatri, L. N. Mishra, On simultaneous approximation for Baskakov- Durrmeyer-Stancu type operators, Journal of Ultra Scientist of Physical Sciences 24 (3A) (2012) 567–577.
  • V. N. Mishra, P. Patel, On generalized integral Bernstein operators based on q-integers, Applied Mathematics and Computation 242 (2014) 931–944.
  • V. Kac, P. Cheung, Quantum calculus, Springer, 2002.
  • A. Il’inskii, S. Ostrovska, Convergence of generalized Bernstein polynomials, Journal of Ap- proximation Theory 116 (1) (2002) 100–112.
  • V. Gupta, Some approxmation properties of q-Durrmeyer operators, Applied Mathematics and Computation 197 (2008) 172–178.
  • G. G. Lorentz, Bernstein polynomials, American Mathematical Society, 1953.
  • X. M. Zeng, D. Lin, L. Li, A note on approximation properties of q-Durrmeyer operators, Applied Mathematics and Computation 216 (3) (2010) 819–821.
  • Current address : Vishnu Narayan Mishra (Corresponding author): Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Ichchhanath Mahadev Du- mas Road, Surat-395 007 (Gujarat), India, L. 1627 Awadh Puri Colony Beniganj, Phase-III, Opposite - Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad, Uttar Pradesh 224 001, India
  • E-mail address : vishnu_narayanmishra@yahoo.co.in; vishnunarayanmishra@gmail.com
  • Current address : Prashantkumar Patel: Department of Applied Mathematics and Humanities, S. V. National Institute of Technology, Ichchhanath Mahadev Dumas Road, Surat-395 007 (Gu- jarat), India, Department of Mathematics, St. Xavier’s College(Autonomous), Ahmedabad-380 009 (Gujarat), India
  • E-mail address : prashant225@gmail.com
There are 45 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Vishnu Narayan Mıshra This is me

Prashantkumar Patel This is me

Publication Date August 1, 2017
Published in Issue Year 2017 Volume: 66 Issue: 2

Cite

APA Narayan Mıshra, V., & Patel, P. (2017). STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 66(2), 263-275. https://doi.org/10.1501/Commua1_0000000817
AMA Narayan Mıshra V, Patel P. STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2017;66(2):263-275. doi:10.1501/Commua1_0000000817
Chicago Narayan Mıshra, Vishnu, and Prashantkumar Patel. “STATISTICAL CONVERGENCE OF MODIFIED Q-DURRMEYER OPERATORS”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66, no. 2 (August 2017): 263-75. https://doi.org/10.1501/Commua1_0000000817.
EndNote Narayan Mıshra V, Patel P (August 1, 2017) STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66 2 263–275.
IEEE V. Narayan Mıshra and P. Patel, “STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 66, no. 2, pp. 263–275, 2017, doi: 10.1501/Commua1_0000000817.
ISNAD Narayan Mıshra, Vishnu - Patel, Prashantkumar. “STATISTICAL CONVERGENCE OF MODIFIED Q-DURRMEYER OPERATORS”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 66/2 (August 2017), 263-275. https://doi.org/10.1501/Commua1_0000000817.
JAMA Narayan Mıshra V, Patel P. STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66:263–275.
MLA Narayan Mıshra, Vishnu and Prashantkumar Patel. “STATISTICAL CONVERGENCE OF MODIFIED Q-DURRMEYER OPERATORS”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 66, no. 2, 2017, pp. 263-75, doi:10.1501/Commua1_0000000817.
Vancouver Narayan Mıshra V, Patel P. STATISTICAL CONVERGENCE OF MODIFIED q-DURRMEYER OPERATORS. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2017;66(2):263-75.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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