Research Article
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Year 2023, Volume: 5 Issue: 4, 293 - 299, 31.12.2023
https://doi.org/10.51537/chaos.1320442

Abstract

References

  • Acu, A. M., 2015 Stancu–Schurer–Kantorovich operators based on q-integers. Applied Mathematics and Computation 259: 896– 907.
  • Acu, A.-M., N. Manav, and D. F. Sofonea, 2018 Approximation properties of λ-Kantorovich operators. Journal of inequalities and applications 2018: 1–12.
  • Agratini, O., 2001 An approximation process of Kantorovich type. Miskolc Mathematical Notes 2: 3–10.
  • Agrawal, P., N. Bhardwaj, and P. Bawa, 2022 Bézier variant of modified α-bernstein operators. Rendiconti del Circolo Matematico di Palermo Series 2 71: 807–827.
  • Agrawal, P., M. Goyal, and A. Kajla, 2015 q-Bernstein-Schurer- Kantorovich type operators. Bollettino dell’Unione Matematica Italiana 8: 169–180.
  • Altomare, F. and M. Campiti, 2011 Korovkin-type approximation theory and its applications, volume 17. Walter de Gruyter.
  • Altomare, F., M. C. Montano, and V. Leonessa, 2013 On a generalization of Szász–Mirakjan–Kantorovich operators. Results in Mathematics 63: 837–863.
  • Andrews, G. E., R. Askey, and R. Roy, 1999 Special functions, volume 71. Cambridge university press.
  • Angeloni, L., D. Costarelli, and G. Vinti, 2020 Approximation properties of mixed sampling-Kantorovich operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115: 1–14.
  • Angeloni, L., G. Vinti, et al., 2005 Rate of approximation for nonlinear integral operators with application to signal processing. Differential and Integral Equations 18: 855–890.
  • Araci, S., A. Kajla, and P. Agarwal, 2019 A Kantorovich variant of a generalized Bernstein operators .
  • Barbosu, D., 2004 Kantorovich-Stancu type operators. J. Inequal. Pure Appl. Math 5: 6.
  • Bardaro, C. and I. Mantellini, 2012 On convergence properties for a class of Kantorovich discrete operators. Numerical functional analysis and optimization 33: 374–396.
  • Bardaro, C., G. Vinti, P. Butzer, and R. Stens, 2007 Kantrovichtype generalized sampling series in the setting of orlicz spaces. Sampling Theory in Signal and Image Processing 6: 29.
  • Bartle, R. G., 1976 The element of real analysis, john willy & sons. Inc., New York .
  • Bawa, P., N. Bhardwaj, and P. Agrawal, 2022 Quantitative voronovskaya type theorems and gbs operators of kantorovich variant of lupa¸s-stancu operators based on pólya distribution. Mathematical Foundations of Computing 5: 269–293.
  • Bernšteın, S., 1912 Démonstration du théoreme de Weierstrass fondée sur le calcul des probabilities. Comm. Soc. Math. Kharkov 13: 1–2.
  • Cai, Q.-B., W.-T. Cheng, and B. Çekim, 2019 Bivariate α, q- Bernstein–Kantorovich operators and gbs operators of bivariate α, q-bernstein–kantorovich type. Mathematics 7: 1161.
  • Cai, Q.-B., B.-Y. Lian, and G. Zhou, 2018 Approximation properties of λ-Bernstein operators. Journal of Inequalities and Applications 2018: 1–11.
  • Cai, Q.-B. and X.-W. Xu, 2018 Shape-preserving properties of a new family of generalized Bernstein operators. Journal of inequalities and applications 2018: 1–14.
  • Chen, X., J. Tan, Z. Liu, and J. Xie, 2017 Approximation of functions by a new family of generalized Bernstein operators. Journal of Mathematical Analysis and Applications 450: 244–261.
  • Cheney, E.W., 1966 Introduction to approximation theory . Cluni, F., D. Costarelli, A. M. Minotti, and G. Vinti, 2013 Multivariate sampling Kantorovich operators: approximation and applications to civil engineering. EURASIP, Proc. SampTA pp. 400–403.
  • Costarelli, D., F. Cluni, A. M. Minotti, and G. Vinti, 2014a Applications of sampling Kantorovich operators to thermographic images for seismic engineering. arXiv preprint arXiv:1411.2584 .
  • Costarelli, D. and G. Vinti, 2011 Approximation by multivariate generalized sampling Kantorovich operators in the setting of orlicz spaces. Bollettino dell’Unione Matematica Italiana 4: 445– 468.
  • Costarelli, D. and G. Vinti, 2013 Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization 34: 819–844.
  • Costarelli, D. and G. Vinti, 2014 Sampling Kantorovich operators and their applications to approximation problems and to digital image processing. In Proceedings of 8th international conference on applied mathematics, simulation, modelling (ASM’14), Florence, Italy November, pp. 22–24.
  • Costarelli, D., G. Vinti, et al., 2014b Order of approximation for sampling Kantorovich operators. Journal of Integral Equations and Applications 26: 345–367.
  • Dalmanog, Ö., O. Dog, et al., 2010 On statistical approximation properties of Kantorovich type q-Bernstein operators. Mathematical and Computer Modelling 52: 760–771.
  • Dalmano˘ glu, Ö., 2007 Approximation by Kantorovich type q- Bernstein operators .
  • de la Cal, J. and A. M. Valle, 2000 A generalization of Bernstein– Kantoroviˇc operators. Journal of mathematical analysis and applications 252: 750–766.
  • Deo, N., M. Dhamija, and D. Micl˘au¸s, 2016 Stancu–Kantorovich operators based on inverse Pólya–Eggenberger distribution. Applied Mathematics and Computation 273: 281–289.
  • Dogru, O. and N. Ozalp, 2001 Approximation by Kantorovich type generalization of meyer-konig and zeller operators. Glasnik matematiˇcki 36: 311–318.
  • Duman, O., M. Özarslan, and O. Do˘gru, 2006 On integral type generalizations of positive linear operators. Studia Mathematica 174: 1–12.
  • Eggenberger, F. and G. Pólya, 1923 Über die statistik verketteter vorgänge. ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik 3: 279–289.
  • Gadjiev, A. and A. Ghorbanalizadeh, 2010 Approximation properties of a new type Bernstein–Stancu polynomials of one and two variables. Applied mathematics and computation 216: 890–901.
  • Gonska, H., M. Heilmann, and I. Ra¸sa, 2011 Kantorovich operators of order k. Numerical functional analysis and optimization 32: 717–738.
  • Hounkonnou, M. N., J. Désiré, and B. Kyemba, 2013 R (p, q)- calculus: differentiation and integration. SUT J. Math 49: 145– 167.
  • ˙Içöz, G., 2012 A Kantorovich variant of a new type Bernstein– Stancu polynomials. Applied Mathematics and Computation 218: 8552–8560.
  • Igoz, G., 2012 A kantorovich variant of a new type bernstein-stancu polynomails. AppI Math Comput 218: 8552–8560.
  • Kac, V. and P. Cheung, 2001 Quantum calculus. Springer Science & Business Media.
  • Kajla, A. and S. Araci, 2017 Blending type approximation by Stancu-Kantorovich operators based on pólya-eggenberger distribution. Open Physics 15: 335–343.
  • Kantorovich, L., 1930 Sur certains développements suivant les polynômes de la forme de s. Bernstein, I, II, CR Acad. URSS 563: 568.
  • Karaca, Y., 2022 Global attractivity, asymptotic stability and blowup points for nonlinear functional-integral equations’solutions and applications in banach space bc (r+) with computational complexity. Fractals 30: 2240188.
  • Karaca, Y., M. Moonis, Y.-D. Zhang, and C. Gezgez, 2019 Mobile cloud computing based stroke healthcare system. International Journal of Information Management 45: 250–261.
  • Katriel, J. and M. Kibler, 1992 Normal ordering for deformed boson operators and operator-valued deformed stirling numbers. Journal of Physics A: Mathematical and General 25: 2683.
  • Lubinsky, D., 1995Weierstrass’theorem in the twentieth century: A selection. Quaestiones Mathematicae 18: 91–130.
  • Lupas, A., 1987 A q-analogue of the Bernstein operator. In Seminar on numerical and statistical calculus, University of Cluj-Napoca, volume 9.
  • Marinkovi´c, S., P. Rajkovi´c, and M. Stankovi´c, 2008 The inequalities for some types of q-integrals. Computers & Mathematics with Applications 56: 2490–2498.
  • Mohiuddine, S., T. Acar, and A. Alotaibi, 2017 Construction of a new family of Bernstein-Kantorovich operators. Mathematical Methods in the Applied Sciences 40: 7749–7759.
  • Muraru, C.-V., 2011 Note on q-Bernstein-Schurer operators. Stud. Univ. Babes-Bolyai Math 56: 489–495.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2015 On (p, q)-analogue of Bernstein operators. Applied Mathematics and Computation 266: 874–882.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2016 Some approximation results for Bernstein-Kantorovich operators based on (p, q)-calculus. UPB Sci. Bull., Ser. A 78: 129–142.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2017 Approximation by kantorovich type q-Bernstein-Stancu operators. Complex Analysis and Operator Theory 11: 85–107.
  • Ostrovska, S., 2016 The q-versions of the Bernstein operator: from mere analogies to further developments. Results in Mathematics 69: 275–295.
  • Ozarslan, M. A. and O. Duman, 2016 Smoothness properties of modified Bernstein-Kantorovich operators .
  • Özarslan, M. A., O. Duman, and H. Srivastava, 2008 Statistical approximation results for Kantorovich-type operators involving some special polynomials. Mathematical and computer modelling 48: 388–401.
  • Özarslan, M. A. and T. Vedi, 2013 q-Bernstein-Schurer-Kantorovich operators. Journal of Inequalities and Applications 2013: 444.
  • Phillips, G. M., 2003 Bernstein polynomials. In Interpolation and Approximation by Polynomials, pp. 247–290, Springer.
  • Pinkus, A., 2000 Weierstrass and approximation theory. Journal of Approximation Theory 107: 1–66.
  • Radu, C., 2008 Statistical approximation properties of Kantorovich operators based on q-integers, creat. math. Inform 17: 75–84.
  • Rashid, S., S. Sultana, Y. Karaca, A. Khalid, and Y.-M. Chu, 2022 Some further extensions considering discrete proportional fractional operators. Fractals 30: 2240026.
  • Ren, M.-Y. and X.-M. Zeng, 2013 On statistical approximation properties of modified q-Bernstein-Schurer operators. Bulletin of the Korean Mathematical Society 50: 1145–1156.
  • Sahai, V. and S. Yadav, 2007 Representations of two parameter quantum algebras and p, q-special functions. Journal of mathematical analysis and applications 335: 268–279.
  • Vinti, G. and L. Zampogni, 2009 Approximation by means of nonlinear Kantorovich sampling type operators in orlicz spaces. Journal of Approximation Theory 161: 511–528.

Different Variants of Bernstein Kantorovich Operators and Their Applications in Sciences and Engineering Field

Year 2023, Volume: 5 Issue: 4, 293 - 299, 31.12.2023
https://doi.org/10.51537/chaos.1320442

Abstract

In this article, we investigate various Bernstein-Kantorovich variants together with their approximation properties. Nowadays, these variants of Bernstein-Kantorovich operators have been a source of inspiration for researchers as it helps to approximate integral functions also which is not feasible in the case of discrete operators. Chaos theory has also been referred to as complexity theory. Using chaos theory complexity is also reduced as in approximation theory. Thus in order to reduce complexity and to have better understanding of images in sciences and engineering field, sampling Kantorovich operators of approximation theory are widely used in this regard for enhancement of images. Thus, we discuss the important applications of Kantorovich operators depicting pragmatic and theoretical aspects of approximation theory.

References

  • Acu, A. M., 2015 Stancu–Schurer–Kantorovich operators based on q-integers. Applied Mathematics and Computation 259: 896– 907.
  • Acu, A.-M., N. Manav, and D. F. Sofonea, 2018 Approximation properties of λ-Kantorovich operators. Journal of inequalities and applications 2018: 1–12.
  • Agratini, O., 2001 An approximation process of Kantorovich type. Miskolc Mathematical Notes 2: 3–10.
  • Agrawal, P., N. Bhardwaj, and P. Bawa, 2022 Bézier variant of modified α-bernstein operators. Rendiconti del Circolo Matematico di Palermo Series 2 71: 807–827.
  • Agrawal, P., M. Goyal, and A. Kajla, 2015 q-Bernstein-Schurer- Kantorovich type operators. Bollettino dell’Unione Matematica Italiana 8: 169–180.
  • Altomare, F. and M. Campiti, 2011 Korovkin-type approximation theory and its applications, volume 17. Walter de Gruyter.
  • Altomare, F., M. C. Montano, and V. Leonessa, 2013 On a generalization of Szász–Mirakjan–Kantorovich operators. Results in Mathematics 63: 837–863.
  • Andrews, G. E., R. Askey, and R. Roy, 1999 Special functions, volume 71. Cambridge university press.
  • Angeloni, L., D. Costarelli, and G. Vinti, 2020 Approximation properties of mixed sampling-Kantorovich operators. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 115: 1–14.
  • Angeloni, L., G. Vinti, et al., 2005 Rate of approximation for nonlinear integral operators with application to signal processing. Differential and Integral Equations 18: 855–890.
  • Araci, S., A. Kajla, and P. Agarwal, 2019 A Kantorovich variant of a generalized Bernstein operators .
  • Barbosu, D., 2004 Kantorovich-Stancu type operators. J. Inequal. Pure Appl. Math 5: 6.
  • Bardaro, C. and I. Mantellini, 2012 On convergence properties for a class of Kantorovich discrete operators. Numerical functional analysis and optimization 33: 374–396.
  • Bardaro, C., G. Vinti, P. Butzer, and R. Stens, 2007 Kantrovichtype generalized sampling series in the setting of orlicz spaces. Sampling Theory in Signal and Image Processing 6: 29.
  • Bartle, R. G., 1976 The element of real analysis, john willy & sons. Inc., New York .
  • Bawa, P., N. Bhardwaj, and P. Agrawal, 2022 Quantitative voronovskaya type theorems and gbs operators of kantorovich variant of lupa¸s-stancu operators based on pólya distribution. Mathematical Foundations of Computing 5: 269–293.
  • Bernšteın, S., 1912 Démonstration du théoreme de Weierstrass fondée sur le calcul des probabilities. Comm. Soc. Math. Kharkov 13: 1–2.
  • Cai, Q.-B., W.-T. Cheng, and B. Çekim, 2019 Bivariate α, q- Bernstein–Kantorovich operators and gbs operators of bivariate α, q-bernstein–kantorovich type. Mathematics 7: 1161.
  • Cai, Q.-B., B.-Y. Lian, and G. Zhou, 2018 Approximation properties of λ-Bernstein operators. Journal of Inequalities and Applications 2018: 1–11.
  • Cai, Q.-B. and X.-W. Xu, 2018 Shape-preserving properties of a new family of generalized Bernstein operators. Journal of inequalities and applications 2018: 1–14.
  • Chen, X., J. Tan, Z. Liu, and J. Xie, 2017 Approximation of functions by a new family of generalized Bernstein operators. Journal of Mathematical Analysis and Applications 450: 244–261.
  • Cheney, E.W., 1966 Introduction to approximation theory . Cluni, F., D. Costarelli, A. M. Minotti, and G. Vinti, 2013 Multivariate sampling Kantorovich operators: approximation and applications to civil engineering. EURASIP, Proc. SampTA pp. 400–403.
  • Costarelli, D., F. Cluni, A. M. Minotti, and G. Vinti, 2014a Applications of sampling Kantorovich operators to thermographic images for seismic engineering. arXiv preprint arXiv:1411.2584 .
  • Costarelli, D. and G. Vinti, 2011 Approximation by multivariate generalized sampling Kantorovich operators in the setting of orlicz spaces. Bollettino dell’Unione Matematica Italiana 4: 445– 468.
  • Costarelli, D. and G. Vinti, 2013 Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing. Numerical Functional Analysis and Optimization 34: 819–844.
  • Costarelli, D. and G. Vinti, 2014 Sampling Kantorovich operators and their applications to approximation problems and to digital image processing. In Proceedings of 8th international conference on applied mathematics, simulation, modelling (ASM’14), Florence, Italy November, pp. 22–24.
  • Costarelli, D., G. Vinti, et al., 2014b Order of approximation for sampling Kantorovich operators. Journal of Integral Equations and Applications 26: 345–367.
  • Dalmanog, Ö., O. Dog, et al., 2010 On statistical approximation properties of Kantorovich type q-Bernstein operators. Mathematical and Computer Modelling 52: 760–771.
  • Dalmano˘ glu, Ö., 2007 Approximation by Kantorovich type q- Bernstein operators .
  • de la Cal, J. and A. M. Valle, 2000 A generalization of Bernstein– Kantoroviˇc operators. Journal of mathematical analysis and applications 252: 750–766.
  • Deo, N., M. Dhamija, and D. Micl˘au¸s, 2016 Stancu–Kantorovich operators based on inverse Pólya–Eggenberger distribution. Applied Mathematics and Computation 273: 281–289.
  • Dogru, O. and N. Ozalp, 2001 Approximation by Kantorovich type generalization of meyer-konig and zeller operators. Glasnik matematiˇcki 36: 311–318.
  • Duman, O., M. Özarslan, and O. Do˘gru, 2006 On integral type generalizations of positive linear operators. Studia Mathematica 174: 1–12.
  • Eggenberger, F. and G. Pólya, 1923 Über die statistik verketteter vorgänge. ZAMM-Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik 3: 279–289.
  • Gadjiev, A. and A. Ghorbanalizadeh, 2010 Approximation properties of a new type Bernstein–Stancu polynomials of one and two variables. Applied mathematics and computation 216: 890–901.
  • Gonska, H., M. Heilmann, and I. Ra¸sa, 2011 Kantorovich operators of order k. Numerical functional analysis and optimization 32: 717–738.
  • Hounkonnou, M. N., J. Désiré, and B. Kyemba, 2013 R (p, q)- calculus: differentiation and integration. SUT J. Math 49: 145– 167.
  • ˙Içöz, G., 2012 A Kantorovich variant of a new type Bernstein– Stancu polynomials. Applied Mathematics and Computation 218: 8552–8560.
  • Igoz, G., 2012 A kantorovich variant of a new type bernstein-stancu polynomails. AppI Math Comput 218: 8552–8560.
  • Kac, V. and P. Cheung, 2001 Quantum calculus. Springer Science & Business Media.
  • Kajla, A. and S. Araci, 2017 Blending type approximation by Stancu-Kantorovich operators based on pólya-eggenberger distribution. Open Physics 15: 335–343.
  • Kantorovich, L., 1930 Sur certains développements suivant les polynômes de la forme de s. Bernstein, I, II, CR Acad. URSS 563: 568.
  • Karaca, Y., 2022 Global attractivity, asymptotic stability and blowup points for nonlinear functional-integral equations’solutions and applications in banach space bc (r+) with computational complexity. Fractals 30: 2240188.
  • Karaca, Y., M. Moonis, Y.-D. Zhang, and C. Gezgez, 2019 Mobile cloud computing based stroke healthcare system. International Journal of Information Management 45: 250–261.
  • Katriel, J. and M. Kibler, 1992 Normal ordering for deformed boson operators and operator-valued deformed stirling numbers. Journal of Physics A: Mathematical and General 25: 2683.
  • Lubinsky, D., 1995Weierstrass’theorem in the twentieth century: A selection. Quaestiones Mathematicae 18: 91–130.
  • Lupas, A., 1987 A q-analogue of the Bernstein operator. In Seminar on numerical and statistical calculus, University of Cluj-Napoca, volume 9.
  • Marinkovi´c, S., P. Rajkovi´c, and M. Stankovi´c, 2008 The inequalities for some types of q-integrals. Computers & Mathematics with Applications 56: 2490–2498.
  • Mohiuddine, S., T. Acar, and A. Alotaibi, 2017 Construction of a new family of Bernstein-Kantorovich operators. Mathematical Methods in the Applied Sciences 40: 7749–7759.
  • Muraru, C.-V., 2011 Note on q-Bernstein-Schurer operators. Stud. Univ. Babes-Bolyai Math 56: 489–495.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2015 On (p, q)-analogue of Bernstein operators. Applied Mathematics and Computation 266: 874–882.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2016 Some approximation results for Bernstein-Kantorovich operators based on (p, q)-calculus. UPB Sci. Bull., Ser. A 78: 129–142.
  • Mursaleen, M., K. J. Ansari, and A. Khan, 2017 Approximation by kantorovich type q-Bernstein-Stancu operators. Complex Analysis and Operator Theory 11: 85–107.
  • Ostrovska, S., 2016 The q-versions of the Bernstein operator: from mere analogies to further developments. Results in Mathematics 69: 275–295.
  • Ozarslan, M. A. and O. Duman, 2016 Smoothness properties of modified Bernstein-Kantorovich operators .
  • Özarslan, M. A., O. Duman, and H. Srivastava, 2008 Statistical approximation results for Kantorovich-type operators involving some special polynomials. Mathematical and computer modelling 48: 388–401.
  • Özarslan, M. A. and T. Vedi, 2013 q-Bernstein-Schurer-Kantorovich operators. Journal of Inequalities and Applications 2013: 444.
  • Phillips, G. M., 2003 Bernstein polynomials. In Interpolation and Approximation by Polynomials, pp. 247–290, Springer.
  • Pinkus, A., 2000 Weierstrass and approximation theory. Journal of Approximation Theory 107: 1–66.
  • Radu, C., 2008 Statistical approximation properties of Kantorovich operators based on q-integers, creat. math. Inform 17: 75–84.
  • Rashid, S., S. Sultana, Y. Karaca, A. Khalid, and Y.-M. Chu, 2022 Some further extensions considering discrete proportional fractional operators. Fractals 30: 2240026.
  • Ren, M.-Y. and X.-M. Zeng, 2013 On statistical approximation properties of modified q-Bernstein-Schurer operators. Bulletin of the Korean Mathematical Society 50: 1145–1156.
  • Sahai, V. and S. Yadav, 2007 Representations of two parameter quantum algebras and p, q-special functions. Journal of mathematical analysis and applications 335: 268–279.
  • Vinti, G. and L. Zampogni, 2009 Approximation by means of nonlinear Kantorovich sampling type operators in orlicz spaces. Journal of Approximation Theory 161: 511–528.
There are 64 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Research Articles
Authors

Parveen Bawa 0000-0002-9645-6787

Neha Bhardwaj 0000-0002-5618-1563

Sumit Kaur Bhatia This is me 0000-0003-1946-0691

Publication Date December 31, 2023
Published in Issue Year 2023 Volume: 5 Issue: 4

Cite

APA Bawa, P., Bhardwaj, N., & Bhatia, S. K. (2023). Different Variants of Bernstein Kantorovich Operators and Their Applications in Sciences and Engineering Field. Chaos Theory and Applications, 5(4), 293-299. https://doi.org/10.51537/chaos.1320442

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science 23830 28903   

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