August  2020, 19(8): 4097-4109. doi: 10.3934/cpaa.2020182

Representations for the inverses of certain operators

1. 

Lucian Blaga University of Sibiu, Department of Mathematics and Informatics, Str. Dr. I. Ratiu, No.5-7, RO-550012 Sibiu, Romania

2. 

Technical University of Cluj-Napoca, Faculty of Automation and Computer Science, Str. Memorandumului nr. 28 Cluj-Napoca, Romania

3. 

Babes-Bolyai University, FSEGA, Department of Statistics-Forecasts-Mathematics, Cluj-Napoca, Romania

* Corresponding author

Received  August 2019 Revised  November 2019 Published  May 2020

Fund Project: The first author is supported by Lucian Blaga University of Sibiu & Hasso Plattner Foundation research grants LBUS-IRG-2019-05

Inverses of certain positive linear operators have been investigated in several recent papers, in connection with problems like decomposition of classical operators, representation of Lagrange-type operators, asymptotic formulas of Voronovskaja type. Motivated by such researches, in this paper we give some representations for the inverses of certain positive linear operators, as Bernstein, Beta, Bernstein - Durrmeyer, genuine Bernstein - Durrmeyer and Kantorovich operators. Moreover, some Voronovskaja type formulas for the inverses of these operators are obtained. Several techniques are used in order to get such results.

Citation: Ana-Maria Acu, Madalina Dancs, Voichiţa Adriana Radu. Representations for the inverses of certain operators. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4097-4109. doi: 10.3934/cpaa.2020182
References:
[1]

U. Abel and M. Ivan, Asymptotic expansion of the multivariate Bernstein polynomials on a simplex, Approx. Theory Appl., 16 (2000), 85-93. 

[2]

F. Alda and B. I. P. Rubinstein, The Bernstein mechanism: function release under differential privacy, in Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17), 1705–1711.

[3]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights, in Approximation Theory and Functional Analysis (eds. C. K. Chui), Boston, Acad. Press, (1991), 25–46.

[4]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights: The cases $p = 1$ and $p = \infty$, in Approximation, Interpolation and Summation (eds. S. Baron and D. Leviatan), Israel Math. Conf. Proc., 4, Ramat Gan: Bar-Ilan Univ., (1991), 51–62.

[5]

W. Chen, On the modified Bernstein-Durrmeyer operator, in Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China, 1987.

[6]

M. M. Derriennic, De La Vallée Poussin and Bernstein-type operators, in Approximation Theory (eds. M. W. Müller, M. Felten and D. H. Mache), Proc. IDoMAT 95, Mathematical Research, Vol. 86, Akademic Verlag, Berlin, (1995), 71–84.

[7]

H. H. Gonska, On the composition and decomposition of positive linear operators, in Approximation Theory and its Applications (Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 31. Kiev. 1999). Kiev: Natsional. Akad. Nauk Ukraini, Inst. Mat. (2000), 161–180.

[8]

H. Gonska, M. Heilmann, A. Lupaș and I. Rașa, On the composition and decomposition of positive linear operators Ⅲ: A non-trivial decomposition of the Bernstein operator, preprint, arXiv: 1204.2723v1. doi: 10.1080/01630563.2014.951772.

[9]

H. H. Gonska and I. Rașa, On the composition and decomposition of positive linear operators (Ⅱ), Stud. Sci. Math. Hung., 47 (2010), 948-461.  doi: 10.1556/SScMath.2009.1144.

[10]

H. Gonska, I. Rașa and E. D. Stănilă, Beta Operators with Jacobi Weights, in Constructive Theory of Functions (eds. K. Ivanov, G. Nikolov and R. Uluchev), Sozopol 2013, 99–112, Prof. Marin Drinov Academic Publishing House, Sofia, 2014.

[11]

H. H. GonskaI. Rașa and E. D. Stănilă, Lagrange-type operators associated with $U^Q_n$, Publications de l'Institut Matheématique, Nouvelle série, 96 (2014), 159-168.  doi: 10.2298/PIM1410159G.

[12]

H. Gonska and R. Păltănea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czechoslovak Math. J., 60 (2010), 783-799.  doi: 10.1007/s10587-010-0049-8.

[13]

H. Gonska and R. Păltănea, Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions, Ukrainian Math. J., 62 (2010), 913-922.  doi: 10.1007/s11253-010-0413-8.

[14]

T. N. T. Goodman and A. Sharma, A modified Bernstein-Schoenberg operator, in Proc. Conf. Constructive Theory of Functions (eds. Bl. Sendov et al.), Varna 1987, Publ. House Bulg. Acad. Sci., Sofia, (1988), 166–173.

[15]

T. N. T. Goodman and A. Sharma, A Bernstein type operator on the simplex, Math. Balkanica, 5 (1991), 129-145. 

[16]

M. Heilmann, F. Nasaireh and I. Rașa, Complements to Voronovskaja's formula, in Mathematics and Computing (eds. D. Ghosh et al.), Chapter 11, Springer Proceedings in Mathematics & Statistics 253, Springer Nature Singapore Pte Ltd., 2018. doi: 10.1007/978-981-13-2095-8.

[17]

M. Heilmann, F. Nasaireh and I. Rașa, Beta and related operators revisited, in Proceedings of Constructive Theory of Functions (eds. K. Ivanov, G. Nikolov and R. Uluchev), Sozopol, (2016), 175–185. Prof. Marin Drinov Academic Publishing House, Sofia, 2018,175–185.

[18]

M. Heilmann and I. Rașa, On the decomposition of Bernstein operators, Numer. Funct. Anal. Optim., 36 (2015), 72-85.  doi: 10.1080/01630563.2014.951772.

[19]

L. V. Kantorovich, Sur certains developpements suivant les polynômes de la forme de S. Bernstein Ⅰ, Ⅱ, Dokl. Akad. Nauk. SSSR, (1930), 563–568,595–600.

[20]

A. Lupaș and L. Lupaș, Polynomials of binomial type and approximation operators, Stud. Univ. Babeș-Bolyai Math., 32 (1987), 61-69. 

[21]

A. Lupaș, The approximation by means of some linear positive operators, in Approximation Theory (eds. M. W. Müller, M. Felten and D.H. Mache), Proc. IDoMAT 95, Methematical Research, Vol. 86,201–229, Academic Verlag, Berlin, 1995.

[22]

G. Mühlbach, Verallgemeinerungen der Bernstein- und der Lagrangepolynome, Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu (Rev. Roumaine Math. Pure Appl.), 15 (1970), 1235–1252.

[23]

G. Mühlbach, Rekursionsformeln für die zentralen Momente der Polya- und der Beta-Verteilung, Metrika, 19 (1972), 171-177.  doi: 10.1007/BF01893292.

[24]

F. Nasaireh and I. Rașa, Another Look at Voronovskaja Type Formulas, J. Math. Inequal., 12 (2018), 95-105.  doi: 10.7153/jmi-2018-12-07.

[25]

F. Nasaireh, Voronovskaja-type formulas and applications, General Math., 25 (2017), 37-43. 

[26]

T. NeerA. M. Acu and P. N. Agrawal, Bezier variant of genuine-Durrmeyer type operators based on Polya distribution, Carpathian J. Math., 33 (2017), 73-86. 

[27]

R. Păltănea, A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca), 5 (2007), 109–117.

[28]

R. Păltănea, Sur un opérateur polynomial defini sur l'ensemble des fonctions intégrables, in Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca), (1983), 101–106.

[29]

G. Szegö, Orthogonal Polynomials, American Mathematical Society, Colloquium Publication, Vol. 23, 1939.

[30]

T. Vladislav and I. Rașa, Analiză Numerică. Elemente introductive, Editura Tehnică, Bucureti, 1997. (in Romanian)

[31]

T. Vladislav, I. Rașa, Analiză Numerică. Aproximare, problema lui Cauchy abstractă, proiectori Altomare, Editura Tehnică, Bucureti, 1999. (in Romanian)

[32]

D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

show all references

References:
[1]

U. Abel and M. Ivan, Asymptotic expansion of the multivariate Bernstein polynomials on a simplex, Approx. Theory Appl., 16 (2000), 85-93. 

[2]

F. Alda and B. I. P. Rubinstein, The Bernstein mechanism: function release under differential privacy, in Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence (AAAI-17), 1705–1711.

[3]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights, in Approximation Theory and Functional Analysis (eds. C. K. Chui), Boston, Acad. Press, (1991), 25–46.

[4]

H. Berens and Y. Xu, On Bernstein-Durrmeyer polynomials with Jacobi weights: The cases $p = 1$ and $p = \infty$, in Approximation, Interpolation and Summation (eds. S. Baron and D. Leviatan), Israel Math. Conf. Proc., 4, Ramat Gan: Bar-Ilan Univ., (1991), 51–62.

[5]

W. Chen, On the modified Bernstein-Durrmeyer operator, in Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China, 1987.

[6]

M. M. Derriennic, De La Vallée Poussin and Bernstein-type operators, in Approximation Theory (eds. M. W. Müller, M. Felten and D. H. Mache), Proc. IDoMAT 95, Mathematical Research, Vol. 86, Akademic Verlag, Berlin, (1995), 71–84.

[7]

H. H. Gonska, On the composition and decomposition of positive linear operators, in Approximation Theory and its Applications (Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 31. Kiev. 1999). Kiev: Natsional. Akad. Nauk Ukraini, Inst. Mat. (2000), 161–180.

[8]

H. Gonska, M. Heilmann, A. Lupaș and I. Rașa, On the composition and decomposition of positive linear operators Ⅲ: A non-trivial decomposition of the Bernstein operator, preprint, arXiv: 1204.2723v1. doi: 10.1080/01630563.2014.951772.

[9]

H. H. Gonska and I. Rașa, On the composition and decomposition of positive linear operators (Ⅱ), Stud. Sci. Math. Hung., 47 (2010), 948-461.  doi: 10.1556/SScMath.2009.1144.

[10]

H. Gonska, I. Rașa and E. D. Stănilă, Beta Operators with Jacobi Weights, in Constructive Theory of Functions (eds. K. Ivanov, G. Nikolov and R. Uluchev), Sozopol 2013, 99–112, Prof. Marin Drinov Academic Publishing House, Sofia, 2014.

[11]

H. H. GonskaI. Rașa and E. D. Stănilă, Lagrange-type operators associated with $U^Q_n$, Publications de l'Institut Matheématique, Nouvelle série, 96 (2014), 159-168.  doi: 10.2298/PIM1410159G.

[12]

H. Gonska and R. Păltănea, Simultaneous approximation by a class of Bernstein-Durrmeyer operators preserving linear functions, Czechoslovak Math. J., 60 (2010), 783-799.  doi: 10.1007/s10587-010-0049-8.

[13]

H. Gonska and R. Păltănea, Quantitative convergence theorems for a class of Bernstein-Durrmeyer operators preserving linear functions, Ukrainian Math. J., 62 (2010), 913-922.  doi: 10.1007/s11253-010-0413-8.

[14]

T. N. T. Goodman and A. Sharma, A modified Bernstein-Schoenberg operator, in Proc. Conf. Constructive Theory of Functions (eds. Bl. Sendov et al.), Varna 1987, Publ. House Bulg. Acad. Sci., Sofia, (1988), 166–173.

[15]

T. N. T. Goodman and A. Sharma, A Bernstein type operator on the simplex, Math. Balkanica, 5 (1991), 129-145. 

[16]

M. Heilmann, F. Nasaireh and I. Rașa, Complements to Voronovskaja's formula, in Mathematics and Computing (eds. D. Ghosh et al.), Chapter 11, Springer Proceedings in Mathematics & Statistics 253, Springer Nature Singapore Pte Ltd., 2018. doi: 10.1007/978-981-13-2095-8.

[17]

M. Heilmann, F. Nasaireh and I. Rașa, Beta and related operators revisited, in Proceedings of Constructive Theory of Functions (eds. K. Ivanov, G. Nikolov and R. Uluchev), Sozopol, (2016), 175–185. Prof. Marin Drinov Academic Publishing House, Sofia, 2018,175–185.

[18]

M. Heilmann and I. Rașa, On the decomposition of Bernstein operators, Numer. Funct. Anal. Optim., 36 (2015), 72-85.  doi: 10.1080/01630563.2014.951772.

[19]

L. V. Kantorovich, Sur certains developpements suivant les polynômes de la forme de S. Bernstein Ⅰ, Ⅱ, Dokl. Akad. Nauk. SSSR, (1930), 563–568,595–600.

[20]

A. Lupaș and L. Lupaș, Polynomials of binomial type and approximation operators, Stud. Univ. Babeș-Bolyai Math., 32 (1987), 61-69. 

[21]

A. Lupaș, The approximation by means of some linear positive operators, in Approximation Theory (eds. M. W. Müller, M. Felten and D.H. Mache), Proc. IDoMAT 95, Methematical Research, Vol. 86,201–229, Academic Verlag, Berlin, 1995.

[22]

G. Mühlbach, Verallgemeinerungen der Bernstein- und der Lagrangepolynome, Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu (Rev. Roumaine Math. Pure Appl.), 15 (1970), 1235–1252.

[23]

G. Mühlbach, Rekursionsformeln für die zentralen Momente der Polya- und der Beta-Verteilung, Metrika, 19 (1972), 171-177.  doi: 10.1007/BF01893292.

[24]

F. Nasaireh and I. Rașa, Another Look at Voronovskaja Type Formulas, J. Math. Inequal., 12 (2018), 95-105.  doi: 10.7153/jmi-2018-12-07.

[25]

F. Nasaireh, Voronovskaja-type formulas and applications, General Math., 25 (2017), 37-43. 

[26]

T. NeerA. M. Acu and P. N. Agrawal, Bezier variant of genuine-Durrmeyer type operators based on Polya distribution, Carpathian J. Math., 33 (2017), 73-86. 

[27]

R. Păltănea, A class of Durrmeyer type operators preserving linear functions, Ann. Tiberiu Popoviciu Sem. Funct. Equat. Approxim. Convex. (Cluj-Napoca), 5 (2007), 109–117.

[28]

R. Păltănea, Sur un opérateur polynomial defini sur l'ensemble des fonctions intégrables, in Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca), (1983), 101–106.

[29]

G. Szegö, Orthogonal Polynomials, American Mathematical Society, Colloquium Publication, Vol. 23, 1939.

[30]

T. Vladislav and I. Rașa, Analiză Numerică. Elemente introductive, Editura Tehnică, Bucureti, 1997. (in Romanian)

[31]

T. Vladislav, I. Rașa, Analiză Numerică. Aproximare, problema lui Cauchy abstractă, proiectori Altomare, Editura Tehnică, Bucureti, 1999. (in Romanian)

[32]

D. X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math., 25 (2006), 323-344.  doi: 10.1007/s10444-004-7206-2.

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