Voronovskaya type results and operators fixing two functions
Abstract
The present paper deals with positive linear operators which fix two functions. The transfer of a given sequence (Ln) of positive linear operators to a new sequence (Kn) is investigated. A general procedure to construct sequences of positive linear operators fixing two functions which form an Extended Complete Chebyshev system is described. The Voronovskaya type formula corresponding to the new sequence which is strongly influenced by the nature of the fixed functions is obtained. In the last section our results are compared with other results existing in literature.
Keyword : positive linear operators, Voronovskaya type theorem, extended complete Chebyshev system, operators fixing two functions
How to Cite
Acu, A. M., Măduţa, A.-I., & Rasa, I. (2021). Voronovskaya type results and operators fixing two functions. Mathematical Modelling and Analysis, 26(3), 395-410. https://doi.org/10.3846/mma.2021.13228
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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D. Cardenas-Morales, P. Garrancho and I. Rasa. Bernstein-type operators which preserve polynomials. Computers & Mathematics with Applications, 62(1):158– 163, 2011. https://doi.org/10.1016/j.camwa.2011.04.063
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H. Gonska, P. Pitul and I. Rasa. General King-type operators. Result. Math., 53:279–286, 2009. https://doi.org/10.1007/s00025-008-0338-9
V. Gupta and A.J. Lopez-Moreno. Phillips operators preserving arbitrary exponential functions, eat, ebt. Filomat, 32(14):5071–5082, 2018. https://doi.org/10.2298/FIL1814071G
M. Heilmann. Erhohung der Konvergenzgeschwindigkeit bei der approximation von funktionen mit hilfe von linearkombinationen spezieller positiver linearer operatoren. Habilitationsschrift, Universitat Dortmund, 1992.
M. Heilmann, F. Nasaireh and I. Rasa. Complements to Voronovskaja’s formula. In Mathematics and Computing, volume 253 of Springer Proceedings in Mathematics & Statistics, pp. 127–134, 2018.
S. Karlin and W.J. Studden. Tchebycheff Systems: with applications in Analysis and Statistics. Interscience Publishers, New York, 1966.
P.J. King. Positive linear operators which preserve x2. Acta Math. Hungar., 99:203–208, 2003.
W. Meyer-Konig and K. Zeller. Bernsteinsche potenzreihen. Studia Math., 19:89–94, 1960.
F. Nasaireh. Voronovskaja-type formulas and applications. General Mathematics, 25(1-2):37–43, 2017.
F. Nasaireh and I. Rasa. Another look at Voronovskaja type formulas. J. Mathem. Inequal., 12(1):95–105, 2018. https://doi.org/10.7153/jmi-2018-12-07
F. Ozsarac and T. Acar. Reconstruction of Baskakov operators preserving some exponential functions. Mathematical Methods in the Applied Sciences, 42(16):5124–5132, 2019. https://doi.org/10.1002/mma.5228
D. Popa. An intermediate Voronovskaja type theorem. RACSAM, 113:2421– 2429, 2019. https://doi.org/10.1007/s13398-018-00623-y
A. Lupaş and M.W. Müller. Approximation properties of the Mn-operators. Aequationes Math., 5:19–37, 1970.
I. Raşa. C0 - semigroups and iterates of positive linear operators: asymptotic behaviour. Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 82:1– 20, 2010.
T. Acar, A. Aral, D Cárdenas-Morales and P. Garrancho. SzászMirakyan type operators which fix exponentials. Results Math, 72:13931404, 2017. https://doi.org/10.1007/s00025-017-0665-9
T. Acar, A. Aral and H. Gonska. On SzászMirakyan operators preserving e2ax, a > 0. Mediterr. J. Math, 14(6), 2016. https://doi.org/10.1007/s00009-016-0804-7
T. Acar, A. Aral and I. Rasa. Positive linear operators preserving τ and τ2. Constructive Mathematical Analysis, 2(3):98–102, 2019. https://doi.org/10.33205/cma.547221
T. Acar, M. C. Montano, P. Garrancho and V. Leonessa. On Bernstein-Chlodovsky operators preserving e−2x. Bulletin of the Belgian Mathematical Society-Simon Stevin, 26(5):681–698, 2019. https://doi.org/10.36045/bbms/1579402817
T. Acar, M.C. Montano, P. Garrancho and V. Leonessa. Voronovskaya type results for Bernstein-Chlodovsky operators preserving e−2x. J. Math. Anal. Appl., 491(1):124307, 2020. https://doi.org/10.1016/j.jmaa.2020.124307
A.M. Acu and V. Gupta. On Baskakov-Szasz-Mirakyan-type operators preserving exponential type functions. Positivity, 22(3):919–929, 2018. https://doi.org/10.1007/s11117-018-0553-x
A.M. Acu, M. Heilmann and I. Rasa. Iterates of convolution-type operators. Positivity, 25(2):495–506, 2020. https://doi.org/10.1007/s11117-020-00773-7
J.A. Adell and J. de la Cal F. German Badia. On the iterates of some Bernsteintype operators. Journal of Mathematical Analysis and Applications, 209:529– 541, 1997. https://doi.org/10.1006/jmaa.1997.5371
J.M. Aldaz, O. Kounchev and H. Render. Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces. Numer. Math., 114(1), 2009. https://doi.org/10.1007/s00211-009-0248-0
F. Altomare and M. Campiti. Korovkin-type approximation theory and its applications. Series: De Gruyter Studies in Mathematics, 17, 1994.
F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa. Markov operators, positive semigroups and approximation processes. De Gruyter Studies in Mathematics, 61, 2014.
F. Altomare and I. Rasa. On some classes of diffusion equations and related approximation problems. In J. Szabados M.G. de Bruin, D.H. Mache(Ed.), Trends and Applications in Constructive Approximation, volume 151 of International Series of Numerical Mathematics, pp. 13–26. Birkhauser Basel, 2005.
A. Aral, D. Cardenas-Morales and P. Garrancho. Bernstein-type operators that reproduce exponential functions. J. Math. Inequal., 12(3):861–872, 2018. https://doi.org/10.7153/jmi-2018-12-64
M. Birou. A proof of a conjecture about the asymptotic formula of a Bernstein type operator. Results Math., 72:1129–1138, 2017. https://doi.org/10.1007/s00025-016-0608-x
M. Birou. Quantitative results for positive linear operators which preserve certain functions. General Mathematics, 17(2):85–95, 2019. https://doi.org/10.2478/gm-2019-0017
D. Cardenas-Morales, P. Garrancho and I. Rasa. Bernstein-type operators which preserve polynomials. Computers & Mathematics with Applications, 62(1):158– 163, 2011. https://doi.org/10.1016/j.camwa.2011.04.063
D. Cardenas-Morales, P. Garrancho and I. Rasa. Asymptotic formulae via a Korovkin-type result. Abstract and Applied Analysis, 2012(217464), 2012. https://doi.org/10.1155/2012/217464
C. Cottin, I. Gavrea, H. Gonska, D. Kacso and D.-X. Zhou. Global smoothness preservation and the variation-diminishing property. J. of lnequal. & Appl., 4:91– 114, 1999.
M.M. Derriennic. De la Vallée Poussin and Bernstein-type operators. In D.H. Mache M.W. Muller, M. Felten(Ed.), Approximation Theory, Proc. IDoMAT 95, volume 86 of Mathematical Research, pp. 71–84, Berlin, 1995. Akademic Verlag.
H. Gonska and P. Pitul. Remarks on an article of J.P.King. Comment. Math. Univ. Carolinae, 46(4):645–652, 2005.
H. Gonska, P. Pitul and I. Rasa. General King-type operators. Result. Math., 53:279–286, 2009. https://doi.org/10.1007/s00025-008-0338-9
V. Gupta and A.J. Lopez-Moreno. Phillips operators preserving arbitrary exponential functions, eat, ebt. Filomat, 32(14):5071–5082, 2018. https://doi.org/10.2298/FIL1814071G
M. Heilmann. Erhohung der Konvergenzgeschwindigkeit bei der approximation von funktionen mit hilfe von linearkombinationen spezieller positiver linearer operatoren. Habilitationsschrift, Universitat Dortmund, 1992.
M. Heilmann, F. Nasaireh and I. Rasa. Complements to Voronovskaja’s formula. In Mathematics and Computing, volume 253 of Springer Proceedings in Mathematics & Statistics, pp. 127–134, 2018.
S. Karlin and W.J. Studden. Tchebycheff Systems: with applications in Analysis and Statistics. Interscience Publishers, New York, 1966.
P.J. King. Positive linear operators which preserve x2. Acta Math. Hungar., 99:203–208, 2003.
W. Meyer-Konig and K. Zeller. Bernsteinsche potenzreihen. Studia Math., 19:89–94, 1960.
F. Nasaireh. Voronovskaja-type formulas and applications. General Mathematics, 25(1-2):37–43, 2017.
F. Nasaireh and I. Rasa. Another look at Voronovskaja type formulas. J. Mathem. Inequal., 12(1):95–105, 2018. https://doi.org/10.7153/jmi-2018-12-07
F. Ozsarac and T. Acar. Reconstruction of Baskakov operators preserving some exponential functions. Mathematical Methods in the Applied Sciences, 42(16):5124–5132, 2019. https://doi.org/10.1002/mma.5228
D. Popa. An intermediate Voronovskaja type theorem. RACSAM, 113:2421– 2429, 2019. https://doi.org/10.1007/s13398-018-00623-y
A. Lupaş and M.W. Müller. Approximation properties of the Mn-operators. Aequationes Math., 5:19–37, 1970.
I. Raşa. C0 - semigroups and iterates of positive linear operators: asymptotic behaviour. Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 82:1– 20, 2010.