Abstract
We consider a Bernstein–Schnabl operator \(L_n\) which commutes with the derivative and consequently can be represented as a differential operator with constant coefficients. The inverse of \(L_n\) restricted to polynomials is investigated. We obtain Voronovskaja type results for \(L_n^{-1}\) and compare them with the corresponding results for \(L_n\). A conjecture in this sense is presented.
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References
Abel, U., Ivan, M.: Asymptotic approximation with a sequence of positive linear operators. J. Comput. Anal. Appl. 3(4), 331–341 (2001)
Abel, U., Ivan, M.: Asymptotic expansion of a sequence of divided differences with application to positive linear operators. J. Comput. Anal. Appl. 7(1), 89–101 (2005)
Abramov, S.A.: Inverse linear difference operators. Comput. Math. Math. Phys. 57(12), 1887–1898 (2017)
Acu, A.M., Dancs, M., Heilmann, M., Paşca, V., Raşa, I.: Voronovskaya type results for special sequences of operators. RACSAM 116(19), 1–13 (2022)
Acu, A.M., Dancs, M., Heilmann, M., Paşca, V., Raşa, I.: A Bernstein–Schnabl type operator: applications to difference equations. Appl. Anal. Discrete Math. 16, 495–507 (2022)
Altomare, F., Campiti, M.: Korovkin-type approximation theory and its applications, Series: De Gruyter Studies in Mathematics, vol. 17 (1994)
Altomare, F., Cappelletti Montano, M., Leonessa, V., Raşa, I.: Markov Operators, Positive Semigroups and Approximation Processes, Walter de Gruyter, Berlin, Munich, Boston (2014)
Bratishchev, A.V., Korobeinik, Y.F.: General form of linear operators which commute with the operation of differentiation. Math. Notes Acad. Sci. USSR 12, 547–551 (1972)
Costabile, F.A., Longo, E.: A determinantal approach to Appell polynomials. J. Comput. Appl. Math. 234(5), 1528–1542 (2010)
Haydock, R.: The inverse of a linear operator. J. Phys. A Math. Nucl. Gen. 7(17), 2120 (1974)
Heilmann, M., Nasaireh, F., Raşa, I.: Complements to Voronovskaya’s Formula. In: Ghosh, D., Giri, D., Mohapatra, R., Sakurai, K., Savas, E., Som, T. (eds.) Mathematics and Computing. ICMC 2018. Springer Proceedings in Mathematics and Statistics, vol. 253. Springer, Singapore (2018)
Ivan, M., Raşa, I.: A Voronovskaya-type theorem. Anal. Numér. Théor Approx. 30(1), 47–54 (2001)
Kahane, C.: On operators commuting with differentiation. Am. Math. Mon. 76(2), 171–173 (1969)
Raşa, I.: Korovkin approximation and parabolic functions. Conf. Sem. Mat. Univ. Bari. 236, 1–25 (1991)
Riordan, J.: An Introduction to Combinatorial Analysis. Princeton, New Jersey (1978)
Weikard, R.: On commuting matrix differential operators. N. Y. J. Math. 8, 9–30 (2002)
Weikard, R.: On commuting differential operators. Electron. J. Differ. Equ. 2000(19), 1–11 (2000)
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Project financed by Lucian Blaga University of Sibiu through the research grant LBUS-IRG-2022-08.
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Acu, AM., Heilmann, M. & Rasa, I. Some results for the inverse of a Bernstein–Schnabl type operator. Anal.Math.Phys. 13, 15 (2023). https://doi.org/10.1007/s13324-022-00777-4
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DOI: https://doi.org/10.1007/s13324-022-00777-4