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  • 1 Kirikkale University, Yahsihan, 71450, Kirikkale, Turkey
  • 2 Technical University of Cluj-Napoca, Romania
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In this paper, we study the k-th order Kantorovich type modication of Szász—Mirakyan operators. We first establish explicit formulas giving the images of monomials and the moments up to order six. Using this modification, we present a quantitative Voronovskaya theorem for differentiated Szász—Mirakyan operators in weighted spaces. The approximation properties such as rate of convergence and simultaneous approximation by the new constructions are also obtained.

  • [1]

    Acar, T., Asymptotic Formulas for Generalized Szász—Mirakyan Operators, Appl. Math. Comput.,, 263 (2015), 223239.

  • [2]

    Acar, T. and Aral, A., On pointwise convergence of q-Bernstein operators and their q-Derivatives, Num. Funct. Anal. Opt.,, 36(3) (2015), 287304.

    • Search Google Scholar
    • Export Citation
  • [3]

    Acar, T., Quantitative q-Voronovskaya and q-Grüss—Voronovskaya-type results for q-Szász Operators, Georgian Math. J., (In Press).

    • Export Citation
  • [4]

    Acar, T., Aral, A. and Raşa, I., The New Forms of Voronovskaya’s Theorem in weighted spaces, Positivity, (In Press), DOI: 10.1007/s11117-015-0338-4.

  • [5]

    Acar, T. and Ulusoy, G., Approximation By Modified Szász—Durrmeyer Operators, Period. Math. Hungar., (In Press), DOI: 10.1007/s10998-015-0091-2.

  • [6]

    Becker, M., Global approximation theorems for Szász—Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J.,, 27(1) (1978), 127142.

    • Search Google Scholar
    • Export Citation
  • [7]

    Butzer, P. L. and Karsli, H., Voronovskaya-type theorems for derivatives of the Bernstein-Chlodowsky polynomials and the Szász—Mirakyan operator, Comm. Math.,, 49(1) (2009), 3358.

    • Search Google Scholar
    • Export Citation
  • [8]

    Floater, M., On the convergence of derivatives of Bernstein approximation, J. Approx. Theory,, 134 (2005), 130135.

  • [9]

    Gonska, H. H., Quantitative Korovkin-type theorems on simultaneous approximation, Math. Z.,, 186 (1984), 419433.

  • [10]

    Gonska, H., Heilmann, M. and Raşa, I., Asymptotic behaviour of differentiated Bernstein polynomials revisited, General Mathematics (Sibiu),, 18 (2010), 4553.

    • Search Google Scholar
    • Export Citation
  • [11]

    Gonska, H. and Păltănea, R., General Voronovskaya and asymptotic theorems in simultaneous approximation, Mediterr. J. Math.,, 7 (2010), 3749.

    • Search Google Scholar
    • Export Citation
  • [12]

    Heilmann, M. and Raşa, I., k-th order Kantorovich type modification of the operators Un ω, J. Appl. Funct. Anal., 9(3–4) (2014), 320334.

    • Search Google Scholar
    • Export Citation
  • [13]

    Ispir, N., On Modified Baskakov Operators on Weighted Spaces, Turk. J. Math., 25 (2001), 355365.

  • [14]

    Kacsó, D., Certain Bernstein–Durrmeyer Operators Preserving Linear Functions. Habilitationsschrift, University of Duisburg-Essen, 2008.

    • Search Google Scholar
    • Export Citation
  • [15]

    Knoop, H. B. and Pottinger, P., Ein Satz vom Korovkin-Typ fur Ck-Raume, Math. Z., 148 (1976), 2332.

  • [16]

    Phillips, G. M., Interpolation and Approximation by Polynomials, Springer-Verlag, 2003.

  • [17]

    Sendov, Bl. and Popov, V. (n.d.), Konvergenz von Ableitungen linearer Operatoren, Hand-written German translation of notes used by the authors at the “Seminar on Interpolation and Convexity,” Cluj-Napoca, September 1–10, 1968.

    • Search Google Scholar
    • Export Citation
  • [18]

    Sendov, Bl. and Popov, V., The convergence of the derivatives of positive linear operators, C.R. Acad. Bulgare Sci., 22 (1969), 507509 (in Russian).

    • Search Google Scholar
    • Export Citation
  • [19]

    Sendov, Bl. and Popov, V., Convergence of the derivatives of positive linear operators, B’lgar. Akad. Nauk. Otdel. Mat. Fiz. Nauk. Izv. Mat. Inst., 11 (1970), 107115 (in Bulgarian).

    • Search Google Scholar
    • Export Citation
  • [20]

    Stancu, D. D., Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl., 13 (1968), 11731194.

    • Search Google Scholar
    • Export Citation