Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials

Abstract

<p style='text-indent:20px;'>Karaisa [<xref ref-type="bibr" rid="b29">29</xref>] presented Jakimovski- Leviatan- Durrmeyer type operators by means of Appell polynomials. In a similar manner, Wani et al. [<xref ref-type="bibr" rid="b43">43</xref>] proposed a sequence of Jakimovski-Leviatan-Durrmeyer type operators involving Brenke type polynomials which include Appell polynomials and Hermite polynomials. We note that the definitions of the operators given in both these papers are not correct. In the present article, we introduce a Stancu variant of the operators considered in [<xref ref-type="bibr" rid="b43">43</xref>] after correcting their definition. The definition of the operator proposed in [<xref ref-type="bibr" rid="b29">29</xref>] may be similarly corrected. We establish the Korovkin type approximation theorem and the rate of convergence by means of the usual modulus of continuity, Peetre's K-functional and the class of Lipschitz type functions for our operators. Next, we discuss the Voronovskaja and Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>ss Voronovskaja type asymptotic theorems. Finally, we study the convergence of these operators in a weighted space and the Korovkin type weighted statistical approximation theorem.</p>