An intermediate Voronovskaja type theorem

We prove the following intermediate Voronovskaja type theorem: Let Vn:Ca,b→Ca,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{n}:C \left[ a,b\right] \rightarrow C\left[ a,b\right] $$\end{document} be a sequence of positive linear operators such that limn→∞Vnf=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }V_{n}\left( f\right) =f$$\end{document} uniformly on a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ a,b\right] $$\end{document} for every f∈Ca,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C\left[ a,b \right] $$\end{document}. Let x∈a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \left[ a,b\right] $$\end{document} and suppose that there exists a sequence λnn∈N⊂0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \lambda _{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right) $$\end{document} such that limn→∞λn=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }\lambda _{n}=\infty $$\end{document} and there exists 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<\infty $$\end{document} such that the sequence λnpVn·-xpxn∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \lambda _{n}^{p}V_{n}\left( \left| \cdot -x\right| ^{p}\right) \left( x\right) \right) _{n\in {\mathbb {N}}}$$\end{document} is bounded. Then, for every continuous function f:a,b→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\left[ a,b\right] \rightarrow {\mathbb {R}}$$\end{document} differentiable at x∈a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \left[ a,b\right] $$\end{document} we have limn→∞λnVnfx-fxVn1x-f′xVn·-xx=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\lambda _{n}\left[ V_{n}\left( f\right) \left( x\right) -f\left( x\right) V_{n}\left( {\mathbf {1}}\right) \left( x\right) -f^{\prime }\left( x\right) V_{n}\left( \left( \cdot -x\right) \right) \left( x\right) \right] =0. \end{aligned}$$\end{document}As an application we prove a Voronovskaja type theorem for the sequences of the operators Vn:C0,1→C0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{n}:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right] $$\end{document} defined by Vnfx=anbn∫01tanf1-tξ+txt+bndt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} V_{n}\left( f\right) \left( x\right) =a_{n}b_{n}\int _{0}^{1}\frac{ t^{a_{n}}f\left( \left( 1-t\right) \xi +tx\right) }{t+b_{n}}dt \end{aligned}$$\end{document}where ξ∈0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in \left[ 0,1\right] $$\end{document}, ann∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( a_{n}\right) _{n\in {\mathbb {N}}}$$\end{document} is a sequence of natural numbers with limn→∞an=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }a_{n}=\infty $$\end{document}, bnn∈N⊂0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( b_{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right) $$\end{document} with limn→∞bn=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }b_{n}=\infty $$\end{document} and limn→∞anbn∈0,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \nolimits _{n\rightarrow \infty }\frac{a_{n}}{b_{n}}\in \left[ 0,\infty \right) $$\end{document}.