The new forms of Voronovskaya's theorem in weighted spaces

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor's formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor's formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Gruss type approximation theorem and the Gruss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.