APPROXIMATION BY GENERALIZED q-SZASZ-MIRAKJAN ´ OPERATORS

In this article, we introduce generalized q−Sz´asz-Mirakjan operators and study their approximation properties. Based on the Voronovskaja’s theorem, we obtain quantitative estimates for these operators.

Voronovskaja’s theorem for functions with exponential growth

In the present paper we establish a general form of Voronovskaja’s theorem for functions defined on an unbounded interval and having exponential growth. The case of approximation by linear combinations is also considered. Applications are given for some Szász–Mirakyan and Baskakov-type operators.

Approximation by the -Szász-Mirakjan Operators

This paper deals with approximating properties of theq-generalization of the Szász-Mirakjan operators in the case . Quantitative estimates of the convergence in the polynomial-weighted spaces and the Voronovskaja's theorem are given. In particular, it is proved that the rate of approximation by theq-Szász-Mirakjan operators ( ) is of order versus 1/nfor the classical Szász-Mirakjan operators.

On generalized Voronovskaja theorem for Bernstein polynomials

We establish a new quantitative variant of Floater’s theorem dealing with the generalization of Voronovskaja’s theorem for Bernstein polynomials. Our estimate improves the recent quantitative version of Floater’s theorem proved in [Gonska, H. and Ras¸a, I., Asymptotic behaviour of differentiated Bernstein polynomials, Mat. Vesnik 61 (2009), 53-60]. Moreover, as a particular case, we recover the main result of [Tachev, G. T., Voronovskaja’s theorem revisited, J. Math. Anal. Appl. 343 (2008), 399-404].