102 research outputs found

    Approximation by Genuine qq-Bernstein-Durrmeyer Polynomials in Compact Disks in the case q>1q > 1

    Full text link
    This paper deals with approximating properties of the newly defined qq-generalization of the genuine Bernstein-Durrmeyer polynomials in the case q>1q>1, whcih are no longer positive linear operators on C[0,1]C[0,1]. Quantitative estimates of the convergence, the Voronovskaja type theorem and saturation of convergence for complex genuine qq-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that for functions analytic in \left\{ z\in\mathbb{C}:\left\vert z\right\vert q, the rate of approximation by the genuine qq-Bernstein-Durrmeyer polynomials (q>1q>1) is of order qβˆ’nq^{-n} versus 1/n1/n for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine qq-Bernstein-Durrmeyer for q>1q>1

    Simultaneous approximation by operators of exponential type

    Full text link
    There are many results on the simultaneous approximation by sequences of special positive linear operators. In the year 1978, Ismail and May as well as Volkov independently studied operators of exponential type covering the most classical approximation operators. In this paper we study asymptotic properties of these class of operators. We prove that under certain conditions, asymptotic expansions for sequences of operators belonging to a slightly larger class of operators, can be differentiated term-by-term. This general theorem contains several results which were previously obtained by several authors for concrete operators. One corollary states, that the complete asymptotic expansion for the Bernstein polynomials can be differentiated term-by-term. This implies a well-known result on the Voronovskaja formula obtained by Floater

    Approximation by q-Szasz operators

    Full text link
    his paper deals with approximating properties of the newly defined qq-generalization of the Sz\'{a}sz operators in the case q>1q>1. 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 the qq-Sz\'{a}sz operators (q>1q>1) is of order qβˆ’nq^{-n} versus 1/n1/n for the classical Sz\'{a}sz--Mirakjan operators
    • …
    corecore