Abstract
In this paper we first use a probabilistic method to construct a linear positive polynomial operatorL α,βm, r Bernstein type, depending on a non-negative integer parameterr and on two real parameters α and β, such that 0≤α≤β. Then we investigate the approximation properties of this operator mapping into itself the Banach spaceC[0,1] of real-valued continuous functions on [0,1]. A special attention is accorded to the case of the operatorL m,r=L 0,0m,r. We prove that the remainder of the approximation formula of a functionfεC[0,1] byL m,r f can be represented either by means of divided differences, or in an integral form, obtained by using a classical theorem of Peano. We give also an asymptotic estimate for this remainder. The operatorL m,r enjoys the variation diminishing property—in the sense of I. J. Schoenberg [15]. By extending the known inequalities of T. Popoviciu [12] and G. G. Lorentz [7], we evaluate the orders of approximation in terms of the modulus of continuity of the functionf or of its derivative. In the last section of this paper we determine the point spectrum of the operatorL m,r and , finally, we present a quadrature formula which can be constructed by means of this operator.
Authors
Dimitrie D. Stancu
“Babes-Bolyai” University, Cluj-Napoca, Romania
Keywords
Quadrature Formula; Point Spectrum; Approximation Formula; Bernstein Polynomial; Divided Difference
Paper coordinates
D.D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (1983) no. 2, 211–229, https://doi.org/10.1007/BF02575593
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