On a Modified Durrmeyer-Bernstein Operator and Applications.pdf

AMRX Applied Mathematics Research eXpress 2005, No. 4 On a Modified Durrmeyer-Bernstein Operator and Applications Germain E. Randriambelosoa 1 Introduction Durrmeyer [6] has introduced a Bernstein-type operator of degree n defined by Mn(f, x) = (n + 1) n∑ i=0 bni (x) ∫1 0 f(u)bni (u)du, (1.1) where f(u) is an integrable function on [0, 1] and bni (t) = ( n i ) ti(1 ? t)n?i, i = 0, . . . , n, are the degree n Bernstein basis polynomials. This operator is a modified kind of the classical Bernstein operator Bn(f, x) = n∑ i=0 f (( i n )) bni (x). (1.2) Many authors have studied the operator Mn(f, x) [1, 3, 4, 5, 10]. However, the op- erator Mn(?, x) does not possess the property of endpoint interpolation which is essen- tial for interpolation problem. For this reason, we consider in this paper a modified kind of the Durrmeyer-Bernstein operator introduced by Goodman and Sharma [11], defined by Un(f, x) = bn0 (x)f(0) + (n ? 1) n?1∑ i=1 aib n i (x) + b n n(x)f(1), (1.3) where x ∈ [0, 1], f is an integrable function on [0, 1], and ai = ∫1 0 f(u)bn?2i?1 (u)du. Received 30 December 2004. Revision received 4 July 2005. 170 Germain E. Randriambelosoa The polynomial Un(f, x) must be compared to the Bernstein-Kantorovich polyno- mial given by Kn(f, x) = (n + 1) n∑ k=0 bnk (t) ∫ (k+1)/(n+1) k/(n+1) f(u)du. (1.4) Moreover, the operator Un(f, x) has the endpoint interpolation property Un(f, 0) = f(0), Un(f, 1) = f(1), (1.5) and other interesting properties [1, 12]. In Section 2, we recall some basic properties of the operator Un given in [12] which constitute the main advantages of our degree reduction and approximation meth- od exposed in next sections. In Section 3, we present a new method for the degree reduction of a Be?zier curve with endpoint interpolation by application of the operatorUn to a vector-valued function f : [0, 1] → Rd. A degree n Be?zier curve is a parametric curve in Rd defined by cn(t) = n∑ i=0 pib n i (t), t ∈ [0, 1], (1.6) where pi ∈ Rd, i = 0, . . . , n, ar