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Article Citation - WoS: 16Citation - Scopus: 16q-bernstein Polynomials of the Cauchy Kernel(Elsevier Science inc, 2008) Ostrovska, SofiyaDue to the fact that in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0, 1], the study of their approximation properties is essentially more difficult than that for 0 < q < 1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0, 1] uniformly approximated by their q-Bernstein polynomials (q > 1) is still open. In this paper, the q-Bernstein polynomials B-n,B-q(f(a); z) of the Cauchy kernel f(a) = 1/(z - a), a is an element of C \ [0, 1] are found explicitly and their properties are investigated. In particular, it is proved that if q > 1, then polynomials B-n,B-q(f(a); z) converge to f(a) uniformly on any compact set K subset of {z : vertical bar z vertical bar < vertical bar a vertical bar}. This result is sharp in the following sense: on any set with an accumulation point in {z : vertical bar z vertical bar > vertical bar a vertical bar}, the sequence {B-n,B-q(f(a); z) is not even uniformly bounded. (C) 2007 Elsevier Inc. All rights reserved.Article Citation - WoS: 8Citation - Scopus: 9Improved convergence criteria for Jacobi and Gauss-Seidel iterations(Elsevier Science inc, 2004) Özban, AY; MathematicsSome simple criteria for the convergence of the Jacobi, Gauss-Seidel and SOR iterations have been proposed in the work of Huang [ZAMM 76-1 (1996) 57-58]. In this study we present some modified forms of the criteria introduced in Huang's work. The new criteria also allow for the norms of the Jacobi iteration matrices to be greater than unity. Numerical examples are also given which show the effectiveness of the criteria. (C) 2003 Elsevier Inc. All rights reserved.
