By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.
Starting from the study of the Shepard nonlinear operator of max-prod type in (Bede, Nobuhara et al., 2006, 2008), in the book (Gal, 2008), Open Problem 5.5.4, pp. 324–326, the Bleimann-Butzer-Hahn max-prod type operator is introduced and the question of the approximation order by this operator is raised. In this paper firstly we obtain an upper estimate of the approximation error of the form . A consequence of this result is that for each compact subinterval , with arbitrary , the order of uniform...
MSC 2010: 41A10, 41A15, 41A25, 41A36 For functions belonging to the classes C2[0; 1] and C3[0; 1], we establish the lower estimate with an explicit constant in approximation by Bernstein polynomials in terms of the second order Ditzian-Totik modulus of smoothness. Several applications to some concrete examples of functions are presented.
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