The monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several complex variables, that is, to extend their quantitative convergence properties to larger sets in the complex plane rather than the real intervals. The operators studied are of the following types: Bernstein, Bernstein Faber, Bernstein Butzer, q-Bernstein, Bernstein Stancu, Bernstein Kantorovich, Favard Sz�sz Mirakjan, Baskakov and Bal�zs Szabados.The second main objective is to provide a study of the approximation and geometric properties of several types of complex convolutions: the de la Vallee Poussin, Fejer, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson Cauchy, Gauss Weierstrass, q-Picard, q-Gauss Weierstrass, Post Widder, rotation-invariant, Sikkema and nonlinear. Several applications to partial differential equations (PDE) also are presented.Many of the open problems encountered in the studies are proposed at the end of each chapter. For further research, the monograph suggests and advocates similar studies for other complex Bernstein-type operators, and for other linear and nonlinear convolutions.

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The material of this book is technical, but the author has organized it very clearly. The book is well written and readable. The author has made an attempt to present the material in an integrated and self-contained fashion and, in my opinion, he has been very successful. The book is addressed to researchers in the fields of complex approximation of functions and its applications, mathematical analysis and numerical analysis. Since most of the proofs use elementary complex analysis, it is accessible to graduate students and suitable for graduate courses in the above fields. --Mathematical Reviews

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