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–Szász–Mirakjan, Baskakov and Balázs–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 Vallée Poussin, Fejér, 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 (PDEs) are also 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.Contents:Bernstein-Type Operators of One Complex Variable:Auxiliary Results in Complex AnalysisBernstein PolynomialsIterates of Bernstein PolynomialsGeneralized Voronovskaja Theorems for Bernstein PolynomialsButzer's Linear Combination of Bernstein Polynomialsq–Bernstein PolynomialsBernstein–Stancu PolynomialsBernstein–Kantorovich Type PolynomialsFavard–Szász–Mirakjan OperatorsBaskakov OperatorsBalázs–Szabados OperatorsBibliographical Notes and Open ProblemsBernstein-Type Operators of Several Complex Variables:IntroductionBernstein PolynomialsFavard–Szász–Mirakjan OperatorsBaskakov OperatorsBibliographical Notes and Open ProblemsComplex Convolutions:Linear Polynomial Convolutions Linear Non-Polynomial ConvolutionsNonlinear Complex ConvolutionsBibliographical Notes and Open ProblemsAppendix: Related Topics:Bernstein Polynomials of Quaternion VariableApproximation of Vector-Valued FunctionsStrong Approximation by Complex Taylor SeriesBibliographical Notes and Open ProblemsReadership: Researchers and graduate students in the field of complex approximation of functions and its applications, mathematical analysis and numerical analysis.
