by Sorin G Gal (University of Oradea, Romania)

Table of Contents (128k)
Preface (103k)
Chapter 1: Bernstein-Type Operators of One Complex Variable (1,102k)

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.

• Bernstein-Type Operators of One Complex Variable:
  • Auxiliary Results in Complex Analysis
  • Bernstein Polynomials
  • Iterates of Bernstein Polynomials
  • Generalized Voronovskaja Theorems for Bernstein Polynomials
  • Butzer's Linear Combination of Bernstein Polynomials
  • q–Bernstein Polynomials
  • Bernstein–Stancu Polynomials
  • Bernstein–Kantorovich Type Polynomials
  • Favard–Szász–Mirakjan Operators
  • Baskakov Operators
  • Balázs–Szabados Operators
  • Bibliographical Notes and Open Problems
• Bernstein-Type Operators of Several Complex Variables:
  • Introduction
  • Bernstein Polynomials
  • Favard–Szász–Mirakjan Operators
  • Baskakov Operators
  • Bibliographical Notes and Open Problems
• Complex Convolutions:
  • Linear Polynomial Convolutions
  • Linear Non-Polynomial Convolutions
  • Nonlinear Complex Convolutions
  • Bibliographical Notes and Open Problems
• Appendix: Related Topics:
  • Bernstein Polynomials of Quaternion Variable
  • Approximation of Vector-Valued Functions
  • Strong Approximation by Complex Taylor Series
  • Bibliographical Notes and Open Problems