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-Szasz-Mirakjan, Baskakov and
Balazs-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 (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.
General
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