This monograph presents a broad treatment of developments in an area of
constructive approximation involving the so-called "max-product" type
operators. The exposition highlights the max-product operators as those
which allow one to obtain, in many cases, more valuable estimates than
those obtained by classical approaches. The text considers a wide
variety of operators which are studied for a number of interesting
problems such as quantitative estimates, convergence, saturation
results, localization, to name several.
Additionally, the book discusses the perfect analogies between the
probabilistic approaches of the classical Bernstein type operators and
of the classical convolution operators (non-periodic and periodic
cases), and the possibilistic approaches of the max-product variants of
these operators. These approaches allow for two natural interpretations
of the max-product Bernstein type operators and convolution type
operators: firstly, as possibilistic expectations of some fuzzy
variables, and secondly, as bases for the Feller type scheme in terms of
the possibilistic integral. These approaches also offer new proofs for
the uniform convergence based on a Chebyshev type inequality in the
theory of possibility.
Researchers in the fields of approximation of functions, signal theory,
approximation of fuzzy numbers, image processing, and numerical analysis
will find this book most beneficial. This book is also a good reference
for graduates and postgraduates taking courses in approximation theory.
