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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.
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.
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