Approximation by Max-Product Type Operators

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. Front Matter....Pages i-xv Introduction and Preliminaries....Pages 1-24 Approximation by Max-Product Bernstein Operators....Pages 25-158 Approximation by Max-Product Favard–Szász–Mirakjan Operators....Pages 159-188 Approximation by Max-Product Baskakov Operators....Pages 189-228 Approximation by Max-Product Bleimann–Butzer–Hahn Operators....Pages 229-243 Approximation by Max-Product Meyer–König and Zeller Operators....Pages 245-279 Approximation by Max-Product Interpolation Operators....Pages 281-325 Approximations by Max-Product Sampling Operators....Pages 327-392 Global Smoothness Preservation Properties....Pages 393-405 Possibilistic Approaches of the Max-Product Type Operators....Pages 407-428 Max-Product Weierstrass Type Functions....Pages 429-447 Back Matter....Pages 449-458