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This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian-Ivanov for linear combinations of Bernstein and Bernstein-Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduateand postgraduate courses as well as a basis for future study and development.
List of contents
1. Moments and Linear Combinations of Positive Linear Operators.- 2. Direct Estimates for Approximation by Linear Combinations.- 3. Inverse Estimates and Saturation Results for Linear Combinations.- 4. Voronovskaja Type Estimates.- 5. Pointwise Estimates for Linear Combinations.- 6. Voronovskaja's Theorem in Terms of Weighted Modulus of Continuity.- 7. Direct Estimates for Some New Operators.- 8. Convergence for Operators Based on P lt anea Basis.- Bibliography.- Index.
Summary
This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian–Ivanov for linear combinations of Bernstein and Bernstein–Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduateand postgraduate courses as well as a basis for future study and development.
Additional text
“This book deals with linear positive operators, one of the central branches of approximation theory, and their combinations, which allows one to increase the degree of approximation. … It is carefully written and organized. In each chapter, remarkable comments and explanatory remarks are expressed that support and extend the basic text. … This significant book can be recommended as a source book to graduate students and researchers in approximation theory.” (Nurhayat İspır, Mathematical Reviews, January, 2018)
“The book under review written by two experts in approximation by positive linear operators (PLO, shortly), consists in eight chapters and contains most of the recent results in the topics announced by the contents of the book, very many belonging to the authors. … This book will be of interest to researchers working in approximation theory and for courses in approximation theory for graduate students.” (Sorin Gheorghe Ga, zbMATH 1371.41035, 2017)
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"This book deals with linear positive operators, one of the central branches of approximation theory, and their combinations, which allows one to increase the degree of approximation. ... It is carefully written and organized. In each chapter, remarkable comments and explanatory remarks are expressed that support and extend the basic text. ... This significant book can be recommended as a source book to graduate students and researchers in approximation theory." (Nurhayat Ispir, Mathematical Reviews, January, 2018)
"The book under review written by two experts in approximation by positive linear operators (PLO, shortly), consists in eight chapters and contains most of the recent results in the topics announced by the contents of the book, very many belonging to the authors. ... This book will be of interest to researchers working in approximation theory and for courses in approximation theory for graduate students." (Sorin Gheorghe Ga, zbMATH 1371.41035, 2017)
