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Quantitative Approximations




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ISBN 9781584882213
Published September 15, 2000 by Chapman and Hall/CRC
624 Pages 150 B/W Illustrations

 
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Book Description

Quantitative approximation methods apply in many diverse fields of research-neural networks, wavelets, partial differential equations, probability and statistics, functional analysis, and classical analysis to name just a few. For the first time in book form, Quantitative Approximations provides a thorough account of all of the significant developments in the area of contemporary quantitative mathematics. It offers readers the unique opportunity of approaching the field under the guidance of an expert.

Among the book's outstanding features is the inclusion of the introductory chapter that summarizes the primary and most useful results. This section serves not only as a more detailed table of contents for those new to an area of application, but also as a quick reference for more seasoned researchers.

The author describes all of the pertinent mathematical entities precisely and concretely. His approach and proofs are straightforward and constructive, making Quantitative Approximations accessible and valuable to researchers and graduate students alike.

Table of Contents

INTRODUCTION
Summary of Results
ON NEURAL NETWORKS
Convergence with Rates of Univariate Neural Network Operators to the Unit Operator
Convergence with Rates of Multivariate Neural Network Operators to the Unit Operator
Asymptotic Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators
Asymptotic Weak Convergence of Squashing Neural Network Operators
ON WAVELETS
Quantitative Monotone and Probabilistic Wavelet Type Approximation
Quantitative Multidimensional High Order Wavelet Type Approximation
More on Shape and Probability Preserving One-Dimensional Wavelet Type Operators
Quantitative Multidimensional High Order Wavelet Type Approximation
Rate of Convergence of Probabilistic Discrete Wavelet Approximation
Asymptotic Non-Orthogonal Wavelet Approximation for Deterministic Signals
Wavelet Type Differentiated Shift-Invariant Integral Operators
ON PARTIAL DIFFERENTIAL EQUATIONS
A Discrete Kac's Formula and Optimal Quantitative Approximation in the Solution of Heat Equation
ON SEMIGROUPS
Quantitative Asymptotic Expansions of the Probabilistic Representation Formulae for (C0 ) m-Parameter Operator Semigroups
ON STOCHASTICS
Quantitative Probability Limit Theorems over Banach Spaces
Quantitative Study of Bias Convergence for Generalized L-Statistics
ON FUNCTIONAL ANALYSIS
Quantitative Korovkin-Type Results for Vector Valued Functions
Quantitative Lp Results for Positive Linear Operators
ON APPROXIMATION THEORY
On Monotone Approximation Theory
Comparisons for Local Moduli of Continuity
Convergence with Rate of Univariate Singular Integrals to the Unit
ON CLASSICAL ANALYSIS
About Univariate Ostrowski Type Inequalities
About Multidimensional Ostrowski Type Inequalities
General Opial Type Inequalities for Linear Differential Operators
Lp Opial Type Inequalities Engaging Fractional Derivatives of Functions
Lp General Fractional Opial Inequalities

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Reviews

"George Anastassiou has done a tremendous job in showing the diversity of quantitative approximation methods…I am pleased to see an enormous collection of essential quantitative results…there is no doubt that this book should be on the desk of any student or specialist with interests in approximation, limit theorems, and more broadly quantitative mathematical methods and their applications"
-Zari Rachev, University of California, Santa Barbara
"The emphasis of the results presented in this monograph is quantitative, thus, for example, all mathematical entities such as operators or constants are defined precisely. This makes the material accessible to a wide audience of graduate students and scientists. The book is well written and contains a very extensive summary of the author's work over the last fifteen years."
--Steven B. Damelin, in Mathematical Reviews, Issue 2001h