1st Edition

Quantitative Approximations

By George Anastassiou Copyright 2001
    624 Pages 150 B/W Illustrations
    by Chapman & Hall

    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.

    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

    Biography

    George Anastassiou

    "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