Quantitative Approximations: 1st Edition (Hardback) book cover

Quantitative Approximations

1st Edition

By George Anastassiou

Chapman and Hall/CRC

624 pages | 150 B/W Illus.

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Hardback: 9781584882213
pub: 2000-09-15
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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.

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

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

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations