Quantitative Approximations

by Anastassiou; George

• ISBN13: 9781584882213
• ISBN10: 1584882212
• Format: Hardcover
• Copyright: 2000-09-15
• Publisher: Chapman & Hall/

Summary

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.

Author Biography

George Anastassiou is a Professor in the Department of Mathematical Sciences at the University of Memphis in Tennessee

Table of Contents

Preface	xiii
Introduction	1	(56)
Convergence with Rates of Univariate Neural Network Operators to the Unit Operator	1	(3)
Convergence with Rates of Multivariate Neural Network Operators to the Unit Operator	4	(3)
Asymptotic Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators	7	(2)
Asymptotic Weak Convergence of Squashing Neural Network Operators	9	(1)
Quantitative Monotone and Probabilistic Wavelet Type Approximation	10	(2)
Quantitative One-Dimensional High Order Wavelet Type Approximation	12	(2)
More on Shape and Probability Preserving One-Dimensional Wavelet Type Operators	14	(1)
Quantitative Multidimensional High Order Wavelet Type Approximation	15	(3)
Rate of Convergence of Probabilistic Discrete Wavelet Approximation	18	(1)
Asymptotic Nonorthogonal Wavelet Approximation for Deterministic Signals	19	(1)
Wavelet Type Differentiated Shift Invariant Integral Operators	20	(2)
A Discrete Kac's Formula and Optimal Quantitative Approximation in the Solution of Heat Equation	22	(3)
Quantitative-Asymptotic Expansions for the Probabilistic Representation Formulae for (CO) m-Parameter Operator Semigroups	25	(2)
Quantitative Probability Limit Theorems over Banach Spaces	27	(3)
Quantitative Study of Bias Convergence for Generalized L-Statistics	30	(3)
Quantitative Korovkin-Type Results for Vector Valued Functions	33	(2)
Quantitative Lp-Results for Positive Linear Operators	35	(3)
On Monotone Approximation Theory	38	(3)
Comparisons for Local Moduli of Continuity	41	(1)
Convergence with Rates of Univariate Singular Integrals to the Unit	42	(2)
About Univariate Ostrowski-Type Inequalities	44	(1)
About Multidimensional Ostrowski-Type Inequalities	45	(2)
General Opial-Type Inequalities for Linear Differential Operators	47	(3)
Lp-Opial Inequalities Engaging Fractional Derivatives of Functions	50	(2)
Lp-General Fractional Opial Inequalities	52	(5)
I On Neural Networks	57	(86)
Convergence with Rates of Univariate Neural Network Operators to the Unit Operator	59	(30)
Convergence with Rates of the Univariate Cardaliaguet-Euvrard Neural Network Operators	60	(16)
The Univariate ``Squashing Operators'' and their Convergence to the Unit with Rates	76	(13)
Convergence with Rates of Multivariate Neural Network Operators to the Unit Operator	89	(32)
Convergence with Rates of Multivariate Cardaliaguet-Euvrard Neural Network Operators	90	(21)
The Multivariate ``Squashing Operators'' and their Convergence to the Unit with Rates	111	(10)
Asymptotic Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators	121	(14)
Background	121	(3)
General Result	124	(6)
Supplement	130	(5)
Asymptotic Weak Convergence of Squashing Neural Network Operators	135	(8)
Background	135	(2)
General Result	137	(6)
II On Wavelets	143	(130)
Quantitative Monotone and Probabilistic Wavelet Type Approximation	145	(20)
Approximation by Ak(f) Operators	146	(10)
Approximation by Bk(f) Operators	156	(3)
More Results on Ak(f) and Bk(f) Operators	159	(6)
Quantitative One-Dimensional High Order Wavelet Type Approximation	165	(20)
Estimates and Sharpness	165	(20)
More on Shape and Probability Preserving One-Dimensional Wavelet Type Operators	185	(16)
Estimates, Maintenance, and Sharpness	186	(15)
Quantitative Multidimensional High Order Wavelet Type Approximation	201	(28)
Estimates and Sharpness	201	(28)
Rate of Convergence of Probabilistic Discrete Wavelet Approximation	229	(6)
Approximation by Lk(f,x) Operators	230	(5)
Asymptotic Nonorthogonal Wavelet Approximation for Deterministic Signals	235	(24)
Introduction	235	(2)
Asymptotic Wavelet Approximation at Resolution 2-k	237	(22)
Wavelet Type Differentiated Shift-Invariant Integral Operators	259	(14)
General Results	259	(10)
Examples	269	(4)
III On Partial Differential Equations	273	(30)
A Discrete Kac's Formula and Optimal Quantitative Approximation in the Solution of Heat Equation	275	(28)
Auxiliary Results	277	(2)
The Dirichlet Problem: Discrete Case	279	(14)
About Approximation on the Grid	293	(6)
On Sharpness of the Error Estimate	299	(4)
IV On Semigroups	303	(30)
Quantitative Asymptotic Expansions of the Probabilistic Representation Formulae for (C0) m-Parameter Operator Semigroups	305	(28)
Background	305	(3)
Basic Results	308	(6)
Quantitative Asymptotic Probabilistic Expansions	314	(9)
Asymptotic Probabilistic Expansions for Continuous Type Formulae	323	(4)
Examples	327	(6)
V On Stochastics	333	(48)
Quantitative Probability Limit Theorems over Banach Spaces	335	(26)
Limit Theorems for Martingales in Banach Spaces	336	(12)
Auxiliary Results	336	(4)
Main Theorems	340	(8)
On Weak Invariance Principle	348	(13)
Background	348	(4)
Central Results	352	(9)
Quantitative Study of Bias Convergence for Generalized L-Statistics	361	(20)
Background	361	(4)
Quantitative Results for L-Statistics with Standard Weights	365	(4)
Quantitative Results for L-Statistics with Nonstandard Weights	369	(10)
More Conclusions	379	(2)
VI On Functional Analysis	381	(68)
Quantitative Korovkin-Type Results for Vector Valued Functions	383	(30)
Background	383	(2)
Auxiliary Results	385	(11)
Quantitative General Theorems	396	(9)
Further Results	405	(6)
Applications	411	(2)
Quantitative Lp-Results for Positive Linear Operators	413	(36)
Background	414	(5)
Univariate Results	419	(14)
Abstract Theory	433	(5)
Multidimensional Theory	438	(6)
Stochastic Theory	444	(5)
VII On Approximation Theory	449	(46)
On Monotone Approximation Theory	451	(22)
High Order Quantitative Monotone Approximation with Linear Differential Operators	451	(4)
Quantitative Monotone Approximation with Pseudo-Polynomials	455	(6)
Quantitative Bivariate Monotone Approximation	461	(4)
Quantitative Spline Monotone Approximation with Linear Differential Operators	465	(8)
Quantitative Monotone Approximation by Polynomial Splines	465	(4)
Quantitative Monotone Approximation by Periodic Polynomial Splines	469	(2)
Quantitative Monotone Approximation by Discrete Polynomial Splines	471	(2)
Comparisons for Local Moduli of Continuity	473	(10)
Univariate Results	473	(6)
A Multivariate Result	479	(4)
Convergence with Rates of Univariate Singular Integrals to the Unit	483	(12)
Background	483	(2)
Quantitative Lp-Approximation, 1 ≤ p < +∞	485	(7)
Quantitative Uniform Approximation by Qn,ξ Operator	492	(3)
VIII On Classical Analysis	495	(90)
About Univariate Ostrowski-Type Inequalities	497	(10)
About Ostrowski's Inequality	497	(2)
About More General Univariate Ostrowski-Type Inequalities	499	(8)
About Multidimensional Ostrowski-Type Inequalities	507	(14)
General Results	507	(14)
General Opial-Type Inequalities for Linear Differential Operators	521	(18)
Setting	522	(1)
General Results	523	(13)
An Application to Differential Equations	536	(3)
Lp-Opial-Type Inequalities Engaging Fractional Derivatives of Functions	539	(28)
Background	539	(6)
General Results	545	(11)
Applications to Differential Equations	556	(11)
Lp-General Fractional Opial Inequalities	567	(18)
General Results	567	(18)
References	585	(14)
List of Symbols	599	(6)
Index	605