洛倫茨的《構造逼近(英文版)》是逼近理論的經
典著作，既是一部教程，也是一部很優秀的參考用書
。在過去的的30年中，逼近理論得到了驚人的發展，
新理論在短時期內也是不斷湧現。本書的初衷是極盡
全力描述該科目的發展，特別是將G. G.
Lorentz,1966年版本《函數逼近》進行了大力擴充。
在1980年R. A. DeVore 和Lorentz的加入為完成這
項使命注入了強動力，產生了1993年版本的《結構逼
近》，也就是這個繫列的303卷；後來M. v.
Golitschek 和Y. Makovoz加入到Lorentz的隊伍中
來，為了目前的這個版本效力，也是第一個版本的延
續。本書的目的並不是追求完美，在一些理論中，隻
節選最重要的表示定理，而在另外一些情況則會繫統
講述。如同前一版本，書中隻講述單變量的函數逼近
，因此，多變函數、復結構和插值並沒有處理。
目次：多項式逼近問題；有約束條件的逼近問題
；不完全多項式；權重多項式；小波和正交展開；樣
條；有理逼近；Stahl定理；Pade逼近；有理逼近中
的Hardy空間方法；Muntz多項式；非線性近似；寬度
Ⅰ型；寬度Ⅱ；熵；算子序列的收斂；函數表示的疊
加原理；附錄：Borsuk定理和Brunn-Minkowski；一
些橢圓積分的估計；Hardy空間和Blaschke乘積；勢
理論和對數容量。
讀者對像：數學專業的本科生、研究生和相關的
科研人員。

Chapter 1. Problems of Polynomial Approximation
1. Examples of Polynomials of Best Approximation
2. Distribution of Alternation Points of Polynomials of Best Approximation
3. Distribution of Zeros of Polynomials of Best Approximation
4. Error of Approximation
5. Approximation on （—∞，∞） by Linear Combinations of Functions （x—c）—1
6. Weighted Approximation by Polynomials on （—∞，∞）
7. Spaces of Approximation Theory
8. Problems and Notes
Chapter 2. Approximation Problems with Constraints
1. Introduction
2. Growth Restrictions for the Coefficients
3. Monotone Approximation
4. Polynomials with Integral Coefficients
5. Determination of the Characteristic Sets
6. Markov—Type Inequalities
7. The Inequality of Remez
8. One—sided Approximation by Polynomials
9. Problems
10. Notes
Chapter 3. Incomplete Polynonuals
1. Incomplete Polynomials
2. Incomplete Chebyshev Polynomials
3. Incomplete Trigonometric Polynomials
4. Sequences of Polynomials with Many Real Zeros
5. Problems
6. Notes
Chapter 4. Weighted Polynomials
1. Essential Sets of Weighted Polynomials
2. Weighted Chebyshev Polynomials
3. The Equilibrium Measure
4. Determination of Minimal Essential Sets
5. Weierstrass Theorems and Oscillations
6. Weierstrass Theorem for Freud Weights
7. Problems
8. Notes
Chapter 5. Wavelets and Orthogonal Expansions
1. Multiresolutions and Wavelets
2. Scaling Functions with a Monotone Majorant
3. Periodization
4. Polynomial Schauder Bases
5. Orthonormal Polynomial Bases
6. Problems and Notes
Chapter 6. Splines
1. General Facts
2. Splines of Best Approximation
3. Periodic Splines
4. Convergence of Some Spline Operators
5. Notes
Chapter 7. Rational Approximation
1. Introduction
2. Best Rational Approximation
3. Rational Approximation of ｜x｜
4. Approximation of ex on （—1，1）
5. Rational Approximation of e—x on （0，∞）
6. Approximation of Classes of Functions
7. Theorems of Popov
8. Properties of the Operator of Best Rational Approximation in C and Lp
9. Approximation by Rational Functions with Arbitrary Powers
10. Problems
11. Notes
Chapter 8. Stahl's Theorem
1. Introduction and Main Result
2. A Dirichlet Problem on （1／2，1／ρn）
3. The Second Approach to the Dirichlet Problem
4. Proof of Theorem 1. 1
5. Notes
Chapter 9. Pade Approximation
1. The Pade Table
2. Convergence of the Rows of the Pade Table
3. The Nuttall—Pommerenke Theorem
4. Problems
5. Notes
Chapter 10. Hardy Space Methods in Rational Approximation
1. Bernstein—Type Inequalities for Rational Functions
2. Uniform Rational Approximation in Hardy Spaces
3. Approximation by Simple Functions
4. The Jackson—Rusak Operator； Rational Approximation of Sums of Simple Functions
5. Rational Approximation on T and on （—1，1）
6. Relations Between Spline and Rational Approximation in the Spaces Lp， 0＜p＜∞
7. Problems
8. Notes
Chapter 11. Muntz Polynomials
1. Definitions and Simple Properties
2. Muntz—Jackson Theorems
3. An Inverse Muntz—Jackson Theorem
4. The Index of Approximation
5. Markov—Type Inequality for Muntz Polynomials
6. Problems
7. Notes
Chapter 12. Nonlinear Approximation
1. Definitions and Simple Properties
2. Varisolvent Families
3. Exponential Sums
4. Lower Bounds for Errors of Nonlinear Approximation
5. Continuous Selections from Metric Projections
6. Approximation in Banach Spaces： Suns and Chebyshev Sets
7. Problems
8. Notes
Chapter 13. Widths I
1. Definitions and Basic Properties
2. Relations Between Different Widths
3. Widths of Cubes and Octahedra
4. Widths in Hilbert Spaces
5. Applications of Borsuk's Theorem
6. Variational Problems and Spectral Functions
7. Results of Buslaev and Tikhomirov
8. Classes of Differentiable Functions on an Interval
9. Classes of Analytic Functions
10. Problems
11. Notes
Chapter 14. Widths H： Weak Asymptotics for Widths of Lipschitz Balls， Random Approximants
1. Introduction
2. Discretization
3. Weak Equivalences for Widths. Elementary Methods
4. Distribution of Scalar Products of Unit Vectors
5. Kashin's Theorems
6. Gaussian Measures
7. Linear Widths of Finite Dimensional Balls
8. Linear Widths of the Lipschitz Classes
9. Problems
10. Notes
Chapter 15. Entropy
1. Entropy and Capacity
2. Elementary Estimates
3. Linear Approximation and Entropy
4. Relations Between Entropy and Widths
5. Entropy of Classes of Analytic Finctions
6. The Birman—Solomyak Theorem
7. Entropy Numbers of Operators
8. Notes
Chapter 16. Convergence of Sequences of Operators
1. Introduction
2. Simple Necessary and Sufficient Conditions
3. Geometric Properties of Dominating Sets
4. Strict Dominating Systems; Minimal Systems; Examples
5. Shadows of Sets of Continuous Functions
6. Shadows in Banach Function Spaces
7. Positive Contractions
8. Contractions
9. Notes
Chapter 17. Representation of Functions by Superpositions
1. The Theorems of Kolmogorov
2. Proof of the Theorems
3. Functions Not Representable by Superpositions
4. Linear Superpositions
5. Notes
Appendix 1. Theorems of Borsuk and of Brunn-Minkowski
1. Borsuk's Theorem
2. The Brunn-Minkowski Inequality
Appendix 2. Estimates of Some Elliptic Integrals
Appendix 3. Hardy Spaces and Blaschke Products
1. Hardy Spaces
2. Conjugate Functions and Cauchy Integrals
3. Atomic Decompositions in Hardy Spaces
4. Blaschke Products
Appendix 4. Potential Theory and Logarithmic Capacity
1. Logarithmic Potentials
2. Equilibrium Distribution and Logarithmic Capacity
3. The Dirichlet Problem and Green's Function
4. Balayage Methods
Bibliography
Author Index
Subject Index