本书是逼近理论的经典著作，既是一部教程，也是一部很优秀的参考用书。在过去的的30年中，逼近理论得到了惊人的发展，新理论在短时期内也是不断涌现。本书的初衷是极尽全力描述该科目的发展，特别是将G. G. Lorentz,1966年版本《函数逼近》进行了大力扩充。在1980年R. A. DeVore 和Lorentz的加入为完成这项使命注入了强动力，产生了1993年版本的《结构逼近》，也就是这个系列的303卷；后来M. v. Golitschek 和Y. Makovoz加入到 Lorentz的队伍中来，为了目前的这个版本效力，也是第一个版本的延续。本书的目的并不是追求完美，在一些理论中，只节选最重要的表示定理，而在另外一些情况则会系统讲述。如同前一版本，书中只讲述单变量的函数逼近，因此，多变函数、复结构和插值并没有处理。
目次：多项式逼近问题；有约束条件的逼近问题；不完全多项式；权重多项式；小波和正交展开；样条；有理逼近；Stahl定理；Pade逼近；有理逼近中的Hardy空间方法；Muntz多项式；非线性近似；宽度Ⅰ型；宽度Ⅱ；熵；算子序列的收敛；函数表示的叠加原理；附录：Borsuk定理和Brunn-Minkowski；一些椭圆积分的估计；Hardy空间和Blaschke乘积；势理论和对数容量。
读者对象：数学专业的本科生、研究生和相关的科研人员。
關於作者：
George G. LorentzG.G.洛伦茨, 美国是国际知名学者，在数学和物理学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。
目錄：
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
Chapter 17.Representation of Functions by Superpositions
Appendix 1.Theorems of Borsuk and of Brunn—Minkowski
Appendix 2.Estimates of Some Elliptic Integrals
Appendix 3.Hardy Spaces and Blaschke Products
Appendix 4.Potential Theory and Logarithmic Capacity
Bibliography
Author Index
Subject Index
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