构造逼近

Chapter 1. Problems of Polynomial Approximation

Chapter 2. Approximation Problems with Constraints

Chapter 3. Incomplete Polynomials

Chapter 4. Weighted Polynomials

Chapter 5. Wavelets and Orthogonal Expansions

Chapter 6. Splines

Chapter 7. Rational Approximation

Chapter 8. Stahl's Theorem

Chapter 9. Pade Approximation

Chapter 10. Hardy Space Methods in Rational Approximation

Chapter 11. Muntz Polynomials

Chapter 12. Nonlinear Approximation

Chapter 13. Widths I

Chapter 14. Widths II: Weak Asymptotics for Widths of Lipschitz Balls, R.andom Approximants

Chapter 15. Entropy

Chapter 16. Convergence of Sequences of Operators

Chapter 17. R,epresentation 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

Most oPfpthe chapters of the present book can be read independently of each other. They fall into groups: Chapters 1-6 deal with polynomial and spline approximation - in some sense they continue the themes of [CA]. Chap- ters 7-10 contain a fairly complete theory of rational approximation. Chapters 12-14 treat widths and entropy of classes of functions. But even within the groups, chapters are more or less independent, except that it is advisable to read Chapter 3 before Chapter 4, while Chapter 7 is indispensable for Chap- ters 8 and 10, and Chapter 13 for Chapter 14. Most of the information about Banach function spaces needed in the two volumes of CA can be found in [CA, Chapter 2], in 7 of Chapter 1 of the present volume, and in the book of Bennett and Sharpley [B-1988]. We also provide a quick new look at some of the important approximation theorems: for polynomials in 7 of Chapter 1, for splines in 1 of Chapter 6.

《构造逼近》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。
　　《构造逼近》读者对象：数学专业的本科生、研究生和相关的科研人员。