Theory of approximation 逼近论

A pioneer of many modern developments in approximation theory, Professor N. I. Achieser designed this graduate-level text from the standpoint of functional analysis. Its clear, cogent treatments range from classical topics such as Weierstrass' first and second theorems to surveys of Hadamard's theorem on determinants, Tchebysheff's approximation by rational functions, de la Vallee-Poussin's approximation theorem, S. N. Bernstein's theorems, and related topics.
　　The first two chapters address approximation problems in linear normalized spaces and the ideas of R L. Tchebysheff. Chapter Ill examines the elements of harmonic analysis, and Chapter IV, integral transcendental functions of the exponential type. The final two chapters explore the best harmonic approximation of functions and Wiener's theorem on approximation. Professor Achieser concludes this exemplary text with an extensive section of problems and applications, including elementary extremal problems, Szego's theorem and some of its applications, the Carathfiodory-Fejer problem, and Soltareff's problems.

CHAPTERⅠAPPROXIMATION PROBLEMS INLINEAR NORMALIZED SPACES
　1. Formulation of the Principal Problem in the Theory of Approximation
　2. The Concept of Metric Space
　3. The Concept of Linear Normalized Space
　4. Examples of Linear Normalized Spaces
　5. The Inequalities of HSlder and Minkowski
　6. Additional Examples of Linear Normalized Spaces
　7. Hilbert Space
　8. The Fundamental Theorem of Approximation Theory in Linear Normalized Spaces
　9. Strictly Normalized Spaces
　10. An Example of Approximation in the Space Lp
　11. Geometric Interpretation
　12. Separable and Complete Spaces
　13. Approximation Theorems in Hilbert Space
　14. An Example of Approximation in Hilbert Space
　15. More About the Approximation Problem in Hilbert Space
　16. Orthonormalized Vector Systems in Hilbert Space
　17. Orthogonalization of Vector Systems
　18. Infinite Orthonormalized Systems
　19. An Example of a Non-Separable System
　20. Weierstrass' First Theorem
　21. Weierstrass' Second Theorem
　22. The Separability of the Space C
　23. The Separability of the Space Lp
　24. Generalization of Weierstrass' Theorem to the Space Lp
　25. The Completeness of the Space Lp
　26. Examples of Complete Orthonormalized Systems in L2
　27. Mintz's Theorem
　28. The Concept of the Linear Functional
　29. F. Riesz's Theorem
　30. A Criterion for the Closure of a Set of Vectors in Linear Normalized Spaces
CHAPTER Ⅱ P. L. TCHEBYSHEFF'S DOMAIN OF IDEAS
　31. Statement of the Problem .
　32. A Generalization of the Theorem of de la Vall6e-Poussln
　33. The Existence Theorem
　34. Tchebysheff's Theorem
　35. A Special Case of Tchebysheff's Theorem
　36. The Tchebysheff Polynomials of Least Deviation from Zero
　37. A Further Example of P. Tchebysheff's Theorem
　38. An Example for the Application of the General Theorem of de la Vallde-Poussin
　39. An Example for the Application of P. L. Tehebysheff's General Theorem
　40. The Passage to Periodic Functions
　41. An Example of Approximating with the Aid of Periodic Functions
　42. The Weierstrass Function
　43. Haar's Problem
　44. Proof of the Necessity of Haar's Condition
　45. Proof of the Sufficiency of Haar's Condition
　46. An Example Related to Haar's Problem
　47. P. L. Tchebysheff's Systems of Functions
　48. Generalization of P. L. Tchebysheff's Theorem
　49. On a Question Pertaining to the Approximation of a Continuous Function in the Space L
　50. A. A. Markoff's Theorem
　51. Special Cases of the Theorem of A. A. Markoff
CHAPTER Ⅲ ELEMENTS OF HARMONIC ANALYSIS
　52. The Simplest Properties of Fourier Series
　53. Fourier Series for Functions of Bounded Variation
　54. The Parseval Equation for Fourier Series
　55. Examples of Fourier Series
　56. Trigonometric Integrals
　57. The Riemann-Lebesgue Theorem
　58. Planeherel's Theory
　59. Watson's Theorem
　60. Plancherel's Theorem
　61. Fejer's Theorem
　62. Integral-Operators of the Fejer Type
　63. The Theorem of Young and Hardy
　64. Examples of Kernels of the Fejer Type
　65. The Fourier Transformation of Integrable Functions
……
CAHPTER Ⅳ CERATIN EXTREMAL PROPERTIES OF INTEGRAL TRANS-CENDENTAL FUNCTIONS OF THE EXPONENTIAL TYPE
CAHPTER ⅤQUESTIONS REGARDING THE BEST HARMONIC APPROXIMATION OF FUNCTIONS
CAHPTER Ⅵ WIENER’S THEOREM ON APPROXIMATION
VARIOUS ADDENDA AND PROBLEMS
Notes
Index