Theory of Approximation

by Achieser, N. I.

• ISBN13: 9780486495439
• ISBN10: 0486495434
• Format: Hardcover
• Copyright: 2004-01-26
• Publisher: Dover Publications

Summary

A pioneer of many modern developments in approximation theory, N. I. Achieser designed this graduate-level text from the standpoint of functional analysis. The first two chapters address approximation problems in linear normalized spaces and the ideas of P. L. Tchebysheff. Chapter III 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 (elementary extremal problems, Szego's theorem, the Carathéodory-Fejér problem, and more).

Table of Contents

CHAPTER I APPROXIMATION PROBLEMS IN LINEAR NORMALIZED SPACES
1. Formulation of the Principal Problem in the Theory of Approximation	1	(1)
2. The Concept of Metric Space	1	(1)
3. The Concept of Linear Normalized Space	2	(1)
4. Examples of Linear Normalized Spaces	3	(1)
5. The Inequalities of Holder and Minkowski	4	(3)
6. Additional Examples of Linear Normalized Spaces	7	(1)
7. Hilbert Space	8	(2)
8. The Fundamental Theorem of Approximation Theory in Linear Normalized Spaces	10	(1)
9. Strictly Normalized Spaces	11	(1)
10. An Example of Approximation in the Space Lp	12	(1)
11. Geometric Interpretation	13	(1)
12. Separable and Complete Spaces	14	(1)
13. Approximation Theorems in Hilbert Space	15	(4)
14. An Example of Approximation in Hilbert Space	19	(2)
15. More About the Approximation Problem in Hilbert Space	21	(1)
16. Orthonormalized Vector Systems in Hilbert Space	22	(1)
17. Orthogonalization of Vector Systems	23	(2)
18. Infinite Orthonormalized Systems	25	(4)
19. An Example of a Non-Separable System	29	(1)
20. Weierstrass' First Theorem	29	(3)
21. Weierstrass' Second Theorem	32	(1)
22. The Separability of the Space C	33	(1)
23. The Separability of the Space Lp	34	(3)
24. Generalization of Weierstrass' Theorem to the Space Lp	37	(1)
25. The Completeness of the Space Lp	38	(2)
26. Examples of Complete Orthonormalized Systems in L2	40	(3)
27. Müntz's Theorem	43	(3)
28. The Concept of the Linear Functional	46	(1)
29. F. Riesz's Theorem	47	(2)
30. A Criterion for the Closure of a Set of Vectors in Linear Normalized Spaces	49	(2)
CHAPTER II P.L. TCHEBYSHEFF'S DOMAIN OF IDEAS
31. Statement of the Problem	51	(1)
32. A Generalization of the Theorem of de la Vallee-Poussin	52	(1)
33. The Existence Theorem	53	(2)
34. Tchebysheff's Theorem	55	(2)
35. A Special Case of Tchebysheff's Theorem	57	(1)
36. The Tehebysheff Polynomials of Least Deviation from Zero	57	(1)
37. A Further Example of P.L. Tchebysheff's Theorem	58	(2)
38. An Example for the Application of the General Theorem of de la Vallee-Poussin	60	(2)
39. An Example for the Application of P.L. Tchebysheff's General Theorem	62	(2)
40. The Passage to Periodic Functions	64	(2)
41. An Example of Approximating with the Aid of Periodic Functions	66	(1)
42. The Weierstrass Function	66	(1)
43. Haar's Problem	67	(1)
44. Proof of the Necessity of Haar's Condition	68	(1)
45. Proof of the Sufficiency of Haar's Condition	69	(3)
46. An Example Related to Haar's Problem	72	(1)
47. P.L. Tchebysheff's Systems of Functions	73	(1)
48. Generalization of P.L. Tchebysheff's Theorem	74	(2)
49. On a Question Pertaining to the Approximation of a Continuous Function in the Space L	76	(6)
50. A.A. Markoff's Theorem	82	(3)
51. Special Cases of the Theorem of A.A. Markoff	85	(4)
CHAPTER III ELEMENTS OF HARMONIC ANALYSIS
52. The Simplest Properties of Fourier Series	89	(4)
53. Fourier Series for Functions of Bounded Variation	93	(4)
54. The Parseval Equation for Fourier Series	97	(1)
55. Examples of Fourier Series	98	(3)
56. Trigonometric Integrals	101	(2)
57. The Riemann-Lebesgue Theorem	103	(1)
58. Plancherel's Theory	104	(2)
59. Watson's Theorem	106	(2)
60. Plancherel's Theorem	108	(2)
61. Fejer's Theorem	110	(3)
62. Integral-Operators of the Fejer Type	113	(3)
63. The Theorem of Young and Hardy	116	(2)
64. Examples of Kernels of the Fejer Type	118	(2)
65. The Fourier Transformation of Integrable Functions	120	(2)
66. The Faltung of two Functions	122	(1)
67. V.A. Stekloff's Functions	123	(2)
68. Multimonotonic Functions	125	(1)
69. Conjugate Functions	126	(4)
CHAPTER IV CERTAIN EXTREMAL PROPERTIES OF INTEGRAL TRANSCENDENTAL FUNCTIONS OF THE EXPONENTIAL TYPE
70. Integral Functions of the Exponential Type	130	(2)
71. The Borel Transformation	132	(2)
72. The Theorem of Wiener and Paley	134	(3)
73. Integral Functions of the Exponential Type which are Bounded along the Real Axis	137	(3)
74. S.N. Bernstein's Inequality	140	(6)
75. B.M. Levitan's Polynomials	146	(6)
76. The Theorem of Fejer and Riesz. A Generalization of This Theorem	152	(2)
77. A Criterion for the Representation of Continuous Functions as Fourier-Stieltjes Integrals	154	(6)
CHAPTER V QUESTIONS REGARDING THE BEST HARMONIC APPROXIMATION OF FUNCTIONS
78. Preliminary Remarks	160	(1)
79. The Modulus of Continuity	161	(1)
80. The Generalization to the Space Lp (p > or equal to 1)	162	(3)
81. An Example of Harmonic Approximation	165	(4)
82. Some Estimates for Fourier Coefficients	169	(4)
83. More about V.A. Stekloff's Functions	173	(2)
84. Two Lemmas	175	(1)
85. The Direct Problem of Harmonic Approximation	176	(7)
86. A Criterion due to B. Sz.-Nagy	183	(4)
87. The Best Approximation of Differentiable Functions	187	(8)
88. Direct Observations Concerning Periodic Functions	195	(4)
89. Jackson's Second Theorem	199	(2)
90. The Generalized Fejer Method	201	(5)
91. Berstein's Theorem	206	(4)
92. Priwaloff's Theorem	210	(1)
93. Generalizations of Bernstein's Theorems to the Space Lp (p > or equal to 1)	211	(3)
94. The Best Harmonic Approximation of Analytic Functions	214	(4)
95. A Different Formulation of the Result of the Preceding Section.	218	(3)
96. The Converse of Bernstein's Theorem	221	(3)
CHAPTER VI WIENER'S THEOREM ON APPROXIMATION
97. Wiener's Problem	224	(1)
98. The Necessity of Wiener's Condition	224	(1)
99. Some Definitions and Notation	225	(2)
100. Several Lemmas	227	(3)
101. The Wiener-Levy Theorem	230	(3)
102. Proof of the Sufficiency of Wiener's Condition	233	(1)
103. Wiener's General Tauber Theorem	234	(1)
104. Weakly Decreasing Functions	235	(2)
105. Remarks on the Terminology	237	(1)
106. Ikehara's Theorem	238	(3)
107. Carleman's Tauber Theorem	241	(2)
VARIOUS ADDENDA AND PROBLEMS
A. Elementary Extremal Problems and Certain Closure Criteria	243	(13)
B. Szego's Theorem and Some of Its Applications	256	(11)
C. Further Examples of Closed Sequences of Functions	267	(3)
D. The Caratheodory-Fejer Problem and Similar Problems	270	(10)
E. Solotareff's Problems and Related Problems	280	(9)
F. The Best Harmonic Approximation of the Simplest Analytic Functions	289	(7)
NOTES	296	(10)
INDEX	306