| 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 |
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