Timan / Sneddon / Ulam Theory of Approximation of Functions of a Real Variable

1. Auflage 2014
ISBN: 978-1-4831-8481-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 34, 644 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Theory of Approximation of Functions of a Real Variable discusses a number of fundamental parts of the modern theory of approximation of functions of a real variable. The material is grouped around the problem of the connection between the best approximation of functions to their structural properties. This text is composed of eight chapters that highlight the relationship between the various structural properties of real functions and the character of possible approximations to them by polynomials and other functions of simple construction. Each chapter concludes with a section containing various problems and theorems, which supplement the main text. The first chapters tackle the Weierstrass's theorem, the best approximation by polynomials on a finite segment, and some compact classes of functions and their structural properties. The subsequent chapters describe some properties of algebraic polynomials and transcendental integral functions of exponential type, as well as the direct theorems of the constructive theory of functions. These topics are followed by discussions of differential and constructive characteristics of converse theorems. The final chapters explore other theorems connecting the best approximations functions with their structural properties. These chapters also deal with the linear processes of approximation of functions by polynomials. The book is intended for post-graduate students and for mathematical students taking advanced courses, as well as to workers in the field of the theory of functions.

Autoren/Hrsg.

Timan, A. F.

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Theory of Approximation of Functions of a Real Variable;4
3;Copyright Page;5
4;Table of Contents;6
5;EDITORIAL PREFACE;10
6;FOREWORD;12
7;CHAPTER I. WEIERSTRASS'S THEOREM;14
7.1;1.1. Approximation of continuous functions by polynomials on a finite segment. The fundamental theorem;14
7.2;1.2. Proof of Weierstrass's theorem;15
7.3;1.3. Generalisation and some particular cases. Periodic functions. Functions of many variables;18
7.4;1.4. Mean approximation of integrable functions by polynomials;19
7.5;1.5. Some stronger forms of Weierstrass's theorem;21
7.6;1.6. Approximation of continuous functions on an infinite interval. Uniform approximation by rational functions;23
7.7;1.7. Uniform approximation on the whole real axis by integral functions of finite degree;25
7.8;1.8. On weighted uniform approximation of continuous functions on the whole real axis;29
7.9;1.9. On mean approximation of integrable functions on an infinite interval;32
7.10;1.10. Various problems and theorems;37
8;CHAPTER II. THE BEST APPROXIMATION;39
8.1;2.1. The best approximation by polynomials on a finite segment;39
8.2;2.2. Generalisation to a linear normed space and some particular cases;41
8.3;2.3. On the uniqueness of the polynomial giving the best uniform approximation. Haar's theorem;48
8.4;2.4. On the uniqueness of the polynomial giving the best integral approximation;51
8.5;2.5. A sequence of best approximations. The fundamental characteristic property;53
8.6;2.6. The best approximation on an infinite interval;58
8.7;2.7. Properties of best polynomial approximations. A theorem of Chebyshev;65
8.8;2.8. On some properties of polynomials of best integral approximations;71
8.9;2.9. Polynomials which deviate least from zero;79
8.10;2.10. Estimation of the best approximation. De la Vallée Poussin's theorem;85
8.11;2.11. Some cases of exact solution of the problem of the best approximation;88
8.12;2.12. On some criteria for the best approximation in an infinite interval;94
8.13;2.13. Various problems and theorems;99
9;CHAPTER III. SOME COMPACT CLASSES OF FUNCTIONS AND THEIR STRUCTURAL CHARACTERISTICS;106
9.1;3.1. Compact classes of functions of a single variable. Criteria of the type of Arzelà's theorem. The concepts of the e-entropy and e-capacity of a compact set;106
9.2;3.2. Moduli of continuity and some of their properties;109
9.3;3.3. Moduli of smoothness of various orders;115
9.4;3.4. On compact classes of functions of many variables;123
9.5;3.5. Compact classes of differentiable functions;127
9.6;3.6. Sets of functions of bounded variation;139
9.7;3.7. Some classes of functions analytic on a finite segment;142
9.8;3.8. Some classes of functions analytic on the whole real axis;145
9.9;3.9. On regularly monotonic functions;154
9.10;3.10. On quasi-analytic classes of functions;162
9.11;3.11. Conjugate classes of functions defined on the whole real axis;167
9.12;3.12. Various problems and theorems;176
10;CHAPTER IV. SOME PROPERTIES OF ALGEBRAIC POLYNOMIALS AND TRANSCENDENTAL INTEGRAL FUNCTIONS OF EXPONENTIAL TYPE;183
10.1;4.1. Interpolation formulae for algebraic polynomials;183
10.2;4.2. Interpolation formulae for trigonometric polynomials;187
10.3;4.3. Interpolation formulae for some classes of transcendental integral functions of exponential type;191
10.4;4.4. On the phenomenon of interference in the behaviour of integral functions of finite degree;200
10.5;4.5. Some integral representations for algebraic polynomials and trigonometric polynomials;207
10.6;4.6. An integral representation for some classes of transcendental integral functions of exponential type. The Wiener—Paley theorem;210
10.7;4.7. Some interpolation and integral identities for the derivatives of algebraic polynomials and transcendental integral functions of exponential type;214
10.8;4.8. Some extremal properties of algebraic polynomials and transcendental integral functions of finite degree;219
10.9;4.9. Integral and interpolation norms of integral functions of finite degree in various metrics;240
10.10;4.10. The connection between trigonometric polynomials and other integral functions of the classes Bs;250
10.11;4.11. Non-negative polynomials and transcendental integral functions of exponential type;253
10.12;4.12. Various problems and theorems;259
11;CHAPTER V. DIRECT THEOREMS OF THE CONSTRUCTIVE THEORY OF FUNCTIONS;267
11.1;5.1. The effect of the differential properties of functions on the rapidity of decrease to zero of their best approximations. Jackson's theorem;267
11.2;5.2. A strong form of Jackson's theorem on the best approximation of continuous functions by algebraic polynomials on a finite segment of the real axis;274
11.3;5.3. Some direct theorems for functions of many variables;286
11.4;5.4. The rapidity of the decrease to zero of the best approximation of analytic functions;293
11.5;5.5. More precise forms of Jackson's theorem. The estimates of N. I. Akhiezer, M. G. Krein and J. Favard for differentiable periodic functions;300
11.6;5.6. On the asymptotic behaviour of the upper bounds of best approximations of classes of functions differentiable a given finite number of times. S. N. Bernstein's theorem;305
11.7;5.7. The best approximation of functions analytic in a strip;317
11.8;5.8. Constructive properties of some quasi-analytic classes of functions;322
11.9;5.9. Estimates of best approximations for some conjugate classes of functions;328
11.10;5.10. Direct theorems in arbitrary Banach spaces;336
11.11;5.11. Various problems and theorems;338
12;CHAPTER VI. CONVERSE THEOREMS. CONSTRUCTIVE CHARACTERISTICS OF SOME CLASSES OF FUNCTIONS;344
12.1;6.1. Differential properties of functions with a given sequence of best approximations;344
12.2;6.2. Constructive characteristics of some classes of continuous functions defined on a finite segment;356
12.3;6.3. Converse theorems for functions of many variables;363
12.4;6.4. Differential properties and best approximations of functions in various metrics. On inclusion theorems for some classes of functions;377
12.5;6.5. The analyticity of functions and their best approximations;382
12.6;6.6. On constructive characteristics of quasi-analytic classes of functions;385
12.7;6.7. Certain converse theorems for conjugate classes of functions;402
12.8;6.8. Converse theorems and some constructive characteristics of compact sets in Banach spaces;405
12.9;6.9 Various problems and theorems;410
13;CHAPTER VII. FURTHER THEOREMS CONNECTING THE BEST APPROXIMATIONS OF FUNCTIONS WITH THEIR STRUCTURAL PROPERTIES;415
13.1;7.1. On the exact order of decrease of best approximations;415
13.2;7.2. Asymptotic properties of the best uniform approximation of some simple functions with singularities;428
13.3;7.3. The best uniform approximation of functions which have a discontinuous derivative of bounded variation;444
13.4;7.4. On the best mean approximation of the simpler functions with singularities;450
13.5;7.5. On the asymptotic behaviour of the best uniform approximation of some analytic functions;456
13.6;7.6. On a constructive property of regularly monotonic functions;464
13.7;7.7. The asymptotic behaviour of the best uniform approximation by polynomials of some transcendental integral functions;468
13.8;7.8. Various problems and theorems;473
14;CHAPTER VIII. LINEAR PROCESSES OF APPROXIMATION OF FUNCTIONS BY POLYNOMIALS AND SOME ESTIMATES CONNECTED WITH THEM;478
14.1;8.1. On the convergence of linear processes of approximation of functions by polynomials;478
14.2;8.2. Lebesgue constants and functions;489
14.3;8.3. On linear methods of approximation by polynomials which give the best order of approximation;513
14.4;8.4. Approximation of functions by arithmetic means of the partial sums of Fourier series;534
14.5;8.5. Further estimates for linear methods of approximation of functions by polynomials;560
14.6;8.6. The approximative properties of orthogonal expansions;584
14.7;8.7. Various problems and theorems;589
15;CHAPTER IX. SOME RESULTS FROM THE THEORY OF FUNCTIONS AND FUNCTIONAL ANALYSIS;600
15.1;9.1. The binomial series;600
15.2;9.2 Abel's transformation;600
15.3;9.3. The theorem of Descartes on the positive roots of the equation with real coefficients;600
15.4;9.4. The Bolzano-Weierstrass principle of compactness;600
15.5;9.5. The diagonal process of Cantor;600
15.6;9.6. Compact sets in metric spaces;601
15.7;9.7. The Borel-Lebesgue lemma;601
15.8;9.8. A bicompact topological space;601
15.9;9.9. A theorem on the continuation of continuous functions defined on closed sets;602
15.10;9.10. Open and closed sets in an n-dimensional euclidean space;602
15.11;9.11. Functions of bounded variation;602
15.12;9.12. The C-property of measurable functions;602
15.13;9.13. The absolute continuity of a Lebesgue integral;603
15.14;9.14. The integral representation of absolutely continuous functions;603
15.15;9.15. Functions satisfying a Lipschitz condition of the first order;603
15.16;9.16. Weierstrass's function;603
15.17;9.17. Fatou's lemma. Passage to the limit under the Lebesgue integral sign;604
15.18;9.18. Fubini's theorem on the multiple Lebesgue integral;604
15.19;9.19. Holder's inequality;604
15.20;9.20. Minkowski's inequality;605
15.21;9.21. The generalised inequality of Minkowski;605
15.22;9.22. A linear normed space;605
15.23;9.24. Linear operators and linear functionals;607
15.24;9.25. Linear functional in the spaces C and Lq;608
15.25;9.26. Sequences of linear operators on Banach spaces;608
15.26;9.27. The integration and differentiation of Fourier series;609
15.27;9.28. The convergence of Fourier series;609
15.28;9.29. The arithmetic mean of the partial sums of a Fourier series;610
15.29;9.30. Parseval's equality;610
15.30;9.31. The Fischer-Riesz theorem;610
15.31;9.32. The uniqueness theorem in the theory of trigonometrical series;610
15.32;9.33. The Taylor series;611
15.33;9.34. Sequences of analytic functions;611
15.34;9.35. The uniqueness theorem in the theory of analytic functions;611
15.35;9.36. Cauchy's theorem for analytic functions and some corollaries from it;612
15.36;9.37. Integral functions. Liouville's theorem;612
15.37;9.38. The order and type of an integral function;612
15.38;9.39. The principle of the maximum modulus and the Phragmén-Lindelöf theorem;613
15.39;9.40. The bilinear function. The Riemann-Schwarz principle of symmetry;613
16;BIBLIOGRAPHY: OF MEMOIRS AND BOOKS REFERRED TO IN THE TEXT;615
17;INDEX;634