Tchebycheff systems: with applications in analysis and statistics
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Wiley
1966
|
| Series: | Pure and applied mathematics
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 586 S. |
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MARC
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| 245 | 1 | 0 | |a Tchebycheff systems |b with applications in analysis and statistics |c Samuel Karlin ; William J. Studden |
| 264 | 1 | |a New York [u.a.] |b Wiley |c 1966 | |
| 300 | |a XVIII, 586 S. | ||
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| 490 | 1 | |a Pure and applied mathematics |v 15 | |
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Record in the Search Index
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| adam_text | Contents
Chapter I. Tchebycheff Systems on a Closed Interval: Definitions,
Examples, and Preliminaries ..... 1
1. Tchebycheff Systems 1
2. Extended Tchebycheff and Weak Tchebycheff
Systems ........ 3
3. Examples ....... 9
4. Equivalent Polynomial Formulation of T Systems. 20
5. Polynomials with Prescribed Zeros ... 27
6. An Interpolation Problem .... 30
Chapter II. Moment Spaces Induced by J Systems and Their Duals 37
1. Definition and General Structure of #„ +1 . . 38
2. Boundary and Extreme Rays of ^n+i . . 41
3. Interior of ^,+ i 44
4. An Extremal Characterization of the Canonical
Measures ....... 49
5. Continuity Properties of the Canonical Represen¬
tations ........ 51
6. An Alternate Approach to the Canonical Measures
and the One Dimensional Sections ... 54
7. Two Dimensional Sections .... 57
8. Monotonicity of the Roots of Certain Principal
Representations ...... 60
9. The Dual Cone ^n+, 64
10. Representation Theorems for Positive Poly¬
nomials ....... 65
11. Extreme Points of ^n+1 77
xi
Xii CONTENTS
Chapter III. The Markov Krein Theorem and Ramifications . . 79
1. The Extremal Values of ^(c°) .... 80
2. The Markov Krein Theorem . . . .81
3. An Alternate Proof of the Markov Krein Theorem 83
4. Extensions ....... 90
5. Sets of Measures Defined by Moment Inequalities 95
6. Extreme Points of the Set of Distribution Func¬
tions with Given Moments .... 101
Chapter IV. Complements and the Classical Moment Spaces . 106
1. Classical Criterion for Moment Points . .106
2. Supporting Hyperplanes and Orthogonal Poly¬
nomials ........ 108
3. The Maximal Mass p(t0) 114
4. Geometry of the Classical Tchebycheff Poly¬
nomials . . . . . . . .118
5. The Simplexes SN and 5 .... 123
A. The Simplex. S 123
B. The Simplex B 124
6. Volumes of M 127
7. Symmetries of the Moment Spaces . . .129
A. Boundary of M 129
B. The Inverse Distribution . . . .130
C. Reversal of the Unit Interval . . .131
D. Moment Spaces of Symmetric Distributions . 134
8. Mechanical Quadrature and Principal Repre¬
sentations 135
9. Spline Functions with Prescribed Zeros . . 140
Chapter V. Tchebycheff Systems and Moment Spaces on the
Interval [0, oo) 144
1. Introduction . . . . . . .144
2. A General Spanning Theorem . . . .144
3. General Structure of #» +i . . . .146
4. Canonical Representations, the Maximal Mass
p(?o) and the Two Dimensional Sections . .152
5. The Markov Krein Theorem . . . .156
6. Another Class of Extremal Problems Whose
Solutions are Canonical Measures . . .159
CONTENTS Xiii
7. Applications of the Canonical Measures to the
Approximations of Laplace and Stieltjes Trans¬
forms . . . . . . . .163
8. Representation Theorems for Positive Polyno¬
mials . . . . . . . .168
9. Extreme Points and Boundary of „+, [0, *) . 171
10. Classical Criterion for Moment Points . .172
11. Relationship Between , + 1 and • „+! . .173
12. Relationship Between A/ [0, a] and M [0, x) . 176
Chapter VI. Moment Spaces of Periodic Functions and 7 Sy stems
on (— oo, oo) 179
1. Definitions 179
2. Basic Properties of Periodic 7 Systems . . 180
3. Two Dimensional Sections . . . .182
4. The Trigonometric Moment Problem . .184
5. Examples . . . . . . .186
6. Representations for Positive Polynomials and the
Extreme Points of . ,+ , 190
A. Extreme Points and Boundary of ^2m», . 195
7. Tchebycheff Systems on ( x, x) . . .195
8. Representations for Polynomials in
^2ol+1( x, x) 197
9. Two Dimensional Sections for {t om on (— x. x) 198
10. Two Dimensional Sections for U lo1 *1 on
( cc, x) 201
11. Relationship Between =?„ + , and urn+1 . . 203
Chapter VII. Moment Spaces for Tchebycheff Systems Defined on
Discrete Sets 204
1. Introduction ....... 204
2. The Index of a Set X c R 207
3. Polynomials with Prescribed Zeros . . .210
4. Boundary and Canonical Representations of
Points in .#„ + ,(£) 211
5. Principal Representations . . . .214
6. Interlacing Properties of the Canonical Represen¬
tations 218
7. The Markov Krein Theorem .... 220
8. Complements ....... 223
xiv CONTENTS
9. Another Class of Extremal Problems Whose
Solutions are Canonical Measures . . . 224
10. Applications to the Theory of Interpolation of
Absolutely Monotonic Functions . . . 228
Chapter VIII. Moment Spaces Generated by Restricted Measures . 233
1. Introduction ...... 233
2. Boundary and General Properties of the Moment
Space *n + 1 234
3. Principal Representation of Interior Points . 237
4. An Alternate Approach to the Construction of the
Principal Representations .... 240
5. Extending a Tchebycheff System Defined on an
Open Interval 241
6. Interlacing Properties of Canonical Representa¬
tions ........ 246
7. Two Dimensional Sections .... 250
8. Tchebycheff Type Inequalities . . . .251
9. Sets of Measures Defined by Moment Inequalities 254
10. Extensions 258
11. A Minimum Problem ..... 260
12. Liapounov Theorem on the Range of a Vector
Measure ....... 265
13. Extensions of the Liapounov Theorem . . 269
14. Applications of the Liapounov Theorem . . 274
Chapter IX. Minimax Approximation, The Markoff Bcrnstein
Inequality, and Related Matters . . . .278
1. Polynomials of Best Approximation . . . 279
2. Polynomials of Best Approximation (Continued). 282
3. Proof of Theorem 1.1 284
4. Examples of Theorem 1.1 . . . . 286
5. Generalized Markoff Bernstein Inequalities . 292
6. Generalized Markoff Bernstein Inequalities for
Infinite Intervals ...... 298
7. Generalized Markoff Bernstein Inequalities for
Periodic Functions ...... 300
8. Examples and Applications .... 304
9. Complements 311
10. Periodic £T Systems and Extremal Problems for
Two Constraints . . . . . .317
CONTENTS XV
Chapter X. Some Problems of Best Interpolating Systems, Maximi¬
zation of Moment Determinants, and Applications to
Optimum Experimental Designs .... 320
1. Introduction ....... 320
2. A General Equivalence Theorem . . . 323
3. Maximization of Certain Determinants . . 328
4. Generalizations of the Fejer Problem . . 334
5. Stable and Economical Interpolation Systems . 338
6. Trigonometric Systems ..... 342
7. Experimental Designs ..... 349
8. Extensions ....... 362
9. Open Problems 373
Chapter XI. Generalized Convex Functions Induced by ET Systems 375
1. Preliminaries ....... 375
2. Properties of the Cone C(u0, uu...,un) . . 381
3. Further Properties of C(u0, «,, ..., «„) . . 393
4. Extreme Rays of the Cones .... 396
5. Dual Cone of C(u0, ...,«„) . . . .404
6. Examples and Applications of the Theory of Dual
Cones 410
7. Discrete Examples . . . . . .421
8. Generalized Absolutely Monotone Functions . 431
9. Interpolation of Functions by Generalized Spline
Polynomials ...... 436
A. Best Mean Square Approximation Properties
of Natural Spline Polynomials . . . 443
10. Best Quadrature Formulas Involving Natural
Spline Polynomials ..... 445
11. Differentiability Properties of Generalized Convex
Functions ....... 454
Chapter XII. Generalized Tchebycheff Inequalities . . . 467
1. Introduction ....... 467
2. A General Theorem 470
3. Examples ....... 475
4. Tchebycheff Inequalities for Classes of Unimodal
Distributions 482
5. Unimodality of Higher Order . . . .498
Chapter XIII. Multivariate TchebychefT Inequalities . . .505
1. Introduction 505
XVi CONTENTS
2. Extreme Points of ssl 508
3. Tchebycheff Inequalities for Pr(max Xt 1) . 512
1 i n
4. Generalizations of the Berge Inequality . .517
5. Tchebycheff Inequalities Where fl(x) is a Sym¬
metric Characteristic Function . . . .519
6. Tchebycheff Inequalities for a Two Dimensional
Rectangle 523
7. Inequalities for Pr(max X, 1) . . . 525
l ,i n
8. Tchebycheff Inequalities Where fl is a Characteris¬
tic Function of a Finite Union of Convex Regions 527
9. Bounds Involving the Minimum Component . 531
Chapter XIV. Tchebycheff Type Inequalities for Sums of Random
Variables and Nonlinear Problems . . . .533
1. Statement of Theorems ..... 533
2. Proofs of Theorems 1.1 1.4 . . . .537
3. A Conjecture of Samuels ..... 542
4. An Inequality for Sums of Nonindependent
Random Variables ..... 547
5. Nonlinear Problems ..... 549
6. Example I: Maximum and Range of a Sample . 552
7. Example II 561
Bibliography 566
Author Index 579
Subject Index 583
|
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515.8 |
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| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| language | English |
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| spelling | Karlin, Samuel 1924-2007 Verfasser (DE-588)118918672 aut Tchebycheff systems with applications in analysis and statistics Samuel Karlin ; William J. Studden New York [u.a.] Wiley 1966 XVIII, 586 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 15 Chebyshev system Čebyšev-Funktionensystem (DE-588)4147439-9 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Čebyšev-Funktionensystem (DE-588)4147439-9 s Statistik (DE-588)4056995-0 s DE-604 Studden, William J. Verfasser aut Pure and applied mathematics 15 (DE-604)BV010179752 15 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001962211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Karlin, Samuel 1924-2007 Studden, William J. Tchebycheff systems with applications in analysis and statistics Pure and applied mathematics Chebyshev system Čebyšev-Funktionensystem (DE-588)4147439-9 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4147439-9 (DE-588)4056995-0 |
| title | Tchebycheff systems with applications in analysis and statistics |
| title_auth | Tchebycheff systems with applications in analysis and statistics |
| title_exact_search | Tchebycheff systems with applications in analysis and statistics |
| title_full | Tchebycheff systems with applications in analysis and statistics Samuel Karlin ; William J. Studden |
| title_fullStr | Tchebycheff systems with applications in analysis and statistics Samuel Karlin ; William J. Studden |
| title_full_unstemmed | Tchebycheff systems with applications in analysis and statistics Samuel Karlin ; William J. Studden |
| title_short | Tchebycheff systems |
| title_sort | tchebycheff systems with applications in analysis and statistics |
| title_sub | with applications in analysis and statistics |
| topic | Chebyshev system Čebyšev-Funktionensystem (DE-588)4147439-9 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Chebyshev system Čebyšev-Funktionensystem Statistik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001962211&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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