Analytic theory of continued fractions:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Mineola, New York
Dover Publications, Inc.
2018
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| Ausgabe: | Dover edition, first published |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Originally published: New York : D. Van Nostrand Company, 1948. - Includes bibliographical references (pages 417-425) and index. |
| Beschreibung: | xiii, 433 pages 23 cm |
| ISBN: | 9780486823690 0486823695 |
Internformat
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| 245 | 1 | 0 | |a Analytic theory of continued fractions |c Hubert Stanley Wall |
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| 264 | 1 | |a Mineola, New York |b Dover Publications, Inc. |c 2018 | |
| 300 | |a xiii, 433 pages |c 23 cm | ||
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| 650 | 4 | |a Continued fractions | |
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Datensatz im Suchindex
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| adam_text | CONTENTS PAGE Preface....................................................................................... vii Introduction......................................................................... 1 Part I: Convergence Theory CHAPTER I: THE CONTINUED FRACTION AS A PRODUCT OF LINEAR FRACTIONAL TRANSFORMATIONS SUCTION 1. 2. 3. 4. Definitions and Formulas................................................. 13 Continued Fractions and Series......................................... 17 Equivalence Transformations ................................................ 19 Even and Odd Parts of a Continued Fraction .................... 20 CHAPTER II : CONVERGENCE THEOREMS 5. 6. 7. 8. Some General Remarks on the Convergence Problem ... 25 Necessary Conditions for Convergence ................................ 27 A Sufficient Condition for Convergence.......................... 33 Convergence of Periodic Continued Fractions .................... 35 CHAPTER in: CONVERGENCE OF CONTINUED FRACTIONS WHOSE PARTIAL DENOMINATORS ARE EQUAL TO UNITY 9. The First Interpretation of the Fundamental Inequalities . 40 10. Worpitzky’s Theorem.......................................................... 42 11. Convergence of Continued Fractions Whose Partial Quotients Հ ք.լ т- (1 ֊ gp-l)gpXp Are ot the b orm---------J........... ............................................... 12. 13. 14. 15. A Convergence Theorem of von Koch.............................. Second Interpretation of the Fundamental Inequalities . The Parabola Theorem...................................................... “Convergence Neighborhoods” of a
Point (1).................. ¡X 45 50 . 52 56 62
x CONTENTS CHAPTER IV. introduction to the theory of positive DEFINITE CONTINUED FRACTIONS SECTION PAGE 16. Definition of a Positive Definite Continued Fraction ... 17. The Nest of Circles................................................................. 18. Positive Definite Continued Fractions and the Parabola Theorem ............................................................................ 19. Chain Sequences..................................................................... 20. Quadratic Forms and Chain Sequences............................... 64 70 75 79 86 CHAPTER v: SOME GENERAL CONVERGENCE THEOREMS 21. 22. 23. 24. 25. 26. 27. Schwarz’s Inequality............................................................. The Theorem of Invariability .............................................. The Indeterminate Case......................................................... Convergence Continuation Theorem .................................. The Determinate Case ......................................................... Bounded J-fractions................................................................. Real J-fractions ..................................................................... 94 96 99 104 109 110 114 CHAPTER VI : STIELTJES TYPE CONTINUED FRACTIONS 28. Convergence and Divergence of the Continued Fraction of Stieltjes................................................................................ 118 29. The Condition (H)................................................................. 122 30. Three Convergence Theorems ..............................................
131 CHAPTER vii: EXTENSIONS OF THE PARABOLA THEOREM 31. 32. 33. 34. 35. A Family of Parabolic Domains .......................................... “Convergence Neighborhoods” of a Point (2) A Theorem of Van Vleck ..................................................... The Cardioid Theorem ......................................................... An F.xtension of a Theorem of Szász .................................. 135 137 138 140 143 CHAPTER Vin: THE VALUE REGION PROBLEM 36. 37. 38. 39. A Sufficient Condition............................................................. 147 The Two-Circle Theorem.......................................................... 148 Circular Element Regions with Centers at the Origin . . . 150 A Family of Parabolic Element Regions.............................. 152
CONTENTS Part II: xi Function Theory CHAPTER IX: J-FRACTION EXPANSIONS FOR RATIONAL FUNCTIONS SECTION PAGE 40. The Expansion Algorithm..................................................... 161 41. Conditions Involving Determinants...................................... 164 42. Relationship Between the J-fraction and the Power Series for ƒi//о................................................................................ 166 43. Rational Fractions with Simple Poles and Positive Residues 167 44. Expansion of Rational Functions into Stieltjes Type Con tinued Fractions................................................................. 170 CHAPTER x: THEORY OF EQUATIONS 45. 46. 47. 48. The Test-Fraction ................................................................. 174 Polygonal Bounds for the Roots of a Polynomial.................... 176 Polynomials Whose Roots Are in a Given Half-Plane ... 178 Determination of the Number of Roots of P(z) in Each of the Half-Planes Oi(z) 0......................................................... 182 49. Computation of the Roots of Polynomials........................... 185 CHAPTER XI: J-FRACTION EXPANSION S FOR POWER SERIES 50. 51. 52. 53. 54. Polynomials Orthogonal Relative to a Sequence .................... 192 Algorithm for Expanding a Power Series into a J-fraction 196 StieltjesType Continued Fraction Expansions for Power Series 200 Stieltjes’ Expansion Theorem .................................................. 202 Convergence Questions..............................................................208 CHAPTER XII : MATRIX THEORY OF
CONTINUED FRACTIONS 55. 56. 57. 58. 59. 60. 61. Linear Forms .............................................................................214 Bilinear Forms.............................................................................216 Bounded Matrices ..................................................................... 218 Bounded Reciprocals ot Bounded Matrices............................... 223 The Bounded Reciprocal of a Bounded J-matrix....................226 Reciprocals of an Arbitrary J-matrix.......................................228 Reciprocals of the J-matrix Associated with a Positive Definite J-fraction ..................................................................230 62. Estimates for the Equivalent Functions................................... 235
xii CONTEXTS CHAPTER хш: CONTINUED FRACTIONS AND DEFINITE INTEGRALS SECTION ГАСЕ 63. 64. 65. 66. The Stieltjes Integral..................................................................239 Sequences of Stieltjes Integrals.................................................. 245 The Stieltjes Inversion Formula ...............................................247 Representation of an Equivalent Function of a Positive Definite J-fraction as a Stieltjes Transform............................250 67. Proper Equivalent Functions...................................................... 254 CHAPTER XIV: THE MOMENT PROBLEM FOR A FINITE INTERVAL 68. 69. 70. 71. 72. Formulation of the Problem...................................................... 258 Solution of the Moment Problem byMeans of S-fractions 260 Some Geometry ......................................................................... 263 Totally Monotone Sequences...................................................... 267 Composition of Moment Sequences........................................... 269 CHAPTER XV: BOUNDED ANALYTIC FUNCTIONS 73. Integral Formulas for Bounded Analytic Functions .... 275 74. Continued Fraction Expansions for Real Analytic Functions 278 75. Continued Fraction Expansions for 1/G(z) and for G[ —z/(l + z)] in Terms of the Expansions for G(z) . . . 280 76. Condition for G(z)/л/1 + z to Be Bounded in the Unit Circle.................................................................................. 283 77. Analytic Functions Bounded in theUnit Circle......................... 285 78. Continued Fraction Expansions for
Arbitrary Functions Which Are Analytic and Have Positive Real Parts in Ext (-l,-oo)..........................................................................288 CHAPTER XVI : HAUSDORFF SUMMABILITY 79. 80. 81. 82. 83. Hausdorff Matrices......................................................................302 A Theorem on (A, ¿įj-Transiormations................................... 304 Hausdorff Means......................................................................... 306 Examples of Hausdorff Means.................................................. 309 The Hausdorff Inclusion Problem...............................................310
CONTENTS xiii CHAPTER xvii: the moment problem for an INFINITE INTERVAL SECTION 84. 85. 86. 87. 88. PAGE Asymptotic Expressions for J-fractions.......................................316 A Theorem of Hamburger..........................................................321 The Moment Problem for the Interval (—»,+») . . . . 325 The Stieltjes Moment Problem.................................................. 327 A Theorem of Carleman..............................................................330 CHAPTER xviii: the continued fraction OF GAUSS 89. 90. 91. 92. General Properties..................................................................... 335 Elementary Functions................................................................. 342 Certain Meromorphic Functions.............................................. 347 A Class of Divergent Series ......................................................349 chapter xix: stieltjes summability 93. Definition and Illustrative Examples.......................................362 94. List of Expansion Formulas......................................................369 chapter 95. 96. 97. 98. 99. 100. 101. 102. xx : the pádé table Definitions.................................................................................... 377 The Normal Pádé Table..............................................................379 The Pádé Table for the Series of Stieltjes............................... 389 General Theorems on the Pádé Table.......................................393
C-fractions.................................................................................... 399 Regular C-fractions and Power Series.......................................405 «-regular C-fractions................................................................. 409 Concluding Remarks on the Pádé Table...................................410 Bibliography................................................................................ 417 Index............................................................................................427
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:17:00Z |
| institution | BVB |
| isbn | 9780486823690 0486823695 |
| language | English |
| lccn | 017051309 |
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| physical | xiii, 433 pages 23 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
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| publisher | Dover Publications, Inc. |
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| spelling | Wall, Hubert S. 1902-1971 (DE-588)17246000X aut Analytic theory of continued fractions Hubert Stanley Wall Dover edition, first published Mineola, New York Dover Publications, Inc. 2018 xiii, 433 pages 23 cm txt rdacontent n rdamedia nc rdacarrier Originally published: New York : D. Van Nostrand Company, 1948. - Includes bibliographical references (pages 417-425) and index. Continued fractions Kettenbruch (DE-588)4030401-2 gnd rswk-swf Kettenbruch (DE-588)4030401-2 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030780982&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wall, Hubert S. 1902-1971 Analytic theory of continued fractions Continued fractions Kettenbruch (DE-588)4030401-2 gnd |
| subject_GND | (DE-588)4030401-2 |
| title | Analytic theory of continued fractions |
| title_auth | Analytic theory of continued fractions |
| title_exact_search | Analytic theory of continued fractions |
| title_full | Analytic theory of continued fractions Hubert Stanley Wall |
| title_fullStr | Analytic theory of continued fractions Hubert Stanley Wall |
| title_full_unstemmed | Analytic theory of continued fractions Hubert Stanley Wall |
| title_short | Analytic theory of continued fractions |
| title_sort | analytic theory of continued fractions |
| topic | Continued fractions Kettenbruch (DE-588)4030401-2 gnd |
| topic_facet | Continued fractions Kettenbruch |
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