The Laplace transform:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ
Princeton University Press
1972
|
| Edition: | Eigth printing |
| Series: | Princeton mathematical series
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 406 Seiten Diagramme |
| ISBN: | 0691079927 |
Staff View
MARC
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| 245 | 1 | 0 | |a The Laplace transform |c by David Vernon Widder |
| 250 | |a Eigth printing | ||
| 264 | 1 | |a Princeton, NJ |b Princeton University Press |c 1972 | |
| 300 | |a 406 Seiten |b Diagramme | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Chapter I
THE STIELTJES INTEGRAL
SECTION PAGE
1. Introduction 3
2. Stieltjes integrals 3
3. Functions of bounded variation 6
4. Existence of Stieltjes integrals 7
5. Properties of Stieltjes integrals 8
6. The Stieltjes integral as a series or a Lebesguc integral 10
7. Further properties of Stieltjes integrals 12
8. Normalization 13
9. Improper Stieltjes integrals 15
10. Laws of the mean 16
11. Change of variable 19
12. Variation of the indefinite integral 20
13. Stieltjes integrals as infinite series; second method 22
14. Further conditions of integrability 24
15. Iterated integrals 25
16. The selection principle 26
17. Weak compactness 33
Chapter II
FUNDAMENTAL FORMULAS
1. Region of convergence 35
2. Abscissa of convergence 38
3. Absolute convergence 46
4. Uniform convergence 50
5. Analytic character of the generating function 57
6. Uniqueness of determining function 59
7. Complex inversion formula 63
8. Integrals of the determining function 70
9. Summability of divergent integrals 75
10. Inversion when the determining function belongs to L2 80
11. Stieltjes resultant 83
12. Classical resultant 91
13. Order on vertical lines 92
14. Generating function analytic at infinity 93
15. Periodic determining function 96
16. Relation to factorial series 97
vii
viii CONTENTS
Chapter III
THE MOMENT PROBLEM
SECTION PAGE
1. Statement of the problem 100
2. Moment sequence 101
3. An inversion operator 107
4. Completely monotonic sequences 108
5. Function of L 109
6. Bounded functions Ill
7. Hausdorff summability 113
8. Statement of further moment problems 125
9. The moment operator 126
10. The Hamburger moment problem 129
11. Positive definite sequences 132
12. Determinant criteria 134
13. The Stieltjes moment problem 136
14. Moments of functions of bounded variation 138
15. A sufficient condition for the solubility of the Stieltjes problem 140
16. Indeterminacy of solution 142
Chapter IV
ABSOLUTELY AND COMPLETELY MONOTONIC FUNCTIONS
1. Introduction 144
2. Elementary properties of absolutely monotonic functions 144
3. Analyticity of absolutely monotonic functions 146
4. Bernstein s second definition 147
5. Existence of one sided derivatives 149
6. Higher differences of absolutely monotonic functions 159
7. Equivalence of Bernstein s two definitions 151
8. Bernstein polynomials 152
9. Definition of Griiss 154
10. Equivalence of Bernstein and Griiss definitions 155
11. Additional properties of absolutely monotonic functions 156
12. Bernstein s theorem 160
13. Alternative proof of Bernstein s theorem 162
14. Interpolation by completely monotonic functions 163
15. Absolutely monotonic functions with prescribed derivatives at a point. . 165
16. Hankel determinants whose elements are the derivatives of an absolutely
monotonic function 167
17. Laguerre polynomials 168
IS. A linear functional 171
19. Bernstein s theorem 175
20. Completely convex functions 177
Chapter V
TAUBERIAN THEOREMS
1. Abelian theorems for the Laplace transform 180
2. Abelian theorems for the Stieltjes transform 183
CONTENTS ix
SECTION PAGE
3. Tauberian theorems 185
4. Karamata s theorem 189
5. Tauberian theorems for the Stieltjes transform 198
6. Fourier transforms 202
7. Fourier transforms of functions of L 204
8. The quotient of Fourier transforms 207
9. A special Tauberian theorem 209
10. Pitt s form of Wiener s theorem 210
11. Wiener s general Tauberian theorem 212
12. Tauberian theorem for the Stieltjes integral 213
13. One sided Tauberian condition 215
14. Application of Wiener s theorem to the Laplace transform 221
15. Another application 222
16. The prime number theorem 224
17. Ikehara s theorem 233
Chapter VI
THE BILATERAL LAPLACE TRANSFORM
1. Introduction 237
2. Region of convergence 238
3. Integration by parts 239
4. Abscissae of convergence 240
5. Inversion formulas 241
6. Uniqueness 243
7. Summability 244
8. Determining function belonging to Ls 245
9. The Mellin transform 246
10. Stieltjes resultant 248
11. Stieltjes resultant at infinity 249
12. Stieltjes resultant completely denned 250
13. Preliminary results 251
14. The product of Fourier Stieltjes transforms 252
15. Stieltjes resultant of indefinite integrals 256
16. Product of bilateral Laplace integrals 257
17. Resultants in a special case 259
18. Iterates of the Stieltjes kernel 262
19. Representation of functions 265
20. Kernels of positive type 270
21. Necessary and sufficient conditions for representation 272
Chapter VII
INVERSION AND REPRESENTATION PROBLEMS FOR THE
LAPLACE TRANSFORM
1. Introduction 276
2. Laplace s asymptotic evaluation of an integral 277
3. Application of the Laplace method 280
4. Uniform convergence 283
x CONTENTS
SECTION PAGE
5. Uniform convergence; continuation 285
6. The inversion operator for the Laplace Lebesgue integral 288
7. The inversion operator for the Laplace Stieltjes integral 290
8. Laplace method for a new integral 296
9. The jump operator 298
10. The variation of the determining function 299
11. A general representation theorem 302
12. Determining function of bounded variation 306
13. Modified conditions for determining functions of bounded variation. . . . 308
14. Determining function non decreasing 310
15. The class L», p 1 312
16. Determining function the integral of a bounded function 315
17. The class L 317
18. The general Laplace Stieltjes integral 320
Chapter VIII
THE STIELTJES TRANSFOEM
1. Introduction 325
2. Elementary properties of the transform 325
3. Asymptotic properties of Stieltjes transforms 329
4. Relation to the Laplace transform 334
5. Uniqueness 336
6. The Stieltjes transform singular at the origin 336
7. Complex inversion formula 338
8. A singular integral 341
9. The inversion operator for the Stieltjes transform with a(t) an integral. . 345
10. The inversion operator for the Stieltjes transform in the general case . . 347
11. The jump operator 351
12. The variation of a(i) 353
13. A general representation theorem 355
14. Order conditions 357
15. General representation theorems 360
16. The function a(t) of bounded variation 361
17. The function a(t) non decreasing and bounded 363
18. The function a(t) non decreasing and unbounded 365
19. The class L», p 1 368
20. The function p(t) bounded 372
21. The class L 374
22. The function a(t) of bounded variation in every finite interval 377
23. Operational considerations 3S1
24. The iterated Stieltjes transform 383
25. Application to the Laplace transform 3S4
26. Solution of an integral equation 387
27. A related integral equation 390
|
| adam_txt |
CONTENTS
Chapter I
THE STIELTJES INTEGRAL
SECTION PAGE
1. Introduction 3
2. Stieltjes integrals 3
3. Functions of bounded variation 6
4. Existence of Stieltjes integrals 7
5. Properties of Stieltjes integrals 8
6. The Stieltjes integral as a series or a Lebesguc integral 10
7. Further properties of Stieltjes integrals 12
8. Normalization 13
9. Improper Stieltjes integrals 15
10. Laws of the mean 16
11. Change of variable 19
12. Variation of the indefinite integral 20
13. Stieltjes integrals as infinite series; second method 22
14. Further conditions of integrability 24
15. Iterated integrals 25
16. The selection principle 26
17. Weak compactness 33
Chapter II
FUNDAMENTAL FORMULAS
1. Region of convergence 35
2. Abscissa of convergence 38
3. Absolute convergence 46
4. Uniform convergence 50
5. Analytic character of the generating function 57
6. Uniqueness of determining function 59
7. Complex inversion formula 63
8. Integrals of the determining function 70
9. Summability of divergent integrals 75
10. Inversion when the determining function belongs to L2 80
11. Stieltjes resultant 83
12. Classical resultant 91
13. Order on vertical lines 92
14. Generating function analytic at infinity 93
15. Periodic determining function 96
16. Relation to factorial series 97
vii
viii CONTENTS
Chapter III
THE MOMENT PROBLEM
SECTION PAGE
1. Statement of the problem 100
2. Moment sequence 101
3. An inversion operator 107
4. Completely monotonic sequences 108
5. Function of L" 109
6. Bounded functions Ill
7. Hausdorff summability 113
8. Statement of further moment problems 125
9. The moment operator 126
10. The Hamburger moment problem 129
11. Positive definite sequences 132
12. Determinant criteria 134
13. The Stieltjes moment problem 136
14. Moments of functions of bounded variation 138
15. A sufficient condition for the solubility of the Stieltjes problem 140
16. Indeterminacy of solution 142
Chapter IV
ABSOLUTELY AND COMPLETELY MONOTONIC FUNCTIONS
1. Introduction 144
2. Elementary properties of absolutely monotonic functions 144
3. Analyticity of absolutely monotonic functions 146
4. Bernstein's second definition 147
5. Existence of one sided derivatives 149
6. Higher differences of absolutely monotonic functions 159
7. Equivalence of Bernstein's two definitions 151
8. Bernstein polynomials 152
9. Definition of Griiss 154
10. Equivalence of Bernstein and Griiss definitions 155
11. Additional properties of absolutely monotonic functions 156
12. Bernstein's theorem 160
13. Alternative proof of Bernstein's theorem 162
14. Interpolation by completely monotonic functions 163
15. Absolutely monotonic functions with prescribed derivatives at a point. . 165
16. Hankel determinants whose elements are the derivatives of an absolutely
monotonic function 167
17. Laguerre polynomials 168
IS. A linear functional 171
19. Bernstein's theorem 175
20. Completely convex functions 177
Chapter V
TAUBERIAN THEOREMS
1. Abelian theorems for the Laplace transform 180
2. Abelian theorems for the Stieltjes transform 183
CONTENTS ix
SECTION PAGE
3. Tauberian theorems 185
4. Karamata's theorem 189
5. Tauberian theorems for the Stieltjes transform 198
6. Fourier transforms 202
7. Fourier transforms of functions of L 204
8. The quotient of Fourier transforms 207
9. A special Tauberian theorem 209
10. Pitt's form of Wiener's theorem 210
11. Wiener's general Tauberian theorem 212
12. Tauberian theorem for the Stieltjes integral 213
13. One sided Tauberian condition 215
14. Application of Wiener's theorem to the Laplace transform 221
15. Another application 222
16. The prime number theorem 224
17. Ikehara's theorem 233
Chapter VI
THE BILATERAL LAPLACE TRANSFORM
1. Introduction 237
2. Region of convergence 238
3. Integration by parts 239
4. Abscissae of convergence 240
5. Inversion formulas 241
6. Uniqueness 243
7. Summability 244
8. Determining function belonging to Ls 245
9. The Mellin transform 246
10. Stieltjes resultant 248
11. Stieltjes resultant at infinity 249
12. Stieltjes resultant completely denned 250
13. Preliminary results 251
14. The product of Fourier Stieltjes transforms 252
15. Stieltjes resultant of indefinite integrals 256
16. Product of bilateral Laplace integrals 257
17. Resultants in a special case 259
18. Iterates of the Stieltjes kernel 262
19. Representation of functions 265
20. Kernels of positive type 270
21. Necessary and sufficient conditions for representation 272
Chapter VII
INVERSION AND REPRESENTATION PROBLEMS FOR THE
LAPLACE TRANSFORM
1. Introduction 276
2. Laplace's asymptotic evaluation of an integral 277
3. Application of the Laplace method 280
4. Uniform convergence 283
x CONTENTS
SECTION PAGE
5. Uniform convergence; continuation 285
6. The inversion operator for the Laplace Lebesgue integral 288
7. The inversion operator for the Laplace Stieltjes integral 290
8. Laplace method for a new integral 296
9. The jump operator 298
10. The variation of the determining function 299
11. A general representation theorem 302
12. Determining function of bounded variation 306
13. Modified conditions for determining functions of bounded variation. . . . 308
14. Determining function non decreasing 310
15. The class L», p 1 312
16. Determining function the integral of a bounded function 315
17. The class L 317
18. The general Laplace Stieltjes integral 320
Chapter VIII
THE STIELTJES TRANSFOEM
1. Introduction 325
2. Elementary properties of the transform 325
3. Asymptotic properties of Stieltjes transforms 329
4. Relation to the Laplace transform 334
5. Uniqueness 336
6. The Stieltjes transform singular at the origin 336
7. Complex inversion formula 338
8. A singular integral 341
9. The inversion operator for the Stieltjes transform with a(t) an integral. . 345
10. The inversion operator for the Stieltjes transform in the general case . . 347
11. The jump operator 351
12. The variation of a(i) 353
13. A general representation theorem 355
14. Order conditions 357
15. General representation theorems 360
16. The function a(t) of bounded variation 361
17. The function a(t) non decreasing and bounded 363
18. The function a(t) non decreasing and unbounded 365
19. The class L», p 1 368
20. The function p(t) bounded 372
21. The class L 374
22. The function a(t) of bounded variation in every finite interval 377
23. Operational considerations 3S1
24. The iterated Stieltjes transform 383
25. Application to the Laplace transform 3S4
26. Solution of an integral equation 387
27. A related integral equation 390 |
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| author_facet | Widder, David Vernon 1898-1990 |
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| id | DE-604.BV022179896 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:20:27Z |
| indexdate | 2024-07-09T20:51:54Z |
| institution | BVB |
| isbn | 0691079927 |
| language | English |
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| oclc_num | 63371694 |
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| owner_facet | DE-706 |
| physical | 406 Seiten Diagramme |
| publishDate | 1972 |
| publishDateSearch | 1972 |
| publishDateSort | 1972 |
| publisher | Princeton University Press |
| record_format | marc |
| series | Princeton mathematical series |
| series2 | Princeton mathematical series |
| spelling | Widder, David Vernon 1898-1990 (DE-588)131819453 aut The Laplace transform by David Vernon Widder Eigth printing Princeton, NJ Princeton University Press 1972 406 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Princeton mathematical series 6 Laplace-Transformation (DE-588)4034577-4 gnd rswk-swf Laplace-Transformation (DE-588)4034577-4 s DE-604 Princeton mathematical series 6 (DE-604)BV000019035 6 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015394643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Widder, David Vernon 1898-1990 The Laplace transform Princeton mathematical series Laplace-Transformation (DE-588)4034577-4 gnd |
| subject_GND | (DE-588)4034577-4 |
| title | The Laplace transform |
| title_auth | The Laplace transform |
| title_exact_search | The Laplace transform |
| title_exact_search_txtP | The Laplace transform |
| title_full | The Laplace transform by David Vernon Widder |
| title_fullStr | The Laplace transform by David Vernon Widder |
| title_full_unstemmed | The Laplace transform by David Vernon Widder |
| title_short | The Laplace transform |
| title_sort | the laplace transform |
| topic | Laplace-Transformation (DE-588)4034577-4 gnd |
| topic_facet | Laplace-Transformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015394643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000019035 |
| work_keys_str_mv | AT widderdavidvernon thelaplacetransform |