Methods of the theory of generalized functions:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London [u.a.]
Taylor & Francis
2002
|
| Edition: | 1. publ. |
| Series: | Analytical methods and special functions
6 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 303 - 308 |
| Physical Description: | XIV, 311 S. graph. Darst. |
| ISBN: | 0415273560 |
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| 245 | 1 | 0 | |a Methods of the theory of generalized functions |c V. S. Vladimirov |
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Record in the Search Index
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| adam_text | METHODS OF THE THEORY OF GENERALIZED FUNCTIONS V.S. VLADIMIROV STEKLOV
MATHEMATICAL INSTITUTE MOSCOW, RUSSIA &/?I FLDED LONDON AND NEW YORK
CONTENTS PREFACE XI SYMBOLS AND DEFINITIONS 1 CHAPTER 1. GENERALIZED
FUNCTIONS AND THEIR PROPERTIES 5 1. TEST AND GENERALIZED FUNCTIONS 5
1.1. INTRODUCTION 5 1.2. THE SPACE OF TEST FUNCTIONS V(O) 6 1.3. THE
SPACE OF GENERALIZED FUNCTIONS V(O) 10 1.4. THE COMPLETENESS OF THE
SPACE OF GENERALIZED FUNCTIONS V (O) 12 1.5. THE SUPPORT OF A
GENERALIZED FUNCTION 13 1.6. REGULAR GENERALIZED FUNCTIONS 15 1.7.
MEASURES 16 1.8. SOCHOZKI FORMULAE 19 1.9. CHANGE OF VARIABLES IN
GENERALIZED FUNCTIONS 21 1.10. MULTIPLICATION OF GENERALIZED FUNCTIONS..
23 2. DIFFERENTIATION OF GENERALIZED FUNCTIONS 25 2.1. DERIVATIVES OF
GENERALIZED FUNCTIONS 25 2.2. THE ANTIDERIVATIVE (PRIMITIVE) OF A
GENERALIZED FUNCTION 27 2.3. EXAMPLES 29 2.4. THE LOCAL STRUCTURE OF
GENERALIZED FUNCTIONS 35 2.5. GENERALIZED FUNCTIONS WITH COMPACT SUPPORT
36 2.6. GENERALIZED FUNCTIONS WITH POINT SUPPORT 37 2.7. GENERALIZED
FUNCTIONS V{-K U X A - 1 ) 39 3. DIRECT PRODUCT OF GENERALIZED
FUNCTIONS 41 3.1. THE DEFINITION OF A DIRECT PRODUCT 41 3.2. THE
PROPERTIES OF A DIRECT PRODUCT 43 3.3. SOME APPLICATIONS 46 3.4.
GENERALIZED FUNCTIONS THAT ARE SMOOTH WITH RESPECT TO SOME OF THE
VARIABLES 48 VI CONTENTS 4. THE CONVOLUTION OF GENERALIZED FUNCTIONS 50
4.1. THE DEFINITION OF CONVOLUTION 50 4.2. THE PROPERTIES OF A
CONVOLUTION 53 4.3. THE EXISTENCE OF A CONVOLUTION 57 4.4. CONES IN M N
59 4.5. CONVOLUTION ALGEBRAS V{T+) AND V(T) 63 4.6. MEAN FUNCTIONS OF
GENERALIZED FUNCTIONS 64 4.7. MULTIPLICATION OF GENERALIZED FUNCTIONS 66
4.8. CONVOLUTION AS A CONTINUOUS LINEAR TRANSLATION- INVARIANT OPERATOR
66 4.9. SOME APPLICATIONS 68 5. TEMPERED GENERALIZED FUNCTIONS 74 5.1.
THE SPACE S OF TEST (RAPIDLY DECREASING) FUNCTIONS 74 5.2. THE SPACE S
OF TEMPERED GENERALIZED FUNCTIONS 77 5.3. EXAMPLES OF TEMPERED
GENERALIZED FUNCTIONS AND ELEMENTARY OPERATIONS IN S 78 5.4. THE
STRUCTURE OF TEMPERED GENERALIZED FUNCTIONS 80 5.5. THE DIRECT PRODUCT
OF TEMPERED GENERALIZED FUNCTIONS 81 5.6. THE CONVOLUTION OF TEMPERED
GENERALIZED FUNCTIONS 82 5.7. HOMOGENEOUS GENERALIZED FUNCTIONS 85
CHAPTER 2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS 89 6. THE
FOURIER TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS 89 6.1. THE FOURIER
TRANSFORM OF TEST FUNCTIONS INS 89 6.2. THE FOURIER TRANSFORM OF
TEMPERED GENERALIZED FUNCTIONS 90 6.3. PROPERTIES OF THE FOURIER
TRANSFORM 92 6.4. THE FOURIER TRANSFORM OF GENERALIZED FUNCTIONS WITH
COMPACT SUPPORT 93 6.5. THE FOURIER TRANSFORM OF A CONVOLUTION 94 6.6.
EXAMPLES 96 6.7. THE MELLIN TRANSFORM 109 7. FOURIER SERIES OF PERIODIC
GENERALIZED FUNCTIONS 113 7.1. THE DEFINITION AND ELEMENTARY PROPERTIES
OF PERIODIC GENERALIZED FUNCTIONS 113 7.2. FOURIER SERIES OF PERIODIC
GENERALIZED FUNCTIONS 116 7.3. THE CONVOLUTION ALGEBRA V T 117 7.4.
EXAMPLES 119 8. POSITIVE DEFINITE GENERALIZED FUNCTIONS 121 8.1. THE
DEFINITION AND ELEMENTARY PROPERTIES OF POSITIVE DEFINITE GENERALIZED
FUNCTIONS 121 8.2. THE BOCHRIER-SCHWARTZ THEOREM 123 8.3. EXAMPLES 125
CONTENTS VII 9. THE LAPLACE TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS
126 9.1. DEFINITION OF THE LAPLACE TRANSFORM 126 9.2. PROPERTIES OF THE
LAPLACE TRANSFORM 128 9.3. EXAMPLES 130 10. THE CAUCHY KERNEL AND THE
TRANSFORMS OF CAUCHY-BOCHNER ANDHILBERT 133 10.1. THE SPACE U S 133
10.2. THE CAUCHY KERNEL K C {Z) 138 10.3. THE CAUCHY-BOCHNER TRANSFORM
144 10.4. THE HILBERT TRANSFORM 146 10.5. HOLOMORPHIC FUNCTIONS OF THE
CLASS TIA {C) 147 10.6. THE GENERALIZED CAUCHY-BOCHNER REPRESENTATION
151 11. POISSON KERNEL AND POISSON TRANSFORM 152 11.1. THE DEFINITION
AND PROPERTIES OF THE POISSON KERNEL 152 11.2. THE POISSON TRANSFORM AND
POISSON REPRESENTATION 155 11.3. BOUNDARY VALUES OF THE POISSON INTEGRAL
157 12. ALGEBRAS OF HOLOMORPHIC FUNCTIONS 159 12.1. THE DEFINITION OF
THE H+{C) AND H{C) ALGEBRAS 160 12.2. ISOMORPHISM OF THE ALGEBRAS
S {C*+) ~ H+(C) AND S {C*) ~ H{C) 160 12.3. THE PALEY-WIENER-SCHWARTZ
THEOREM AND ITS GENERALIZATIONS 165 12.4. THE SPACE H A {C) IS THE
PROJECTIVE LIMIT OF THE SPACES H A (C) 166 12.5. THE SCHWARTZ
REPRESENTATION 168 12.6. A GENERALIZATION OF THE PHRAGMEN-LINDELOF
THEOREM 171 13. EQUATIONS IN CONVOLUTION ALGEBRAS 171 13.1. DIVISORS OF
UNITY IN THE H+(C) AND H(Q ALGEBRAS 171 13.2. ON DIVISION BY A
POLYNOMIAL IN THE H{C) ALGEBRA 172 13.3. ESTIMATES FOR HOLOMORPHIC
FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T C 174 13.4. DIVISORS OF
UNITY IN THE ALGEBRA W(C) 177 13.5. EXAMPLE 177 14. TAUBERIAN THEOREMS
FOR GENERALIZED FUNCTIONS 179 14.1. PRELIMINARY RESULTS 179 14.2.
GENERAL TAUBERIAN THEOREM 183 14.3. ONE-DIMENSIONAL TAUBERIAN THEOREMS
186 14.4. TAUBERIAN AND ABELIAN THEOREMS FOR NONNEGATIVE MEASURES 187
14.5. TAUBERIAN THEOREMS FOR HOLOMORPHIC FUNCTIONS OF BOUNDED ARGUMENT
188 VIII CONTENTS CHAPTER 3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
191 15. DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS 191 15.1.
FUNDAMENTAL SOLUTIONS INL 191 15.2. TEMPERED FUNDAMENTAL SOLUTIONS 194
15.3. A DESCENT METHOD 196 15.4. EXAMPLES 199 15.5. A COMPARISON OF
DIFFERENTIAL OPERATORS 207 15.6. ELLIPTIC AND HYPOELLIPTIC OPERATORS 210
15.7. HYPERBOLIC OPERATORS 212 15.8. THE SWEEPING PRINCIPLE 212 16. THE
CAUCHY PROBLEM 213 16.1. THE GENERALIZED CAUCHY PROBLEM FOR A HYPERBOLIC
EQUATION 213 16.2. WAVE POTENTIAL 216 16.3. SURFACE WAVE POTENTIALS 220
16.4. THE CAUCHY PROBLEM FOR THE WAVE EQUATION 222 16.5. A STATEMENT OF
THE GENERALIZED CAUCHY PROBLEM FOR THE HEAT EQUATION 224 16.6. HEAT
POTENTIAL 224 16.7. SOLUTION OF THE CAUCHY PROBLEM FOR THE HEAT EQUATION
228 17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T C 229
17.1. PRELIMINARY REMARKS 229 17.2. PROPERTIES OF FUNCTIONS OF THE CLASS
V+(T C ) 231 17.3. ESTIMATES OF THE GROWTH OF FUNCTIONS OF THE CLASS
H+(T C ) 238 17.4. SMOOTHNESS OF THE SPECTRAL FUNCTION. 240 17.5.
INDICATOR OF GROWTH OF FUNCTIONS OF THE CLASS V. .{T C ) 242 17.6. AN
INTEGRAL REPRESENTATION OF FUNCTIONS OF THE CLASS H + (T C ) 245 18.
HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T N 249 18.1.
LEMMAS 249 18.2. FUNCTIONS OF THE CLASSES H + (T 1 ) AND V+ {T 1 ) 254
18.3. FUNCTIONS OF THE CLASS V+(T N ) 258 18.4. FUNCTIONS OF THE CLASS
H+(T N ) 263 19. POSITIVE REAL MATRIX FUNCTIONS IN T C 266 19.1.
POSITIVE REAL FUNCTIONS IN T C 267 19.2. POSITIVE REAL MATRIX FUNCTIONS
IN T C 269 20. LINEAR PASSIVE SYSTERNS 271 20.1. INTRODUCTION 271 20.2.
COROLLARIES TO THE CONDITION OF PASSIVITY 273 20.3. THE NECESSARY AND
SUFFICIENT CONDITIONS FOR PASSIVITY 277 20.4. MULTIDIMENSIONAL
DISPERSION RELATIONS 282 CONTENTS IX 20.5. THE FUNDAMENTAL SOLUTION AND
THE CAUCHY PROBLEM 285 20.6. WHAT DIFFERENTIAL AND DIFFERENCE OPERATORS
ARE PASSIVE OPERATORS? 287 20.7. EXAMPLES 290 20.8. QUASIASYMPTOTICS OF
THE SOLUTIONS OF SYSTEMS OF EQUATIONS IN CONVOLUTIONS 294 21. ABSTRACT
SCATTERING OPERATOR 295 21.1. THE DEFINITION AND PROPERTIES OF AN
ABSTRACT SCATTERING MATRIX 295 21.2. A DESCRIPTION OF ABSTRACT
SCATTERING MATRICES 298 21.3. THE RELATIONSHIP BETWEEN PASSIVE OPERATORS
AND SCATTERING OPERATORS 299 BIBLIOGRAPHY 303 INDEX 309
|
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| indexdate | 2024-07-09T19:04:56Z |
| institution | BVB |
| isbn | 0415273560 |
| language | English |
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| spelling | Vladimirov, Vasilij S. 1923-2012 Verfasser (DE-588)119416662 aut Methods of the theory of generalized functions V. S. Vladimirov 1. publ. London [u.a.] Taylor & Francis 2002 XIV, 311 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Analytical methods and special functions 6 Literaturverz. S. 303 - 308 Distributions, Théorie des (Analyse fonctionnelle) Physique mathématique Transformations intégrales Mathematische Physik Integral transforms Mathematical physics Theory of distributions (Functional analysis) Distribution Funktionalanalysis (DE-588)4070505-5 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 s DE-604 Analytical methods and special functions 6 (DE-604)BV011932737 6 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009955681&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vladimirov, Vasilij S. 1923-2012 Methods of the theory of generalized functions Analytical methods and special functions Distributions, Théorie des (Analyse fonctionnelle) Physique mathématique Transformations intégrales Mathematische Physik Integral transforms Mathematical physics Theory of distributions (Functional analysis) Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| subject_GND | (DE-588)4070505-5 |
| title | Methods of the theory of generalized functions |
| title_auth | Methods of the theory of generalized functions |
| title_exact_search | Methods of the theory of generalized functions |
| title_full | Methods of the theory of generalized functions V. S. Vladimirov |
| title_fullStr | Methods of the theory of generalized functions V. S. Vladimirov |
| title_full_unstemmed | Methods of the theory of generalized functions V. S. Vladimirov |
| title_short | Methods of the theory of generalized functions |
| title_sort | methods of the theory of generalized functions |
| topic | Distributions, Théorie des (Analyse fonctionnelle) Physique mathématique Transformations intégrales Mathematische Physik Integral transforms Mathematical physics Theory of distributions (Functional analysis) Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| topic_facet | Distributions, Théorie des (Analyse fonctionnelle) Physique mathématique Transformations intégrales Mathematische Physik Integral transforms Mathematical physics Theory of distributions (Functional analysis) Distribution Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009955681&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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