Integral transforms of generalized functions and their applications:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Amsterdam
Gordon and Breach
1997
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 415 S. |
| ISBN: | 9056995545 |
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Record in the Search Index
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| adam_text | INTEGRAL TRANSFORMS OF GENERALIZED FUNCTIONS AND THEIR APPLICATIONS RAM
SHANKAR PATHAK BANARAS HINDU UNIVERSITY VARANASI, INDIA GORDON AND
BREACH SCIENCE PUBLISHERS AUSTRALIA * CANADA * CHINA * FRANCE * GERMANY
* INDIA * JAPAN * LUXEMBOURG * MALAYSIA * THE NETHERLANDS * RUSSIA
SINGAPORE * SWITZERLAND * THAILAND * UNITED KINGDOM CONTENTS PREFACE XV
1 NORMED AND COUNTABLY NORMED SPACES 1 1.1 INTRODUCTION 1 1.2 SEMINORMS
AND LOCALLY CONVEX SPACES 1 1.3 INDUCTIVE LIMITS AND UNION OF
MULTINORMED SPACES 4 1.4 THE TEST FUNCTION SPACE 3) 4 1.5 THE TEST
FUNCTION SPACE S 5 1.6 THE TEST FUNCTION SPACE SF 5 1.7 THE TEST
FUNCTION SPACES Z(A) AND Z 6 1.8 THE TEST FUNCTION SPACE W{I) 7 1.9
SPACES OF GENERALIZED FUNCTIONS 8 1.9.1 SOME SPECIAL GENERALIZED
FUNCTION SPACES 9 1.9.2 CONVERGENCE OF GENERALIZED FUNCTIONS 13 1.9.3
WEAK CONVERGENCE AND STRONG CONVERGENCE 14 1.9.4 A BOUNDEDNESS PROPERTY
15 1.9.5 LINEAR OPERATORS 16 1.10 OPERATIONS ON GENERALIZED FUNCTIONS 17
1.10.1 DIFFERENTIATION OF GENERALIZED FUNCTIONS 18 1.11 THE
MULTIPLICATION AND DIVISION OF GENERALIZED FUNCTIONS 20 1.12 STRUCTURES
OF GENERALIZED FUNCTIONS 22 1.13 THE TENSOR PRODUCT OF DISTRIBUTIONS 24
1.14 THE CONVOLUTION OF GENERALIZED FUNCTIONS 26 VLLL CONTENTS 1.15
FUNDAMENTAL SOLUTIONS OF LINEAR DIFFERENTIAL OPERATORS 27 PROBLEMS 29 2
FOURIER TRANSFORMS OF DISTRIBUTIONS 33 2.1 INTRODUCTION 33 2.2 THE
FOURIER TRANSFORM OF TEMPERED DISTRIBUTIONS 34 2.2.1 THE GENERALIZED
FOURIER TRANSFORM 36 2.2.2 FOURIER TRANSFORMS IN L 2 36 2.2.3 EXAMPLES
37 2.3 PROPERTIES OF THE FOURIER TRANSFORM 39 2.3.1 THE FOURIER
TRANSFORM OF A CONVOLUTION OF GENERALIZED FUNCTIONS 40 2.4 APPLICATIONS
41 2.4.1 APPLICATION OF THE FOURIER TRANSFORM TO DIFFERENTIAL EQUATIONS
41 2.4.2 APPLICATION OF THE FOURIER TRANSFORM TO A CONVOLUTION EQUATION
42 2.5 DUAL INTEGRAL EQUATIONS WITH TRIGONOMETRIC KERNELS ... 42 2.6
EXISTENCE OF FUNDAMENTAL SOLUTIONS OF A CERTAIN BOUNDARY VALUE PROBLEM
50 2.7 THE FOURIER TRANSFORM ON 3) 54 2.8 DISTRIBUTIONAL BOUNDARY
VALUES OF ANALYTIC, FUNCTIONS .. 58 2.9 THE FOURIER TRANSFORM ON & 62
2.10 THE STRUCTURE OF GENERALIZED FUNCTIONS IN Z 66 2.11 THE CAUCHY
PROBLEM FOR THE TWO DIMENSIONAL DIFFUSION EQUATION .. .*. 67 PROBLEMS 70
3 FOURIER TRANSFORMS OF ULTRADISTRIBUTIONS 73 3.1 INTRODUCTION 73 3.2
ULTRADIFFERENTIABLE FUNCTIONS 74 3.2.1 THE FUNCTION M(P) 75 3.3 THE
SPACE S{M P Q) 77 3.4 THE SPACE (M P ;Q) 78 CONTENTS IX 3.5 SOME
OPERATIONS ON TEST FUNCTIONS 80 3.6 THE FOURIER TRANSFORM OF
ULTRADIFFERENTIABLE FUNCTIONS 82 3.7 ULTRADISTRIBUTIONS 84 3.7.1
CONVOLUTION OF ULTRADISTRIBUTIONS 85 3.7.2 STRUCTURE OF
ULTRADISTRIBUTIONS 88 3.8 THE FOURIER TRANSFORM OF ULTRADISTRIBUTIONS 91
3.8.1 PALEY-WIENER THEOREM 92 3.9 THE CAUCHY INTEGRAL REPRESENTATION FOR
ULTRADISTRIBUTIONS OF COMPACT SUPPORT 95 3.10 THE POISSON INTEGRAL
REPRESENTATION 103 3.11 TEMPERED ULTRADISTRIBUTIONS 106 3.12 THE SPACES
9 R (M P ; W) AND 2 R {M P ; W) ILL PROBLEMS 114 LAPLACE TRANSFORM 119
4.1 INTRODUCTION 119 4.2 THE TEST FUNCTION SPACE AND ITS DUAL A 120
4.3 THE LAPLACE TRANSFORM OF GENERALIZED FUNCTIONS 121 4.3.1 EXAMPLES
122 4.3.2 ANALYTICITY 122 4.3.3 IMPORTANT PROPERTIES 123 4.4 INVERSION
124 4.5 CONVOLUTION 130 4.6 AN OPERATIONAL CALCULUS 133 4.7 A CAUCHY
PROBLEM FOR THE DIFFUSION EQUATION 135 4.8 LAPLACE TRANSFORM VIA FOURIER
TRANSFORM 136 4.9 ORDINARY LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT
COEFFICIENTS ...; 137 4.10 THE NON-HOMOGENEOUS HEAT EQUATION 139 4.11 AN
INTEGRO-DIFFERENTIAL EQUATION 140 4.12 ASYMPTOTIC BEHAVIOUR OF THE
LAPLACE TRANSFORM OF FUNCTIONS 141 CONTENTS 4.13 ASYMPTOTIC BEHAVIOUR OF
THE LAPLACE TRANSFORM OF GENERALIZED FUNCTIONS 144 PROBLEMS 147
STIELTJES TRANSFORM 151 5.1 INTRODUCTION 151 5.2 THE TEST FUNCTION
SPACES S X (I) AND S A (I) 152 5.3 PRELIMINARY LEMMAS 154 5.4 THE
DISTRIBUTIONAL STIELTJES TRANSFORM 159 5.4.1 ASYMPTOTIC BEHAVIOUR OF F
{M X) 161 5.5 A COMPLEX INVERSION THEOREM 162 5.6 A REAL INVERSION
THEOREM 166 5.7 ITERATION OF THE LAPLACE TRANSFORM 172 5.8 ABELIAN
THEOREMS FOR THE STIELTJES TRANSFORM OF FUNCTIONS 174 5.9
QUASI-ASYMPTOTIC AND ABELIAN THEOREMS 178 PROBLEMS 182 HILBERT TRANSFORM
185 6.1 INTRODUCTION 185 6.2 THE SCHWARTZ TEST FUNCTION SPACE Q) U AND
ITS DUAL 9 V . 186 6.2.1 THE HILBERT TRANSFORM ON 2 U 187 6.3- - THE
HILBERT TRANSFORM OF GENERALIZED FUNCTIONS IN 3 U . 188 6.4
APPROXIMATE HILBERT TRANSFORM 190 6.5 DISTRIBUTIONAL REPRESENTATION OF
ANALYTIC FUNCTIONS .... 193 6.6 THE HILBERT PROBLEM FOR GENERALIZED
FUNCTIONS 196 6.7 EXISTENCE AND UNIQUENESS OF THE SOLUTION TO A
DIRICHLET PROBLEM 197 6.8 HILBERT TRANSFORM VIA FOURIER TRANSFORM 199
6.9 THE HILBERT TRANSFORM OF ULTRADISTRIBUTIONS 202 6.10 MODIFIED
HILBERT TRANSFORMS 203 6.11 THE SPACES H X (S L ,) AND K X {2I LP ) 205
6.12 DISTRIBUTIONAL MODIFIED HILBERT TRANSFORMS 206 CONTENTS XI 6.13
FINITE HILBERT TRANSFORM 207 PROBLEMS 209 7 MELLIN AND WATSON TRANSFORMS
211 7.1 INTRODUCTION 211 7.2 THE TEST FUNCTION SPACES M A AND M(W,Z) AND
THEIR DUALS 213 7.3 THE MELLIN TRANSFORM 214 7.4 CONVOLUTION 217 7.5 AN
OPERATIONAL CALCULUS 219 7.6 THE SPACE T(X,N) AND THE MELLIN TRANSFORM
219 7.7 THE WATSON TRANSFORM ON T(X,N) 223 7.8 EXAMPLES AND APPLICATIONS
226 7.9 PRODUCT CONVOLUTIONS AND FRACTIONAL INTEGRAL OPERATORS ON T{X,N)
229 7.10 A DUAL DISTRIBUTIONAL EQUATION OF TITCHMARSH TYPE .... 234
PROBLEMS 235 8 HANKEL TRANSFORMS OF DISTRIBUTIONS 239 8.1 INTRODUCTION
239 8.2 THE TEST FUNCTION SPACE H^ AND ITS DUAL H ^ 240 8.2.1 OPERATIONS
ON #* AND HJ, 242 8.3 HANKEL TRANSFORMS ON H^ AND H ^ 244 8.4 THE
N-DIMENSIONAL DISTRIBUTIONAL HANKEL TRANSFORM .... 246 8.5 EXISTENCE OF
FUNDAMENTAL SOLUTIONS OF A CERTAIN BOUNDARY VALUE PROBLEM 248 8.6
FRACTIONAL INTEGRALS AND HANKEL TRANSFORMS 253 8.7 DUAL INTEGRAL
EQUATIONS OF TITCHMARSH TYPE 258 8.7.1 EXISTENCE 260 8.7.2 REGULARITY
263 8.7.3 UNIQUENESS 264 8.8 AN EXTENSION OF §* BY KERNEL METHOD 264
8.8.1 THE HANKEL TRANSFORM OF GENERALIZED FUNCTIONS IN G *(/) 266 XII
CONTENTS 8.9 INVERSION AND UNIQUENESS 270 8.10 AN OPERATIONAL CALCULUS
274 8.11 AXISYMMETRIC DIRICHLET PROBLEM FOR A THICK PLATE 277 PROBLEMS
279 9 HANKEL TRANSFORMS OF ULTRADISTRIBUTIONS 283 9.1 INTRODUCTION 283
9.2 TEST FUNCTION SPACES AND THEIR DUALS 283 9.3 SOME OPERATIONS ON TEST
FUNCTION SPACES 289 9.3.1 OPERATION OF MULTIPLICATION BY X 290 9.3.2
MULTIPLICATION BY AN INFINITELY DIFFERENTIABLE FUNCTION 291 9.3.3 THE
DIFFERENTIAL OPERATOR D 291 9.3.4 SOME OTHER DIFFERENTIAL AND INTEGRAL
OPERATORS .... 292 9.3.5 OPERATIONS IN DUAL SPACES 294 9.4 THE HANKEL
TRANSFORM OF TEST FUNCTIONS 294 9.5 THE GENERALIZED HANKEL TRANSFORM OF
ULTRADISTRIBUTIONS 296 9.6 THE HANKEL TRANSFORM OF ARBITRARY ORDER 297
9.7 SOME OPERATIONAL FORMULAE 300 9.8 THE HANKEL TRANSFORM OF
ULTRADISTRIBUTIONS OF RAPID GROWTH 303 - 9.8.1 THE TEST FUNCTION SPACES
*;/ AND K 304 9.8.2 THE TEST FUNCTION SPACES Y Z AND Y A 305
9.8.3 THE HANKEL TRANSFORM OF ARBITRARY ORDER 308 9.9 AN OPERATIONAL
CALCULUS 313 9.10 A DIRICHLET PROBLEM IN CYLINDRICAL COORDINATES 314
PROBLEMS 317 10 KONTOROVICH-LEBEDEV TRANSFORM 319 10.1 INTRODUCTION 319
10.2 THE TEST FUNCTION SPACE JF (J) AND ITS DUAL 321 CONTENTS XU1 10.3
THE KONTOROVICH-LEBEDEV TRANSFORM OF GENERALIZED FUNCTIONS 322 10.4 THE
FUNCTION G N (T,X) 323 10.5 INVERSION 329 10.6 AN OPERATIONAL CALCULUS
335 10.7 DIRICHLET S PROBLEM FOR A WEDGE WITH A DISTRIBUTIONAL BOUNDARY
CONDITION 337 10.8 THE HANKEL-FORM OF THE KONTOROVICH-LEBEDEV TRANSFORM
339 PROBLEMS 340 11 GENERALIZED MEHLER-FOCK TRANSFORM 343 11.1
INTRODUCTION 343 11.2 THE TEST FUNCTION SPACE M F (I) AND ITS DUAL 345
11.3 THE DISTRIBUTIONAL GENERALIZED MEHLER-FOCK TRANSFORM . 346 11.4 THE
FUNCTION F N {T,X) 348 11.5 INVERSION 353 11.6 AN OPERATIONAL CALCULUS
358 11.7 A DIRICHLET PROBLEM WITH A DISTRIBUTIONAL BOUNDARY CONDITION
360 11.8 CERTAIN DUAL INTEGRAL EQUATIONS 364 PROBLEMS 365 12
EIGENFUNCTION EXPANSION OF GENERALIZED FUNCTIONS 367 12.1 INTRODUCTION
367 12.2 THE TEST FUNCTION SPACE JV{1) 368 12.3 THE STURM-LIOUVILLE
EXPANSION OF GENERALIZED FUNCTIONS IN JT 1) 369 12.4 SPECIAL CASES 370
12.4.1 EXPANSION IN FOURIER SERIES 371 12.4.2 EXPANSION IN A SERIES OF
JACOBI POLYNOMIALS .... 371 12.4.3 EXPANSION IN A SERIES OF LEGENDRE
POLYNOMIALS .. 371 12.4.4 EXPANSION IN SERIES OF SPHERICAL HARMONICS 373
12.4.5 EXPANSION IN SERIES OF BESSEL FUNCTIONS 373 XIV CONTENTS 12.5 AN
OPERATIONAL CALCULUS 375 12.6 DIRICHLET PROBLEM FOR THE INTERIOR OF A
UNIT SPHERE 376 12.7 TEMPERATURE IN A LONG CYLINDER 378 12.8 A
MATHEMATICAL MODEL OF VOLCANOES 382 PROBLEMS 384 BIBLIOGRAPHY 387 INDEX
OF SYMBOLS 401 AUTHOR INDEX 405 SUBJECT INDEX 409
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| spelling | Pathak, Ram Shankar Verfasser aut Integral transforms of generalized functions and their applications Ram Shankar Pathak Amsterdam Gordon and Breach 1997 XVI, 415 S. txt rdacontent n rdamedia nc rdacarrier Integral transforms Theory of distributions (Functional analysis) Integraltransformation (DE-588)4027235-7 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 s Integraltransformation (DE-588)4027235-7 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008128515&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pathak, Ram Shankar Integral transforms of generalized functions and their applications Integral transforms Theory of distributions (Functional analysis) Integraltransformation (DE-588)4027235-7 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| subject_GND | (DE-588)4027235-7 (DE-588)4070505-5 |
| title | Integral transforms of generalized functions and their applications |
| title_auth | Integral transforms of generalized functions and their applications |
| title_exact_search | Integral transforms of generalized functions and their applications |
| title_full | Integral transforms of generalized functions and their applications Ram Shankar Pathak |
| title_fullStr | Integral transforms of generalized functions and their applications Ram Shankar Pathak |
| title_full_unstemmed | Integral transforms of generalized functions and their applications Ram Shankar Pathak |
| title_short | Integral transforms of generalized functions and their applications |
| title_sort | integral transforms of generalized functions and their applications |
| topic | Integral transforms Theory of distributions (Functional analysis) Integraltransformation (DE-588)4027235-7 gnd Distribution Funktionalanalysis (DE-588)4070505-5 gnd |
| topic_facet | Integral transforms Theory of distributions (Functional analysis) Integraltransformation Distribution Funktionalanalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008128515&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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