Handbook of function and generalized function transformations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
1996
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| Series: | Mathematical sciences reference series
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 643 S. graph. Darst. |
| ISBN: | 0849378516 |
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MARC
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| 084 | |a MAT 440b |2 stub | ||
| 100 | 1 | |a Zayed, Ahmed I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of function and generalized function transformations |c Ahmed I. Zayed |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 1996 | |
| 300 | |a XXII, 643 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Mathematical sciences reference series | |
| 650 | 4 | |a Distribution <Funktionalanalysis> | |
| 650 | 4 | |a Funktionaltransformation | |
| 650 | 4 | |a Functions -- Handbooks, manuals, etc | |
| 650 | 4 | |a Transformations (Mathematics) -- Handbooks, manuals, etc | |
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Record in the Search Index
| _version_ | 1804125739010752512 |
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| adam_text | 1 PRELIMINARIES 1 1.1 NOTATION 1 1.2 FUNCTION SPACES 2 1.3 ANALYTIC
FUNCTIONS 4 1.4 LOCAL BEHAVIOR OF FUNCTIONS AND INTEGRALS 8 1.5
ORTHOGONAL EXPANSIONS IN A HILBERT SPACE 11 1.6 ORTHOGONAL POLYNOMIALS
16 1.7 ORTHOGONAL FUNCTIONS AND SERIES EXPANSIONS 18 1.8 STURM-LIOUVILLE
SERIES EXPANSIONS 22 1.9 MORE GENERAL ORTHOGONAL EXPANSIONS 26 1.10
NON-ORTHOGONAL EXPANSIONS 30 1.10.1 FRAMES AND NON-ORTHOGONAL BASIS 30
1.10.2 REPRODUCING-KERNEL HILBERT SPACES 34 1.11 SUMMABILITY OF SERIES
AND INTEGRALS 35 1.11.1 SUMMABILITY OF SERIES 36 1.11.2 SUMMABILITY OF
INTEGRALS 40 2 SPECIAL FUNCTIONS 43 2.1 THE GAMMA AND BETA FUNCTIONS 43
2.2 THE ZETA AND MITTAG-LEFFLER FUNCTIONS 46 2.2.1 THE ZETA-FUNCTION 46
2.2.2 THE MITTAG-LEFFLER FUNCTION 48 2.3 THE HYPERGEOMETRIC FUNCTIONS 49
2.3.1 THE GENERALIZED HYPERGEOMETRIC FUNCTIONS 49 2.3.2 THE GAUSS
HYPERGEOMETRIC FUNCTION 50 2.3.3 THE CONFLUENT HYPERGEOMETRIC FUNCTION
54 2.4 CLASSICAL ORTHOGONAL POLYNOMIALS 59 2.4.1 THE JACOBI POLYNOMIALS
L4 AJJ) (X) (A, /? -1) . . . . 59 2.4.2 THE LAGUERRE POLYNOMIALS
L%(X)(A -1) 61 2.4.3 THE HERMITE POLYNOMIALS H N (X) 6 2 2.5 THE
BESSEL FUNCTIONS 63 2.6 THE E-FUNCTION AND MEIJER S G-FUNCTION 68 2.6.1
MACROBERT S E-FUNCTION 68 2.6.2 MEIJER S G-FUNCTION 69 3 GENERALIZED
FUNCTIONS 73 3.1 COUNTABLY-NORMED SPACES AND THEIR DUALS 74 3.1.1
COUNTABLY-NORMED SPACES 74 3.1.2 THE DUAL SPACES OF COUNTABLY-NORMED
SPACES ... 76 3.1.3 EXAMPLES OF COUNTABLY NORMED SPACES 79 3.2
COUNTABLE-UNION SPACES AND THEIR DUALS . 81 3.2.1 COUNTABLE-UNION SPACES
81 3.2.2 THE DUAL SPACES OF COUNTABLE-UNION SPACES .... 82 3.2.3
EXAMPLES OF COUNTABLE-UNION SPACES 83 3.3 GENERALIZED FUNCTIONS 85 3.3.1
TESTING-FUNCTION SPACES AND THEIR DUALS 85 3.3.2 A SPECIAL TYPE OF
GENERALIZED FUNCTIONS 87 3.3.3 SPACES OF GENERALIZED FUNCTIONS 89 3.4
OPERATIONS WITH GENERALIZED FUNCTIONS 90 3.4.1 OPERATIONS AND
TRANSFORMATIONS 90 3.4.2 EXAMPLES OF OPERATIONS ON GENERALIZED FUNCTIONS
91 3.4.3 ORDINARY AND GENERALIZED DERIVATIVES 93 3.5 LOCAL BEHAVIOR OF
GENERALIZED FUNCTIONS 94 3.5.1 THE ORDER OF A GENERALIZED FUNCTION 94
3.5.2 THE VALUE OF A GENERALIZED FUNCTION AT A POINT ... 95 3.6 ANALYTIC
REPRESENTATIONS OF GENERALIZED FUNCTIONS 97 4 FUNCTION TRANSFORMATIONS
99 4.1 INTRODUCTION 99 4.2 FUNCTION TRANSFORMATIONS 101 4.3 RELATIONSHIP
BETWEEN CONTINUOUS AND DISCRETE TRANSFORMS 103 4.4 LINEAR INTEGRAL
TRANSFORMATIONS 105 4.5 LINEAR INTEGRAL TRANSFORMATIONS OF GENERALIZED
FUNCTIONS 107 4.5.1 GENERAL METHODS 107 4.5.2 THE EMBEDDING METHOD 108
4.6 DISCRETE TRANSFORMS OF GENERALIZED FUNCTIONS ILL R THE LAPLACE
TRANSFORM 115 5.1 DEFINITIONS 116 5.2 EXISTENCE AND DOMAIN OF
CONVERGENCE 116 5.3 DOMAIN OF ABSOLUTE AND UNIFORM CONVERGENCE 118 5.4
UNIQUENESS . . . .- 120 5.5 ELEMENTARY PROPERTIES 120 5.6 ANALYTIC
PROPERTIES AND ASYMPTOTIC BEHAVIOR 124 5.7 CHARACTERIZATION OF THE
LAPLACE TRANSFORM 125 5.8 INVERSION FORMULA 126 5.8.1 INVERSION BY
INTEGRATION 126 5.8.2 INVERSION BY PARTIAL FRACTIONS 127 5.8.3 INVERSION
BY DIFFERENTIATION 129 5.9 CONVOLUTION THEOREMS 130 5.9.1 CONVOLUTION IN
THE T-PLANE 130 5.9.2 CONVOLUTION IN THE S-PLANE 131 5.10 PARSEVAL S
RELATION 131 5.11 ABELIAN AND TAUBERIAN THEOREMS 132 5.11.1 ABELIAN
THEOREMS 132 5.11.2 TAUBERIAN THEOREM 133 5.12 LAPLACE TRANSFORM OF
GENERALIZED FUNCTIONS 133 5.12.1 DEFINITION AND PROPERTIES 133 5.12.2
CHARACTERIZATION AND ASYMPTOTIC BEHAVIOR 136 5.12.3 INVERSION FORMULA
136 5.12.4 CONVOLUTION 136 5.13 APPLICATIONS 137 5.13.1 DIFFERENTIAL
EQUATIONS (INITIAL-VALUE PROBLEMS) ... 137 5.13.2 INTEGRAL EQUATIONS 138
5.13.3 INTEGRODIFFERENTIAL EQUATION 139 5.13.4 SYSTEMS OF LINEAR
DIFFERENTIAL EQUATIONS 140 5.13.5 PARTIAL DIFFERENTIAL EQUATIONS 142
5.13.6 ORDINARY DIFFERENTIAL DELAY EQUATIONS 143 5.13.7 FRACTIONAL
DIFFERENTIAL EQUATIONS 143 5.13.8 CONVOLUTION EQUATIONS 144 5.13.9
DIFFERENTIAL AND INTEGRAL EQUATIONS INVOLVING GENERALIZED FUNCTIONS 145
5.13.10 DIFFERENCE EQUATIONS 146 6 THE TWO-SIDED LAPLACE TRANSFORM 147
6.1 DEFINITION 147 6.2 EXISTENCE AND DOMAIN OF CONVERGENCE 148 6.2.1
DOMAIN OF CONVERGENCE 148 6.2.2 SUFFICIENT CONDITIONS FOR EXISTENCE 150
6.2.3 UNIQUENESS 150 6.3 SUMMABILITY 150 6.4 INVERSION FORMULA 151 6.5
CHARACTERIZATION AND REPRESENTATION 151 6.6 CONVOLUTION 152 6.7
PARSEVAL S RELATION 153 6.8 GENERALIZED FUNCTIONS 154 6.8.1 DEFINITION
(THE EMBEDDING METHOD) 154 6.8.2 ELEMENTARY PROPERTIES 155 6.8.3
INVERSION FORMULA 156 6.8.4 UNIQUENESS 156 6.8.5 CHARACTERIZATION AND
REPRESENTATION 156 6.8.6 CONVOLUTION 157 6.8.7 THE ADJOINT METHOD 158
6.9 APPLICATIONS 159 7 THE BOREL TRANSFORM 163 7.1 DEFINITIONS 164 7.2
EXISTENCE AND DOMAIN OF CONVERGENCE 165 7.3 SERIES REPRESENTATION OF THE
BOREL TRANSFORM 166 7.4 POLYA S THEOREM 167 7.5 INVERSION FORMULA 168
7.6 ANALYTIC PROPERTIES OF THE BOREL TRANSFORM 170 7.7 THE BOREL
TRANSFORM OF GENERALIZED FUNCTIONS 170 7.8 APPLICATIONS 172 8 THE
STIELTJES TRANSFORM 175 8.1 DEFINITIONS 175 8.2 EXISTENCE AND DOMAIN OF
CONVERGENCE 176 8.2.1 DOMAIN OF CONVERGENCE 176 8.2.2 UNIQUENESS 177 8.3
ANALYTIC PROPERTIES 178 8.4 ELEMENTARY PROPERTIES 179 8.5 RELATIONSHIPS
WITH OTHER TRANSFORMS 180 8.5.1 THE LAPLACE TRANSFORM 180 8.5.2 THE
CONVOLUTION TRANSFORM 182 8.6 INVERSION FORMULAS 183 8.6.1
COMPLEX-INVERSION FORMULA 183 8.6.2 REAL INVERSION FORMULA 183 8.7
ASYMPTOTIC BEHAVIOR 184 8.8 ABELIAN AND TAUBERIAN THEOREMS 184 8.8.1
ABELIAN THEOREM 184 8.8.2 TAUBERIAN THEOREM 185 8.9 STIELTJES TRANSFORM
OF GENERALIZED FUNCTIONS 185 8.9.1 DEFINITION AND PROPERTIES 185 8.9.2
INVERSION FORMULA 187 8.10 APPLICATIONS 187 9 THE LAMBERT TRANSFORM 191
9.1 DEFINITIONS 192 9.2 EXISTENCE 193 9.3 RELATIONSHIPS WITH OTHER
TRANSFORMS 194 9.4 INVERSION FORMULAS 197 9.5 GENERALIZED FUNCTIONS 198
9.6 APPLICATIONS 198 10 THE MELLIN TRANSFORM 201 10.1 DEFINITIONS 201
10.2 EXISTENCE AND DOMAIN OF CONVERGENCE 203 10.2.1 DOMAIN OF
CONVERGENCE 203 10.2.2 UNIQUENESS 204 10.3 ELEMENTARY PROPERTIES 204
10.4 INVERSION FORMULA 205 10.5 SUMMABILITY 206 10.6 CONVOLUTION 207
10.7 MELLIN TRANSFORM OF L 2 -FUNCTIONS 209 10.8 MELLIN TRANSFORM OF
GENERALIZED FUNCTIONS 210 10.8.1 DEFINITION 210 10.8.2 UNIQUENESS 211
10.8.3 INVERSION FORMULA 211 10.8.4 CHARACTERIZATION 211 10.9
APPLICATIONS 212 10.9.1 DIFFERENTIAL EQUATIONS AND INITIAL-VALUE
PROBLEMS . . 212 10.9.2 PARTIAL DIFFERENTIAL EQUATIONS 213 10.9.3
INTEGRAL EQUATIONS 214 11 FINITE MELLIN-TYPE TRANSFORMS 215 11.1
DEFINITIONS 215 11.2 EXISTENCE 216 11.3 ELEMENTARY PROPERTIES 216 11.4
RELATIONSHIP WITH THE MELLIN TRANSFORM 217 11.5 CONVOLUTION 218 11.6
INVERSION FORMULAS 218 11.7 APPLICATIONS 219 12 THE FOURIER TRANSFORM
221 12.1 DEFINITIONS 222 12.2 EXISTENCE AND UNIQUENESS 227 12.2.1
EXISTENCE 227 12.2.2 UNIQUENESS 228 12.3 ANALYTIC PROPERTIES 228 12.4
ELEMENTARY PROPERTIES 229 12.5 SUMMABILITY 230 12.6 PARSEVAL S RELATION
232 12.7 THE UNCERTAINTY PRINCIPLE 233 12.8 INVERSION FORMULA 234 12.9
CONVOLUTION 236 12.9.1 CONVOLUTION FOR THE FOURIER TRANSFORM 236 12.9.2
CONVOLUTION FOR THE FOURIER COSINE AND SINE TRANSFORMS 238 12.10POISSON
SUMMATION FORMULAS 239 12.10.1 POISSON SUMMATION FORMULA FOR UNIFORM
SAMPLING . 239 12.10.2 POISSON SUMMATION FORMULA FOR NON-UNIFORM SAM-
PLING 241 12.H THE FOURIER TRANSFORM OF L 2 -FUNCTIONS 242 12.12ANALYTIC
REPRESENTATION OF THE FOURIER TRANSFORM 248 12.13GENERALIZED FUNCTIONS
249 12.13.1 THE EMBEDDING METHOD 249 12.13.2 THE ADJOINT METHOD 251
12.14APPLICATIONS 254 12.14.1 ORDINARY DIFFERENTIAL EQUATIONS 255
12.14.2 INTEGRAL EQUATIONS 256 12.14.3 DIFFERENCE EQUATIONS 257 12.14.4
DELAYED DIFFERENTIAL EQUATIONS 258 12.14.5 DIFFERENTIAL-DIFFERENCE
EQUATIONS 259 12.14.6 CONVOLUTION EQUATIONS 261 12.14.7 PARTIAL
DIFFERENTIAL EQUATIONS 261 13 THE HARTLEY TRANSFORM 265 13.1 DEFINITIONS
266 13.2 ELEMENTARY PROPERTIES 267 13.3 RELATIONSHIP WITH OTHER
TRANSFORMS 267 13.3.1 THE FOURIER TRANSFORM 267 13.3.2 THE HARDY
TRANSFORM 268 13.4 PARSEVAL S RELATION 269 13.5 INVERSION FORMULA 269
13.6 CONVOLUTION 270 13.7 GENERALIZED FUNCTIONS AND APPLICATIONS 271
13.7.1 GENERALIZED FUNCTIONS 271 13.7.2 APPLICATIONS 271 I I THE HILBERT
TRANSFORM 273 14.1 MOTIVATION 273 14.2 DEFINITION 275 14.3 EXISTENCE 275
14.4 ELEMENTARY PROPERTIES 276 14.5 RELATIONSHIPS WITH OTHER TRANSFORMS
277 14.5.1 THE FOURIER TRANSFORM 277 14.5.2 THE STIELTJES TRANSFORM 278
14.5.3 THE BOAS TRANSFORM 279 14.5.4 THE CONVOLUTION TRANSFORM 279
14.5.5 THE CAUCHY TRANSFORM 280 14.6 INVERSION FORMULA 280 14.7
GENERALIZED FUNCTIONS 282 14.8 APPLICATIONS 284 14.8.1 SIGNAL ANALYSIS
284 14.8.2 ANALYTIC SIGNALS 286 14.8.3 AIRFOIL DESIGN 287 15 THE BOAS
TRANSFORM 291 15.1 MOTIVATION 291 15.2 DEFINITIONS 292 15.3 EXISTENCE
293 15.4 THE FUNDAMENTAL RELATION FOR THE BOAS TRANSFORM 294 15.5
ELEMENTARY PROPERTIES 296 15.6 RELATIONSHIPS WITH OTHER TRANSFORMS 296
15.7 PARSEVAL S RELATION 297 15.8 INVERSION FORMULAS 298 15.9
CHARACTERIZATION OF THE BOAS TRANSFORM 298 15.10BOAS THEOREMS 299 15.11
APPLICATIONS 300 16 THE MITTAG-LEFFLER TRANSFORM 301 16.1 DEFINITIONS
302 16.2 EXISTENCE 303 16.3 INVERSION FORMULA 303 16.4 TRANSFORMS OF L 2
-FUNCTIONS 304 16.5 APPLICATIONS 306 17 THE CONVOLUTION TRANSFORM 309
17.1 DEFINITIONS 311 17.2 EXISTENCE 314 17.3 ELEMENTARY PROPERTIES 315
17.4 RELATIONSHIPS WITH OTHER TRANSFORMS 315 17.4.1 THE LAPLACE
TRANSFORM 315 17.4.2 THE STIELTJES TRANSFORM 316 17.4.3 THE GENERALIZED
STIELTJES TRANSFORM 316 17.4.4 THE K (MEIJER) TRANSFORM 317 17.4.5
FOURIER-TYPE INTEGRAL TRANSFORMS 317 17.5 INVERSION FORMULA 317 17.6
GENERALIZED FUNCTIONS 321 17.7 APPLICATIONS 323 18 THE WEIERSTRASS
TRANSFORM 325 18.1 DEFINITIONS 325 18.2 EXISTENCE 330 18.3 ELEMENTARY
PROPERTIES 330 18.4 RELATIONSHIPS WITH OTHER TRANSFORMS 330 18.4.1 THE
TWO-SIDED LAPLACE TRANSFORM 330 18.4.2 THE CONVOLUTION TRANSFORM 331
18.5 INVERSION FORMULAS 331 18.6 GENERALIZED FUNCTIONS 333 18.7
APPLICATIONS 335 19 THE ABEL TRANSFORM 337 19.1 DEFINITIONS 338 19.2
EXISTENCE 339 19.3 ELEMENTARY PROPERTIES 339 19.4 RELATIONSHIPS WITH
OTHER TRANSFORMS 340 19.5 INVERSION FORMULA 341 19.6 THE ABEL TRANSFORM
OF GENERALIZED FUNCTIONS 343 19.7 APPLICATIONS 343 20 THE
RIEMANN-LIOUVILLE AND WEYL FRACTIONAL INTEGRALS 345 20.1 INTRODUCTION
345 20.2 DEFINITIONS 348 20.3 EXISTENCE 352 20.4 ELEMENTARY PROPERTIES
354 20.4.1 ELEMENTARY PROPERTIES OF I A AND W A 354 20.4.2 ELEMENTARY
PROPERTIES OF THE ERDELYI-KOBER FRACTIONAL INTEGRALS 355 20.5 INVERSION
FORMULAS 356 20.6 RELATIONSHIPS WITH OTHER TRANSFORMS 360 20.6.1 THE
LAPLACE TRANSFORM 360 20.6.2 THE MELLIN TRANSFORM 361 20.7 FRACTIONAL
DIFFERENTIATION AND INTEGRATION OF GENERALIZED FUNCTIONS 362 20.7.1
RIEMANN-LIOUVILLE INTEGRALS OF GENERALIZED FUNCTIONS 362 20.7.2 THE
ERDELYI-KOBER-TYPE INTEGRALS OF GENERALIZED FUNC- TIONS 364 20.8
APPLICATIONS 366 20.8.1 ORDINARY DIFFERENTIAL EQUATIONS WITH FRACTIONAL
ORDERS 366 20.8.2 PARTIAL DIFFERENTIAL EQUATIONS 367 20.8.3 INTEGRAL
EQUATIONS 368 20.8.4 SPECIAL FUNCTIONS 368 20.8.5 EVALUATION OF DEFINITE
INTEGRALS 370 21 THE HANKEL AND HANKEL-TYPE TRANSFORMS 371 21.1
DEFINITIONS 372 21.2 EXISTENCE 374 21.3 ELEMENTARY PROPERTIES 375 21.4
INVERSION FORMULAS 375 21.4.1 THE HANKEL, Y, AND H TRANSFORMS 375 21.4.2
THE I TRANSFORM 379 21.4.3 THE HANKEL POTENTIAL TRANSFORM 379 21.5
PARSEVAL S RELATION 380 21.6 CONVOLUTION 380 21.7 GENERALIZED FUNCTIONS
382 21.7.1 THE HANKEL TRANSFORM 382 21.7.2 THE I TRANSFORM 386 21.8
APPLICATIONS 386 21.8.1 AXISYMMETRIC DIRICHLET PROBLEM FOR A HALF-SPACE
. . 387 21.8.2 HANKEL HARMONIC FUNCTIONS 388 22 THE HARDY TRANSFORM 389
22.1 DEFINITIONS 389 22.2 EXISTENCE 390 22.3 ELEMENTARY PROPERTIES 391
22.4 ASYMPTOTIC BEHAVIOR AND ABELIAN THEOREMS 391 22.5 INVERSION FORMULA
391 22.6 THE HARDY TRANSFORM OF GENERALIZED FUNCTIONS 394 22.6.1
DEFINITION 394 22.6.2 INVERSION FORMULA 395 22.7 APPLICATIONS 395 23 THE
K (MEIJER) TRANSFORM 397 23.1 DEFINITIONS 397 23.2 EXISTENCE 398 23.3
ELEMENTARY PROPERTIES 398 23.4 RELATIONSHIPS WITH OTHER TRANSFORMS 399
23.4.1 THE LAPLACE TRANSFORM 399 23.5 INVERSION FORMULA 400 23.6 THE K
TRANSFORM OF L 2 -FUNCTIONS . . . 400 23.7 GENERALIZED FUNCTIONS 401
23.7.1 DEFINITIONS 401 23.7.2 CHARACTERIZATION 402 23.7.3 OPERATIONAL
CALCULUS 403 23.7.4 INVERSION FORMULA 403 23.8 APPLICATIONS 403 24 THE
KONTOROVICH-LEBEDEV TRANSFORM 405 24.1 DEFINITIONS 405 24.2 EXISTENCE
406 24.3 RELATIONSHIP WITH OTHER TRANSFORMS 406 24.3.1 THE FOURIER
TRANSFORM 406 24.3.2 THE LAPLACE TRANSFORM 408 24.4 INVERSION FORMULAS
409 24.5 PARSEVAL S RELATION 410 24.6 GENERALIZED FUNCTIONS 410 24.7
APPLICATIONS 412 25 THE MEHLER-FOCK TRANSFORM 415 25.1 DEFINITIONS 416
25.2 EXISTENCE 416 25.3 ELEMENTARY PROPERTIES 417 25.4 RELATIONSHIP WITH
OTHER TRANSFORMS 417 25.4.1 THE FOURIER AND WEYL FRACTIONAL INTEGRALS
417 25.4.2 THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL 418 25.5 INVERSION
FORMULA 418 25.6 PARSEVAL-TYPE RELATION 420 25.7 GENERALIZED FUNCTIONS
421 25.7.1 DEFINITION 421 25.7.2 CONVOLUTION 422 25.8 APPLICATIONS 422
25.8.1 SOLUTIONS OF LAPLACE S EQUATION IN SPHEROIDAL COORDINATES 422
25.8.2 SOLUTIONS OF LAPLACE S EQUATION IN TOROIDAL COORDINATES 424 26
THE STURM-LIOUVILLE-TYPE TRANSFORMS 427 26.1 THE REGULAR CASE 428 26.1.1
THE DISCRETE TRANSFORM 428 26.1.2 THE CONTINUOUS TRANSFORM 429 26.1.3
INVERSION OF THE CONTINUOUS TRANSFORM 431 26.2 THE SINGULAR CASE ON A
HALF-LINE 432 M).;I THE SINGULAR CASE ON THE WHOLE LINE 435 MI/I
SUMMABILITY 437 (;.R EXAMPLES 437 26.5.1 STURM-LIOUVILLE TRANSFORM
PAIRS 437 26.5.2 CONTINUOUS TRANSFORMATIONS ASSOCIATED WITH DISCRETE
SPECTRA 440 - (I.6 APPLICATIONS 444 26.6.1 THE DISCRETE STURM-LIOUVILLE
TRANSFORM 444 26.6.2 THE FINITE CONTINUOUS JACOBI TRANSFORM 446 */
MISCELLANEOUS TRANSFORMS 449 27.1 THE HYPERGEOMETRIC TRANSFORMS 450 27.2
THE E TRANSFORM 454 27.3 THE G TRANSFORM 457 M THE BARGMANN TRANSFORM
461 28.1 DEFINITIONS 462 28.2 EXISTENCE 462 28.3 ELEMENTARY PROPERTIES
463 28.4 RELATIONSHIP WITH OTHER TRANSFORMS 463 28.5 THE
BARGMANN-SEGAL-FOCH SPACE 464 28.6 PARSEVAL S RELATION 467 28.7
INVERSION FORMULA 468 28.8 GENERALIZED FUNCTIONS 469 28.9 APPLICATIONS
471 ?M THE ZAK TRANSFORM 473 29.1 DEFINITIONS 474 29.2 EXISTENCE 474
29.3 ELEMENTARY PROPERTIES 475 29.4 ANALYTIC PROPERTIES 476 29.5
INVERSION FORMULAS 477 29.6 RELATIONSHIPS WITH OTHER TRANSFORMS 477 29.7
PRODUCT AND CONVOLUTION OF ZAK TRANSFORMS 478 29.8 THE ZAK TRANSFORM OF
L 2 -FUNCTIONS 481 29.9 THE ZAK TRANSFORM OF GENERALIZED FUNCTIONS 486
29.9.1 DEFINITION 486 29.9.2 CHARACTERIZATION 487 29.10 APPLICATIONS 488
:T() THE GABOR TRANSFORM 491 30.1 DEFINITIONS 492 30.2 EXISTENCE 494
30.3 ELEMENTARY PROPERTIES 496 30.4 PARSEVAL S RELATION 497 30.5
INVERSION FORMULA 497 30.6 L 2 -THEORY 498 30.7 RELATIONSHIPS WITH OTHER
TRANSFORMS 501 30.8 THE GABOR REPRESENTATION PROBLEM 502 30.9
GENERALIZED FUNCTIONS 503 30.10 APPLICATIONS 505 31 THE AMBIGUITY
TRANSFORMATION AND WIGNER DISTRIBUTION 507 31.1 DEFINITIONS 508 31.2
EXISTENCE 509 31.3 ELEMENTARY PROPERTIES 510 31.4 RELATIONSHIPS WITH
OTHER TRANSFORMS 513 31.4.1 THE FOURIER TRANSFORM 513 31.4.2 THE ZAK
TRANSFORM 516 31.5 PARSEVAL S RELATION 517 31.6 THE AMBIGUITY
TRANSFORMATION 519 31.7 THE WIGNER DISTRIBUTION OF BAND-LIMITED
FUNCTIONS 521 31.8 CONVOLUTION 522 31.9 GENERALIZED FUNCTIONS 523 31.10
APPLICATIONS 526 32 THE WAVELET TRANSFORM 531 32.1 DEFINITIONS 532 32.2
THE CONTINUOUS WAVELET TRANSFORM 533 32.2.1 EXISTENCE 533 32.2.2
ELEMENTARY PROPERTIES 534 32.2.3 RELATIONSHIP WITH THE FOURIER TRANSFORM
534 32.2.4 PARSEVAL S RELATION 534 32.2.5 INVERSION FORMULA 537 32.2.6
REPRODUCING-KERNEL-HILBERT SPACE ASSOCIATED WITH THE CONTINUOUS WAVELET
TRANSFORM 537 32.2.7 LOCAL PROPERTIES OF THE CONTINUOUS WAVELET
TRANSFORM 538 32.3 THE DISCRETE WAVELET TRANSFORM 539 32.3.1 ORTHONORMAL
BASIS AND FRAMES OF WAVELETS 540 32.3.2 MULTIRESOLUTION ANALYSIS 542
32.3.3 ANOTHER APPROACH TO MULTIRESOLUTION ANALYSIS . . . 548 32.4
ORTHONORMAL WAVELETS WITH COMPACT SUPPORT 550 32.5 EXAMPLES OF WAVELETS
552 32.6 GENERALIZED FUNCTIONS 554 32.7 APPLICATIONS 555 :*:$ THE RADON
TRANSFORM 557 33.1 DEFINITIONS 558 33.2 EXISTENCE AND UNIQUENESS 562
33.3 ELEMENTARY PROPERTIES 562 33.4 CONVOLUTION 565 33.5 RELATIONSHIP
WITH OTHER TRANSFORMS 566 33.5.1 THE X-RAY AND K-PLANE TRANSFORMS 566
33.5.2 THE FOURIER TRANSFORM 567 33.5.3 THE GEGENBAUER TRANSFORM 567
33.6 INVERSION FORMULA 570 33.7 THE ADJOINT RADON TRANSFORMATION 572
33.8 PARSEVAL S RELATION 573 33.9 GENERALIZED FUNCTIONS 575 33.10
APPLICATIONS 577 33.10.1 PHYSICAL APPLICATIONS 577 33.10.2 MATHEMATICAL
APPLICATIONS 579 34 SEQUENCE TRANSFORMATIONS 581 34.1 THE DISCRETE
FOURIER TRANSFORM (DFT) 582 34.1.1 DEFINITION AND INVERSION FORMULA 582
34.1.2 ELEMENTARY PROPERTIES 583 34.1.3 CONVOLUTION AND PARSEVAL S
RELATION 584 34.2 THE FAST FOURIER TRANSFORM (FFT) 585 34.3 THE Z-
TRANSFORM 586 34.3.1 DEFINITIONS 586 34.3.2 ELEMENTARY PROPERTIES 588
34.3.3 CONVOLUTION, PRODUCT, AND PARSEVAL S RELATION . . . 589 34.3.4
INVERSION FORMULAS 589 34.4 THE WALSH FUNCTIONS 592 34.4.1 DEFINITIONS
592 34.5 BASIC PROPERTIES 593 34.6 THE DISCRETE WALSH TRANSFORM 596
APPENDICES A SIGNAL ANALYSIS TERMINOLOGY 597 B TABLES OF INTEGRAL
TRANSFORMS 601 C LIST OF SYMBOLS 613 REFERENCES 617 INDEX 635
|
| any_adam_object | 1 |
| author | Zayed, Ahmed I. |
| author_facet | Zayed, Ahmed I. |
| author_role | aut |
| author_sort | Zayed, Ahmed I. |
| author_variant | a i z ai aiz |
| building | Verbundindex |
| bvnumber | BV011238160 |
| callnumber-first | Q - Science |
| callnumber-label | QA331 |
| callnumber-raw | QA331.Z39 1996 |
| callnumber-search | QA331.Z39 1996 |
| callnumber-sort | QA 3331 Z39 41996 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 450 SK 700 |
| classification_tum | MAT 440b |
| ctrlnum | (OCoLC)247047844 (DE-599)BVBBV011238160 |
| dewey-full | 515/.72320 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.723 20 |
| dewey-search | 515/.723 20 |
| dewey-sort | 3515 3723 220 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011238160 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:06:20Z |
| institution | BVB |
| isbn | 0849378516 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007542273 |
| oclc_num | 247047844 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-706 DE-521 DE-11 DE-188 DE-739 |
| owner_facet | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-706 DE-521 DE-11 DE-188 DE-739 |
| physical | XXII, 643 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Mathematical sciences reference series |
| spelling | Zayed, Ahmed I. Verfasser aut Handbook of function and generalized function transformations Ahmed I. Zayed Boca Raton [u.a.] CRC Press 1996 XXII, 643 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical sciences reference series Distribution <Funktionalanalysis> Funktionaltransformation Functions -- Handbooks, manuals, etc Transformations (Mathematics) -- Handbooks, manuals, etc Funktionaltransformation (DE-588)4123244-6 gnd rswk-swf Funktionaltransformation (DE-588)4123244-6 s DE-604 OEBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007542273&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zayed, Ahmed I. Handbook of function and generalized function transformations Distribution <Funktionalanalysis> Funktionaltransformation Functions -- Handbooks, manuals, etc Transformations (Mathematics) -- Handbooks, manuals, etc Funktionaltransformation (DE-588)4123244-6 gnd |
| subject_GND | (DE-588)4123244-6 |
| title | Handbook of function and generalized function transformations |
| title_auth | Handbook of function and generalized function transformations |
| title_exact_search | Handbook of function and generalized function transformations |
| title_full | Handbook of function and generalized function transformations Ahmed I. Zayed |
| title_fullStr | Handbook of function and generalized function transformations Ahmed I. Zayed |
| title_full_unstemmed | Handbook of function and generalized function transformations Ahmed I. Zayed |
| title_short | Handbook of function and generalized function transformations |
| title_sort | handbook of function and generalized function transformations |
| topic | Distribution <Funktionalanalysis> Funktionaltransformation Functions -- Handbooks, manuals, etc Transformations (Mathematics) -- Handbooks, manuals, etc Funktionaltransformation (DE-588)4123244-6 gnd |
| topic_facet | Distribution <Funktionalanalysis> Funktionaltransformation Functions -- Handbooks, manuals, etc Transformations (Mathematics) -- Handbooks, manuals, etc |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007542273&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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