Integral geometry and convolution equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht
Kluwer Academic Publishers
2003
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and indexes. - Erscheint: November 2003 |
| Beschreibung: | p. cm. |
| ISBN: | 140201628X |
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| adam_text | INTEGRAL GEOMETRY AND CONVOLUTION EQUATIONS BY V.V. VOLCHKOV DONETSK
NATIONAL UNIVERSITY, DEPARTMENT OF MATHEMATICS, DONETSK, UKRAINE KLUWER
ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON TABLE OF CONTENTS TABLE
OF CONTENTS V PREFACE XI PART 1. PRELIMINARIES 1 1 SETS AND MAPPINGS 1
1.1. SOME CONCEPTS RELATED TO SETS 1 1.2. MAPPINGS 3 2 SOME CLASSES OF
FUNCTIONS 5 2.1. INTEGRABLE FUNCTIONS 5 2.2. SPACES OF SMOOTH FUNCTIONS
6 2.3. QUASI-ANALYTIC CLASSES OF FUNCTIONS 8 2.4. ENTIRE FUNCTIONS 9 3
DISTRIBUTIONS 12 3.1. SIMPLEST PROPERTIES OF DISTRIBUTIONS 12 3.2.
TENSOR PRODUCT AND CONVOLUTION 13 3.3. REGULARIZATION 15 4 SOME SPECIAL
FUNCTIONS 16 4.1. GAMMA FUNCTION 16 4.2. HYPERGEOMETRIC FUNCTION 16 4.3.
THE LEGENDRE FUNCTIONS 17 4.4. THE GEGENBAUER POLYNOMIALS 19 4.5. THE
CYLINDRICAL FUNCTIONS 21 5 SOME RESULTS RELATED TO SPHERICAL HARMONICS
26 5.1. SIMPLEST PROPERTIES OF SPHERICAL HARMONICS 26 5.2. EXPANSIONS
OVER SPHERICAL HARMONICS 28 5.3. THE FUNCTIONS $^, TF*;J, 32 5.4. SOME
PROPERTIES OF SOLUTIONS OF CONVOLUTION EQUATIONS 34 6 FOURIER TRANSFORM
AND RELATED QUESTIONS 37 6.1. THE FOURIER TRANSFORM OF INTEGRABLE
FUNCTIONS 37 6.2. THE FOURIER-LAPLACE TRANSFORM 39 6.3. ANALYTIC WAVE
FRONT SET 43 VI TABLE OF CONTENTS 7 PARTIAL DIFFERENTIAL EQUATIONS 46
7.1. SOME GENERAL RESULTS 46 7.2. THE EQUATION (A + A 2 ) M / = 0 47 8
RADON TRANSFORM OVER HYPERPLANES 49 8.1. SIMPLEST PROPERTIES OF THE
RADON TRANSFORM OVER HYPERPLANES . . 49 8.2. FUNCTIONS WITH VANISHING
INTEGRALS OVER HYPERPLANES 50 9 COMMENTS AND OPEN PROBLEMS 55 PART 2.
FUNCTIONS WITH ZERO INTEGRALS OVER BALLS OF A FIXED RADIUS 57 1
FUNCTIONS WITH ZERO AVERAGES OVER BALLS ON SUBSETS OF THE SPACE R N 57
1.1. EXAMPLES OF FUNCTIONS WITH ZERO AVERAGES OVER BALLS AND THEIR
SIMPLEST PROPERTIES 57 1.2. UNIQUENESS THEOREMS 61 1.3. DESCRIPTION OF
FUNCTIONS IN THE CLASS V R (BN) 65 1.4. A LOCAL TWO-RADII THEOREM 69
1.5. A LOCAL ONE-RADIUS THEOREM 81 1.6. DESCRIPTION OF FUNCTIONS IN THE
CLASS V R (B A ^) 86 1.7. OO-RADII THEOREM 89 1.8. THE SUPPORT PROBLEM
FOR AVERAGES OVER BALLS 91 1.9. FUNCTIONS WITH ZERO INTEGRALS OVER BALLS
ON HALF-SPACE 95 2 AVERAGES OVER BALLS ON HYPERBOLIC SPACES 100 2.1.
AUXILIARY CONSTRUCTIONS RELATED TO HYPERBOLIC SPACE 100 2.2. HYPERBOLIC
ANALOGUES OF THE CLASSES V R (BN) 108 2.3. UNIQUENESS THEOREM AND
DESCRIPTION OF THE CLASS V T {BR) IN THE HYPERBOLIC CASE ILL 2.4. A
LOCAL TWO-RADII THEOREM ON HYPERBOLIC SPACE 115 3 FUNCTIONS WITH ZERO
INTEGRALS OVER SPHERICAL CAPS 122 3.1. SOME AUXILIARY ASSERTIONS 122
3.2. SPHERICAL ANALOGUES OF THE CLASSES V T {BR) 127 3.3. THEOREMS ON
AVERAGES OVER SPHERICAL CAPS 131 4 COMMENTS AND OPEN PROBLEMS 137 PART
3. CONVOLUTION EQUATION ON DOMAINS IN M. N 143 1 ONE-DIMENSIONAL CASE
143 1.1. AUXILIARY CONSTRUCTIONS 143 TABLE OF CONTENTS VII 1.2. PARTIAL
SOLUTIONS OF CONVOLUTION EQUATION 148 1.3. UNIQUENESS THEOREM 150 1.4.
GENERAL SOLUTION OF CONVOLUTION EQUATION 152 1.5. SOLUTIONS OF
CONVOLUTION EQUATIONS WITH GROWTH RESTRICTIONS AT INFINITY 157 1.6.
SYSTEMS OF CONVOLUTION EQUATIONS 159 1.7. GENERALIZATIONS OF LOCAL
TWO-RADII THEOREM IN ONE-DIMENSIONAL CASE 162 2 GENERAL SOLUTION OF
CONVOLUTION EQUATION IN DOMAINS WITH SPHERICAL SYMMETRY 169 2.1.
AUXILIARY CONSTRUCTIONS 169 2.2. PARTIAL SOLUTIONS OF CONVOLUTION
EQUATION 172 2.3. UNIQUENESS THEOREMS 176 2.4. GENERAL SOLUTION OF
CONVOLUTION EQUATION IN A BALL 180 2.5. GENERAL SOLUTION OF CONVOLUTION
EQUATION IN A SPHERICAL ANNULUS . 185 2.6. THEOREM ON A REMOVABLE
SINGULARITY FOR SOLUTIONS OF CONVOLUTION EQUATION 189 3 BEHAVIOR OF
SOLUTIONS OF CONVOLUTION EQUATION AT INFINITY 191 3.1. CONVOLUTION
EQUATION ON R N , N ^ 2 191 3.2. CONVOLUTION EQUATION ON EXTERIOR OF
COMPACT SET 196 3.3. CONVOLUTION EQUATION ON HALF-SPACE 199 4 SYSTEMS OF
CONVOLUTION EQUATIONS 201 4.1. SYSTEMS OF CONVOLUTION EQUATIONS ON
DOMAINS WITH SPHERICAL SYMMETRY 201 4.2. GENERALIZATIONS OF LOCAL
TWO-RADII THEOREM 204 5 COMMENTS AND OPEN PROBLEMS 212 PART 4. EXTREMAL
VERSIONS OF THE POMPEIU PROBLEM 214 1 SETS WITH THE POMPEIU PROPERTY 214
1.1. POMPEIU PROBLEM . . . ., 214 1.2. THE LOCAL POMPEIU PROPERTY 219
1.3. UPPER AND LOWER ESTIMATES FOR TZ(A) 223 2 FUNCTIONS WITH VANISHING
INTEGRALS OVER PARALLELEPIPEDS 226 2.1. STATEMENT OF CENTRAL RESULT 226
2.2. A FUNCTIONAL EQUATION 227 2.3. ESTIMATES OF SOME QUANTITIES RELATED
TO 230 2.4. EXAMPLES OF FUNCTIONS IN THE CLASS ^(A, B R ) 233 2.5.
PROPERTIES OF FUNCTIONS IN THE CLASS TY(A, B R ) 237 2.6. PROOF OF
THEOREM 2.1 246 VIII TABLE OF CONTENTS 3 POLYHEDRA WITH LOCAL POMPEIU
PROPERTY 250 3.1. A FUNCTIONAL EQUATION , 250 3.2. UPPER ESTIMATE TZ(A)
FOR POLYHEDRA 255 3.3. VALUE U{A) FOR POLYGONS 259 3.4. VALUE TZ(A) FOR
CONVEX POLYHEDRA 264 3.5. THE REGULAR POLYHEDRA 269 4 FUNCTIONS WITH
VANISHING INTEGRALS OVER ELLIPSOIDS 271 4.1. INTEGRATION OVER ELLIPSOIDS
WITH A COMMON CENTER 271 4.2. SOME AUXILIARY RESULTS ON DISPOSITIONS OF
ELLIPSOIDS INCLUDED TO THE BALL 278 4.3. EXISTENCE RESULT FOR SOLUTIONS
OF AUXILIARY INTEGRAL EQUATION . . . 282 4.4. EXAMPLES OF FUNCTIONS WITH
ZERO INTEGRALS OVER ELLIPSOIDS 287 4.5. FUNCTIONS WITH ZERO INTEGRALS
OVER CONGRUENT ELLIPSOIDS 292 4.6. VALUE H(E A ) FOR ELLIPSOID WHICH IS
NOT A BALL 298 5 OTHER SETS WITH LOCAL POMPEIU PROPERTY 303 5.1. VALUE
K(A) FOR A HALF-BALL 303 5.2. VALUE TZ(A) FO R CIRCULAR SECTOR AND
REULEAUX TRIANGLE 306 6 THE THREE SQUARES PROBLEM AND RELATED
QUESTIONS 311 6.1. STATEMENT OF THE PROBLEM AND SOME GENERALIZATIONS 311
6.2. LOCAL VERSIONS OF THE THREE SQUARES PROBLEM 313 6.3. SOME
HYPERBOLIC ANALOGUES 316 7 INJECTIVITY SETS OF THE POMPEIU TRANSFORM 320
7.1. MAIN PROBLEMS 320 7.2. SUFFICIENT CONDITIONS FOR INJECTIVITY V? 321
7.3. CHARACTERIZATION OF INJECTIVITY SETS OF VF FOR SOME T 323 8
COMMENTS AND OPEN PROBLEMS 334 PART 5. FIRST APPLICATIONS AND RELATED
QUESTIONS 339 1 INJECTIVITY SETS FOR SPHERICAL RADON TRANSFORM 339 1.1.
SPHERICAL RADON TRANSFORM 339 1.2. CONICAL INJECTIVITY SETS 341 1.3.
DESCRIPTION OF THE KERNEL OF 7?. WITH RESPECT TO SETS WITH SPHERICAL
SYMMETRY 344 1.4. TWO-RADII THEOREMS FOR THE SPHERICAL RADON TRANSFORM
351 1.5. INJECTIVITY SETS OF THE TRANSFORM H ON COMPACTLY SUPPORTED
FUNCTIONS 355 TABLE OF CONTENTS IX 2 SOME QUESTIONS OF APPROXIMATION
THEORY 359 2.1. IP ANALOGUES OF THE WIENER-TAUBERIAN THEOREM ON SUBSETS
OF THE REAL LINE 359 2.2. MULTIDIMENSIONAL L P ANALOGUES OF THE
WIENER-TAUBERIAN THEOREM 361 2.3. APPROXIMATION BY SPHERICAL WAVES 364 3
GAP THEOREMS 366 3.1. UNIQUENESS THEOREM FOR SEQUENCE OF EXPONENTIAL
POLYNOMIALS . . 366 3.2. LOCAL THEOREMS FOR GENERALIZED TRIGONOMETRICAL
SERIES 371 3.3. GAP THEOREMS FOR THE .SEQUENCES OF LINEAR COMBINATIONS
OF THE CYLINDRICAL FUNCTIONS 375 4 MORERA TYPE THEOREMS 378 4.1. AN
EXTREMAL PROBLEM RELATED TO MORERA S THEOREM 378 4.2. ZALCMAN S PROBLEM
AND RELATED QUESTIONS 382 4.3. MORERA TYPE THEOREMS IN DOMAINS WITH THE
WEAK CONE CONDITION 387 4.4. REFINEMENT OF DZYADYK S THEOREM 388 5 MEAN
VALUE CHARACTERIZATION OF VARIOUS CLASSES OF FUNCTIONS 390 5.1. MEAN
VALUE PROPERTY FOR HARMONIC FUNCTIONS 390 5.2. THEOREMS ON WEIGHTED
SPHERICAL MEANS FOR SOME DIFFERENTIAL EQUATIONS 399 5.3. MEAN VALUE
THEOREMS FOR POLYANALYTIC FUNCTIONS 402 5.4. MEAN VALUE THEOREM FOR
HARMONIC POLYNOMIALS 404 6 APPLICATIONS TO PARTIAL DIFFERENTIAL
EQUATIONS 408 6.1. UNIQUENESS THEOREMS FOR THE DARBOUX EQUATION 408 6.2.
UNIQUENESS THEOREMS FOR THE WAVE EQUATION 409 6.3. STATIONARY SETS FOR
THE HEAT EQUATION 414 7 SOME QUESTIONS OF MEASURE THEORY 416 7.1.
MEASURE PRESERVING TRANSFORMATIONS 416 7.2. MEASURES ARE DETERMINED BY
THEIR VALUES ON BALLS 418 8 FUNCTIONS WITH ZERO INTEGRALS IN PROBLEMS OF
THE DISCRETE GEOMETRY 420 8.1. SOME GENERAL ESTIMATES FOR DENSITIES OF
PACKINGS 420 8.2. PACKINGS BY /-ADMISSIBLE SETS 423 8.3. INSCRIBED SETS
425 9 COMMENTS AND OPEN PROBLEMS 427 BIBLIOGRAPHY 430 AUTHOR INDEX 449
TABLE OF CONTENTS SUBJECT INDEX 450 BASIC NOTATIONS 452
|
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| id | DE-604.BV017467780 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:18:21Z |
| institution | BVB |
| isbn | 140201628X |
| language | English |
| lccn | 2003058728 |
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| spelling | Volchkov, Valeri V. Verfasser aut Integral geometry and convolution equations by V.V. Volchkov Dordrecht Kluwer Academic Publishers 2003 p. cm. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and indexes. - Erscheint: November 2003 Convolutions (Mathematics) Integral geometry Integralgeometrie (DE-588)4161911-0 gnd rswk-swf Faltungsgleichung (DE-588)4368138-4 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Radon-Transformation (DE-588)4479199-9 gnd rswk-swf Integralgeometrie (DE-588)4161911-0 s DE-604 Faltungsgleichung (DE-588)4368138-4 s Radon-Transformation (DE-588)4479199-9 s Partielle Differentialgleichung (DE-588)4044779-0 s GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010522712&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Volchkov, Valeri V. Integral geometry and convolution equations Convolutions (Mathematics) Integral geometry Integralgeometrie (DE-588)4161911-0 gnd Faltungsgleichung (DE-588)4368138-4 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Radon-Transformation (DE-588)4479199-9 gnd |
| subject_GND | (DE-588)4161911-0 (DE-588)4368138-4 (DE-588)4044779-0 (DE-588)4479199-9 |
| title | Integral geometry and convolution equations |
| title_auth | Integral geometry and convolution equations |
| title_exact_search | Integral geometry and convolution equations |
| title_full | Integral geometry and convolution equations by V.V. Volchkov |
| title_fullStr | Integral geometry and convolution equations by V.V. Volchkov |
| title_full_unstemmed | Integral geometry and convolution equations by V.V. Volchkov |
| title_short | Integral geometry and convolution equations |
| title_sort | integral geometry and convolution equations |
| topic | Convolutions (Mathematics) Integral geometry Integralgeometrie (DE-588)4161911-0 gnd Faltungsgleichung (DE-588)4368138-4 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Radon-Transformation (DE-588)4479199-9 gnd |
| topic_facet | Convolutions (Mathematics) Integral geometry Integralgeometrie Faltungsgleichung Partielle Differentialgleichung Radon-Transformation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010522712&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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