Spherical harmonics and approximations on the unit sphere: an introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Lecture notes in mathematics
2044 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | IX, 244 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642259821 3642259820 9783642259838 |
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Record in the Search Index
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IMAGE 1
C O N T E N T S
1 PRELIMINARIES 1
1.1 NOTATIONS 3
1.2 THE R-FURICTION 5
1.3 BASIC RESULTS RELATED TO THE SPHERE 6
2 SPHERICAL HARMONICS 11
2.1 SPHERICAL HARMONICS THROUGH PRIMITIVE SPACES 11
2.1.1 SPACES OF HOMOGENEOUS POLYNOMIALS 13
2.1.2 LEGENDRE HARMONIC AND LEGENDRE POLYNOMIAL 17 2.1.3 SPHERICAL
HARMONICS 19
2.2 ADDITION THEOREM AND ITS CONSEQUENCES 20
2.3 A PROJECTION OPERATOR 26
2.4 RELATIONS AMONG POLYNOMIAL SPACES 30
2.5 THE FUNK-HECKE FORMULA 34
2.6 LEGENDRE POLYNOMIALS: REPRESENTATION FORMULAS 36
2.6.1 RODRIGUES REPRESENTATION FORMULA 36
2.6.2 INTEGRAL REPRESENTATION FORMULAS 39
2.7 LEGENDRE POLYNOMIALS: PROPERTIES 41
2.7.1 INTEGRALS, ORTHOGONALITY 42
2.7.2 DIFFERENTIAL EQUATION AND DISTRIBUTION OF ROOTS 43
2.7.3 RECURSION FORMULAS 45
2.7.4 GENERATING FUNCTION 52
2.7.5 VALUES AND BOUNDS 56
2.8 COMPLETENESS 60
2.8.1 COMPLETENESS IN C ( S D _ 1 ) 60
2.8.2 COMPLETENESS IN C(S D ~ 1 ) VIA THE POISSON IDENTITY . 64 2.8.3
CONVERGENCE OF FOURIER-LAPLACE SERIES 66
2.8.4 COMPLETENESS IN I 2 ( S D _ 1 ) 69
2.9 THE GEGENBAUER POLYNOMIALS 71
V I I
HTTP://D-NB.INFO/1017147515
IMAGE 2
VIII
C O N T E N T S
2.10 THE ASSOCIATED LEGENDRE FUNCTIONS 74
2.10.1 DEFINITION AND REPRESENTATION FORMULAS 74
2.10.2 PROPERTIES 77
2.10.3 NORMALIZED ASSOCIATED LEGENDRE FUNCTIONS 80 2.11 GENERATING
ORTHONORMALIZED BASES FOR SPHERICAL HARMONIC SPACES 81
3 DIFFERENTIATION AND INTEGRATION OVER T H E SPHERE 87
3.1 THE LAPLACE-BELTRAMI OPERATOR 87
3.2 A FORMULA FOR THE LAPLACE-BELTRAMI OPERATOR 94
3.3 SPHERICAL HARMONICS AS EIGENFUNCTIONS OF THE LAPLACE-BELTRAMI
OPERATOR 96
3.4 SOME INTEGRATION FORMULAS 100
3.5 SOME DIFFERENTIATION FORMULAS 104
3.6 SOME INTEGRAL IDENTITIES FOR SPHERICAL HARMONICS 106
3.7 INTEGRAL IDENTITIES THROUGH THE FUNK-HECKE FORMULA 113 3.7.1 A
FAMILY OF INTEGRAL IDENTITIES FOR SPHERICAL HARMONICS 114
3.7.2 SOME EXTENSIONS 118
3.8 SOBOLEV SPACES ON THE UNIT SPHERE 119
3.9 POSITIVE DEFINITE FUNCTIONS 124
4 APPROXIMATION THEORY 131
4.1 SPHERICAL POLYNOMIALS 132
4.2 BEST APPROXIMATION ON THE UNIT SPHERE 135
4.2.1 THE APPROACH TO BEST APPROXIMATION OF DAI AND XU 136
4.2.2 THE APPROACH TO BEST APPROXIMATION OF RAGOZIN . 142 4.2.3 BEST
SIMULTANEOUS APPROXIMATION INCLUDING DERIVATIVES 149
4.2.4 LEBESGUE CONSTANTS 150
4.2.5 BEST APPROXIMATION FOR A PARAMETERIZED FAMILY 152 4.3
APPROXIMATION ON THE UNIT DISK 154
4.3.1 ORTHOGONAL POLYNOMIALS 155
4.3.2 PROPERTIES OF ORTHOGONAL POLYNOMIALS OVER B 2 161 4.3.3 ORTHOGONAL
EXPANSIONS 162
5 NUMERICAL QUADRATURE 165
5.1 THE USE OF UNIVARIATE FORMULAS 166
5.2 COMPOSITE METHODS 172
5.2.1 THE CENTROID METHOD 174
5.2.2 GENERAL COMPOSITE METHODS 176
5.2.3 ERROR ANALYSIS 178
5.3 HIGH ORDER GAUSS-TYPE METHODS 185
5.3.1 EFFICIENCY OF A HIGH-ORDER FORMULA 188
5.3.2 THE CENTROID METHOD 189
5.3.3 AN ALTERNATIVE APPROACH 190
IMAGE 3
C O N T E N T S IX
5.4 INTEGRATION OF SCATTERED DATA 193
5.5 INTEGRATION OF SINGULAR FUNCTIONS 195
5.5.1 SINGULAR INTEGRANDS 199
5.6 QUADRATURE OVER THE UNIT DISK 201
5.6.1 A PRODUCT GAUSS FORMULA 202
5.7 DISCRETE ORTHOGONAL EXPANSIONS 203
5.7.1 HYPERINTERPOLATION OVER § 2 204
5.7.2 HYPERINTERPOLATION OVER THE UNIT DISK 210
6 APPLICATIONS: SPECTRAL M E T H O D S 211
6.1 A BOUNDARY INTEGRAL EQUATION 212
6.1.1 CONVERGENCE THEORY 214
6.1.2 QUADRATURE 217
6.1.3 A NUMERICAL EXAMPLE 220
6.2 A SPECTRAL METHOD FOR A PARTIAL DIFFERENTIAL EQUATION 224
6.2.1 IMPLEMENTATION 229
6.2.2 A NUMERICAL EXAMPLE 230
6.3 A GALERKIN METHOD FOR A BELTRAMI-TYPE EQUATION 232
REFERENCES 237
INDEX 243 |
| any_adam_object | 1 |
| author | Atkinson, Kendall E. 1940- Han, Weimin 1963- |
| author_GND | (DE-588)12286977X (DE-588)121177971 |
| author_facet | Atkinson, Kendall E. 1940- Han, Weimin 1963- |
| author_role | aut aut |
| author_sort | Atkinson, Kendall E. 1940- |
| author_variant | k e a ke kea w h wh |
| building | Verbundindex |
| bvnumber | BV039963602 |
| classification_rvk | SI 850 |
| classification_tum | MAT 337f MAT 652f MAT 410f MAT 335f |
| ctrlnum | (OCoLC)781826006 (DE-599)DNB1017147515 |
| dewey-full | 518.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.6 |
| dewey-search | 518.6 |
| dewey-sort | 3518.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| institution | BVB |
| isbn | 9783642259821 3642259820 9783642259838 |
| language | English |
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| physical | IX, 244 S. Ill., graph. Darst. 235 mm x 155 mm |
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| spelling | Atkinson, Kendall E. 1940- Verfasser (DE-588)12286977X aut Spherical harmonics and approximations on the unit sphere an introduction Kendall Atkinson ; Weimin Han Berlin [u.a.] Springer 2012 IX, 244 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2044 Approximation (DE-588)4002498-2 gnd rswk-swf Einheitssphäre (DE-588)4151316-2 gnd rswk-swf Kugelfunktionen-Methode (DE-588)4487746-8 gnd rswk-swf Kugelfunktionen-Methode (DE-588)4487746-8 s Einheitssphäre (DE-588)4151316-2 s Approximation (DE-588)4002498-2 s DE-604 Han, Weimin 1963- Verfasser (DE-588)121177971 aut Erscheint auch als Online-Ausgabe Spherical Harmonics and Approximations on the Unit Sphere: An Introduction Lecture notes in mathematics 2044 (DE-604)BV000676446 2044 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3911640&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024821286&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Atkinson, Kendall E. 1940- Han, Weimin 1963- Spherical harmonics and approximations on the unit sphere an introduction Lecture notes in mathematics Approximation (DE-588)4002498-2 gnd Einheitssphäre (DE-588)4151316-2 gnd Kugelfunktionen-Methode (DE-588)4487746-8 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4151316-2 (DE-588)4487746-8 |
| title | Spherical harmonics and approximations on the unit sphere an introduction |
| title_auth | Spherical harmonics and approximations on the unit sphere an introduction |
| title_exact_search | Spherical harmonics and approximations on the unit sphere an introduction |
| title_full | Spherical harmonics and approximations on the unit sphere an introduction Kendall Atkinson ; Weimin Han |
| title_fullStr | Spherical harmonics and approximations on the unit sphere an introduction Kendall Atkinson ; Weimin Han |
| title_full_unstemmed | Spherical harmonics and approximations on the unit sphere an introduction Kendall Atkinson ; Weimin Han |
| title_short | Spherical harmonics and approximations on the unit sphere |
| title_sort | spherical harmonics and approximations on the unit sphere an introduction |
| title_sub | an introduction |
| topic | Approximation (DE-588)4002498-2 gnd Einheitssphäre (DE-588)4151316-2 gnd Kugelfunktionen-Methode (DE-588)4487746-8 gnd |
| topic_facet | Approximation Einheitssphäre Kugelfunktionen-Methode |
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