Representation of Lie groups and special functions: 2 Class I representations, special functions, and integral transforms
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Academic Publications
1993
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| Series: | Mathematics and its applications. Soviet series
74 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004231610&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XVIII, 607 S. |
| ISBN: | 0792314921 0792314948 |
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| 049 | |a DE-384 |a DE-91G |a DE-739 |a DE-824 |a DE-29T |a DE-83 | ||
| 100 | 1 | |a Vilenkin, Naum Ja. |d 1920-1991 |e Verfasser |0 (DE-588)127328122 |4 aut | |
| 245 | 1 | 0 | |a Representation of Lie groups and special functions |n 2 |p Class I representations, special functions, and integral transforms |c by N. Ja. Vilenkin and A. U. Klimyk |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Academic Publications |c 1993 | |
| 300 | |a XVIII, 607 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications. Soviet series |v 74 | |
| 490 | 0 | |a Mathematics and its Applications / Soviet Series |v ... | |
| 700 | 1 | |a Klimyk, Anatolij U. |d 1939-2008 |e Verfasser |0 (DE-588)115774580 |4 aut | |
| 773 | 0 | 8 | |w (DE-604)BV005437238 |g 2 |
| 830 | 0 | |a Mathematics and its applications. Soviet series |v 74 |w (DE-604)BV004708148 |9 74 | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-004231610 | |
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| adam_text | Table of Contents
List of the Most Important Notations xvi
Chapter 9:
Special Functions Connected with SO(n) and with
Related Groups
9.1. Groups Related to SO(n) and Corresponding
Homogeneous Spaces 1
9.1.1. The groups SO(n), SOo(n — 1,1) and corresponding spaces . . 1
9.1.2. The Lie algebra of SO0(n 1,1) 4
9.1.3. The groups ISO(n 1) and ISO0(n 2,1) 6
9.1.4. The group SO0 (p, q) and related homogeneous spaces 7
9.1.5. Coordinate systems on S 1 and H 1 10
9.1.6. Coordinate systems on tf 1 and C 1 14
9.1.7. Spherical coordinates on Hg and C« 17
9.1.8. The Laplace operators 19
9.1.9. Invariant measures 22
9.2. Class 1 Representations of SO(n) and of Related Groups 25
9.2.1. The representations Tna of the group SO0(n 1,1) 25
9.2.2. Finite dimensional representations of the group SO(n) 27
9.2.3. Realizations of representations of the groups SO(n) and
SOo(n — 1,1) in spaces of harmonic and Q harmonic
functions 28
9.2.4. The representations TnR of the groups ISO(n 1) and
ISO0(n 2,1) 31
9.2.5. Infinitesimal operators of representations 32
9.2.6. Irreducibility 33
9.2.7. Intertwining operators for the representations Tna
of the group SO0(n 1,1) 36
9.2.8. Unitary representations 39
9.2.9. Representations of the group SOo(p, q) 40
9.2.10. Discrete series representations of the group
SOo(p,q)onHlq 43
9.3. Zonal Spherical Functions of Representations of SO(n)
and of Related Groups 44
9.3.1. The orthonormal basis of the space £2(5 1) and
spherical functions of representations of SO(n) 44
9.3.2. Evaluation of zonal spherical functions 46
9.3.3. Integral representations of special functions 48
9.3.4. The connections with other functions 50
9.3.5. Differential equations and integral representations 51
9.3.6. Analogs of the Rodrigues formula 55
vii
viii Table of Contents
9.3.7. Generating functions 59
9.3.8. Orthogonality relations for Gegenbauer polynomials 62
9.4. Associated Spherical Functions and Their Properties 65
9.4.1. The matrices of the representations Tni, Tna and T R 65
9.4.2. Evaluation of associated spherical functions 67
9.4.3. Addition theorems 71
9.4.4. Generalizations of the addition theorems 73
9.4.5. Product formulas 76
9.4.6. Generalized product theorems 80
9.4.7. The Banach algebras 84
9.4.8. Raising and lowering operators 86
9.4.9. Relations between spherical functions for the groups
of different dimensionalities 89
9.4.10. Asymptotic properties of spherical functions of
the group SO0(n 1,1) 92
9.4.11. Dougall s formula for Gegenbauer polynomials 94
9.4.12. Functional relations for Chebyshev polynomials 99
9.5. Matrix Elements of Class 1 Representations and Generalization
of Gegenbauer Polynomials, Legendre and Bessel Functions . . 100
9.5.1. Matrix elements of class 1 representations 100
9.5.2. Evaluation of the functions P££,(cos 0) 103
9.5.3. Evaluation of the functions V kmj;(cosh8) and Jtmj(x) • • • • 1°5
9.5.4. Expansion of P£^ (cos 0) into Fourier series 106
9.5.5. Symmetry properties of the functions ?Pi** (a:), Pkmj(x)
and Jt»m (z) 108
9.5.6. The functions ^*ra(x), P m(*) and Jt mm(x) 109
9.5.7. The expression for P^(0) in terms of Wilson polynomials . . Ill
9.5.8. Boundedness properties 114
9.5.9. The orthogonality relation for P^7(x) 117
9.5.10. Addition theorems and product formulas 118
9.5.11. Generating functions 120
9.5.12. Characters of Tnl and the functions P^ (i) 123
9.5.13. The functions J£ as the limit of P£l
andofVj i 124
9.5.14. Infinitesimal operators of the representations 124
9.6. The Groups O(oo), /O(co), the Infinite Dimensional Laplace
Operator and Hermite Polynomials 127
9.6.1. The Gauss measure. The group O(oo) 127
9.6.2. The projective limit of spheres 129
9.6.3. The infinite dimensional Laplace operator 133
9.6.4. The Hilbert space £c($) 136
9.6.5. Irreducible representations of O(oo) 138
9.6.6. Matrix elements of the representations T 139
Table of Contents ix
9.6.7. Hermite polynomials as the limit of Gegenbauer polynomials . 141
9.6.8. Properties of Hermite polynomials 143
9.6.9. The Wiener transform 148
9.6.10. Representations of the group JO(oo) 149
9.6.11. Matrix elements of the representations Tc 151
9.6.12. Other properties of Hermite polynomials 153
Chapter 10:
Representations of Groups, Related to SO{ n 1), in Non
Canonical Bases, Special Functions, and Integral Transforms
10.1. Decompositions of Quasi Regular Representations and
Integral Transforms 159
10.1.1. Decomposition of the quasi regular representation of the
group IS0{n 1) 159
10.1.2. Decomposition of the quasi regular representation of the
group 500(n 1,1) in £2(C; ]) 160
10.1.3. The Gel fand Graev transform 162
10.1.4. Decomposition of the quasi regular representation Q^T1
of the group SO0(n 1,1) 166
10.1.5. Restrictions of the representation Tn r of SO0(n 1,1)
onto subgroups 169
10.1.6. Decomposition of the quasi regular representation
of SO0(p,q) 171
10.2. The Funk Hecke Theorem and its Analogs. Continuous Bases
and Integral Transforms 173
10.2.1. The Funk Hecke theorem 173
10.2.2. The analog of the Funk Hecke theorem for
D harmonic functions 176
10.2.3. The analog of the Funk Hecke theorem for the
group ISO(n 1) 180
10.2.4. The Bochner theorem and its corollaries 181
10.2.5. The continuous basis in the space ^(R 1) 184
10.2.6. The Fourier Bessel transform 186
10.2.7. The continuous basis in the space ^(ff 1) 189
10.2.8. The generalized Fock Mehler transform 190
10.3. The Poisson Transforms and Special Functions 193
10.3.1. The Poisson transforms 193
10.3.2. The Poisson transforms of the bases of the space 58 and
integral transforms on the hyperboloid 198
10.3.3. Expansions of Poisson kernels 201
10.3.4. Poisson transforms and addition theorems for
special functions 203
x Table of Contents
10.4. Spherical Functions in Cylindrical Coordinates and
Special Functions 207
10.4.1. Harmonic polynomials in bispherical coordinates 207
10.4.2. Associated Kpq spherical functions on the sphere 210
10.4.3. Associated Kpq sph. cica functions on the hyperboloid H™ . . 212
10.4.4. Differential equations and integral representations
for the functions Y?f(0) and Y™ (6) 214
10.4.5. Addition and product theorems 217
10.4.6. The Poisson transform of the basis of 33 , corresponding
to the cylindrical section of the cone 220
10.4.7. Integral transforms on H ~ related to the subgroup
SO(p) x SO0(q 1,1) 221
10.5. The Tree Method 223
10.5.1. The tree method and polyspherical coordinates 223
10.5.2. The Laplace operator and the invariant measure on 5 1 . . 226
10.5.3. Trees and orthonormal bases in ^(S 1) 227
10.5.4. Relations between orthogonal bases 230
10.5.5. T coefficients for the transplantation of an edge 232
10.5.6. T coefficients for the transplantation of an edge
(degenerate cases) 236
10.5.7. The tree method and invariant harmonic polynomials .... 237
10.5.8. The tree method and coordinates on the hyperboloid H™ . . 247
10.6. Transition Coefficients for Bases on the Cone and
Special Functions 248
10.6.1. Maijer G functions ^ 248
10.6.2. Transition coefficients for the bases {Hj)^}
and {E^2} ^ 252
10.6.3. Transition coefficients for the bases {^™m}
and {§» } ^ 254
10.6.4. Transition coefficients for the bases {H^J}
and{H^} 255
10.6.5. The coefficients Cna{m,R,mi) and Maijer G functions ... 257
10.6.6. The coefficients D^Xv, R, m) and Maijer G functions .... 262
10.6.7. The coefficients E (m,u,mi) and associated Legendre
functions 264
10.7. Representations of the Group ISO0(n 2,1) and
Special Functions 265
10.7.1. Irreducible representations of the group ISOo(n — 2,1) ... 265
10.7.2. Representations of the group ISOo(n — 2,1)
by integral operators 266
10.7.3. Addition and product theorems for Macdonald functions . . . 269
10.7.4. Evaluation of the kernel Km(a,cr ;z) for the general case . . . 271
Table of Contents xi
10.7.5. Class 1 representations of the group HSO(n) and the
function jFi 272
Chapter 11:
Special Functions Connected with the Groups U{ n),
f/(n l,l) and IU{n )
11.1. The groups U(n), U(n 1,1), IU(n 1) and Related
Homogeneous Spaces 278
11.1.1. The groups U(n), U(n — 1,1) and related homogeneous
spaces 278
11.1.2. The Lie algebras of the groups U(n) and U(n 1,1) .... 280
11.1.3. Subgroups of U(n 1,1) 284
11.1.4. The group IU(n 1) 285
11.1.5. Coordinate systems on S£~* aX1^ H^ 1 285
11.1.6. Coordinate systems on C£ 1 288
11.1.7. Laplace operators 289
11.1.8. Invariant measures 291
11.2. Class 1 Representations of the Groups U(n), U(n 1,1)
and IU(n 1) 292
11.2.1. Harmonic polynomials on C 292
11.2.2. The representations Tn of U(n) 295
11.2.3. Spectral decompositions of the representations Tnlt 299
11.2.4. The representations Tnak of U(n 1,1) 301
11.2.5. Other realizations of Tn and Tn rk 304
11.2.6. The representations TnkR of the group JU(n 1) 306
11.3. Zonal and Associated Spherical Functions 307
11.3.1. The orthonormal basis in £2(S£ 1) 307
11.3.2. Evaluation of zonal spherical functions 308
11.3.3. Asymptotic properties of zonal spherical functions 312
11.3.4. Associated spherical functions 313
11.3.5. Evaluation of associated spherical functions 316
11.3.6. Special cases 321
11.3.7. Symmetry relations 321
11.3.8. Raising and lowering operators 322
11.3.9. Relations between spherical functions for groups of
different dimensionalities 323
11.3.10. Relations for zonal spherical functions 325
11.3.11. The connection of spherical functions of the groups
U{n) and IU(n — 1) with spherical functions of
the groups 50(2n) and ISO(2n 1) 326
11.3.12. JfM spherical functions on the complex sphere 327
xii Table of Contents
11.4. Functional Relations for Jacobi Polynomials and Functions
and for Bessel Functions 332
11.4.1. Integral representations 332
11.4.2. Addition theorems for Jacobi polynomials 333
11.4.3. Addition theorems for Jacobi and Bessel functions 336
11.4.4. Expansions in zonal spherical functions of the group
U(n) 339
11.4.5. Product formulas for Jacobi and Laguerre polynomials .... 340
11.4.6. Product formulas for Jacobi and Bessel functions 348
11.4.7. Generating functions 353
11.4.8. Expansion in zonal spherical functions of the group
U(n 1,1) 356
11.5. Orthogonal Polynomials on the Disk 359
11.5.1. The definition 359
11.5.2. Integral representation and differential equations 360
11.5.3. The addition theorem and the product formula 361
11.5.4. Special functions on the exterior of the disk 363
11.5.5. Integral representation and differential equations
for/ff t 365
11.6. Matrix Elements of Class 1 Representations 365
11.6.1. Integral representation for matrix elements of the
operators Tnak(ffUi(O) 365
11.6.2. Infinitesimal operators of the representations Tnak 366
11.6.3. Irreducibility of the representations Tnak for
non integral a 368
11.6.4. Infinitesimal operators of the representations T
of the group U(n) 370
11.6.5. Infinitesimal operators of the representations TnkR
of the group JU{n 1) 370
11.6.6. Matrix elements of finite dimensional representations
of the group GL(n,C) # 372
11.6.7. Matrix elements of the representations Tnte
of the group U(n) 374
11.6.8. Matrix elements of the representations Tnak
of the group U(n 1,1) 376
11.6.9. Matrix elements of the representations TnkR
of the group JU(n 1) 379
11.6.10. Symmetry relations 381
11.6.11. Relations between matrix elements of representations
for groups of different dimensionalities 384
11.6.12. The functions *n/0(Sn_i(fl)) 386
11.6.13. Special cases of matrix elements of the operators
TB Wi(0)) 387
Table of Contents xiii
11.6.14. Special cases of matrix elements of the operators
T ^g n At)) 388
11.6.15. Special cases of matrix elements of the operators
TnkR(gr) 389
11.6.16. Matrix elements of the operators Tn ( 7n_i (f)) 390
11.7. Zonal and Associated Spherical Functions of the Groups
Sp(n) and Sp(n 1,1) 392
11.7.1. The group Sp(n) and spheres in the quaternion space .... 392
11.7.2. The group Sp(n — 1,1), the hyperboloid and the cone
in the quaternion space 395
11.7.3. Invariant measures and Laplace operators 397
11.7.4. Spherical functions of representations of the group Sp(n) . . . 398
11.7.5. The representations Tn of the group Sp(n 1,1) 402
11.7.6. Spherical functions of the representations Tna 404
11.7.7. Expansion in zonal spherical functions of the group
Sp(n 1,1) 407
11.7.8. Decomposition of the quasi regular representation
of the group Sp(n — 1,1) 408
Chapter 12:
Representations of the Heisenberg Group and
Special Functions
12.1. Representations of the Heisenberg Group, Hermite and
Laguerre Polynomials 410
12.1.1. The Heisenberg group 410
12.1.2. The Lie algebra of the Heisenberg group 411
12.1.3. The exponential mapping 412
12.1.4. Unitary representations 413
12.1.5. The orthonormal basis 414
12.1.6. Matrix elements of the representations Rx 415
12.1.7. The Fock realization of the representations Rx 418
12.1.8. Addition formulas for Laguerre polynomials 420
12.1.9. Relations between Laguerre and Hermite polynomials .... 421
12.1.10. Orthogonal polynomials on C 423
12.1.11. Decomposition of the regular representation of
the Heisenberg group 424
12.2. The Group of Automorphisms of the Heisenberg Group and the
Weyl Representation 427
12.2.1. The group of automorphisms for N 427
12.2.2. The group of automorphisms for Nn 428
12.2.3. The group Sp{n,R) x Nn 429
12.2A. The Weyl representation of the group 51(2, R) 430
12.2.5. The integral form of the Weyl representation 434
xiv Table of Contents
12.2.6. Infinitesimal operators of the Weyl representation 436
12.3. The Weyl Representation of Sp(n,R) and Bases of
the Carrier Space 437
12.3.1. The Weyl representation of the group Sp(n, R) 437
12.3.2. New bases of £2(R ) 440
12.3.3. The restriction of the Weyl representation
onto SO(n) x SL(2, R) 442
12.3.4. Transition coefficients for the bases 443
12.3.5. The transition coefficients and Clebsch Gordan coefficients . . 446
12.3.6. The Weyl transform and the Heisenberg group 447
12.4. Representations of the Group U(n)Nn and Orthogonal
Polynomials 451
12.4.1. Representations of the group U(n)N+n 451
12.4.2. Matrix elements of the representations T 453
12.4.3. The addition theorem and the product formula for
Laguerre polynomials 454
12.4.4. Relations between Laguerre and Jacobi polynomials 455
12.5. The Compact Realization of the Representation of the
Heisenberg Group and Orthogonal Polynomials 457
12.5.1. The Weil Brezin operator and a new realization of the
representation R2* 457
12.5.2. The Poisson summation formula 460
12.5.3. Summation formulas for Hermite polynomials 461
12.5.4. The expansion in the functions sine 463
12.5.5. Schempp s summation formula 465
12.6. Harmonic Polynomials on the Heisenberg Group 467
12.6.1. The operator Ly 467
12.6.2. The operator Ly and the group U{n + 1,1) 470
12.6.3. Harmonic polynomials on H(n, R) 474
12.6.4. The polynomials C{na 0)(() 477
12.6.5. The polynomials Cn (C) and representations
of the group SO(n) 478
12.6.6. Orthogonality relations for ^ (C) 481
12.6.7. The addition theorem for C{naJ)(C) 483
Chapter 13:
Representations of the Discrete Groups and Special
Functions of Discrete Argument
13.1. Representations of the Symmetric Group, Krawtchouk and
Hahn Polynomials 485
13.1.1. Introduction 485
13.1.2. Irreducible representations of the symmetric group 485
Table of Contents xv
13.1.3. Zonal spherical functions on the space Xmn 487
13.1.4. Semidirect products of symmetric groups and
their representations 488
13.1.5. Homogeneous graphs and special functions 490
13.1.6. Addition theorems for Krawtchouk and Hahn polynomials . . 494
13.1.7. Functions, invariant with respect to Sn x Sm x Sk,
and orthogonal polynomials 499
13.2. Groups of Linear Transformations over Finite Fields and
^ Analogs of Special Functions 508
13.2.1. Linear spaces over finite fields and
Chevalley groups 508
13.2.2. Basic hypergeometric functions and g analogs
of orthogonal polynomials 510
13.2.3. Irreducible representations of Chevalley groups and
g analogs of orthogonal polynomials 518
13.2.4. Functional relations for g analogs of
orthogonal polynomials 524
13.2.5. Functions, invariant with respect to subgroups of block
triangular matrices and ^ analogs of
orthogonal polynomials in two variables 530
13.3. Special Functions, Related to Locally Compact Totally
Disconnected Fields, and Group Representations 540
13.3.1. Characters of locally compact abelian groups 540
13.3.2. The field of p adic numbers and other totally disconnected
locally compact fields 544
13.3.3. The Gamma and the Beta functions, related to a field K . . 549
13.3.4. Quadratic extensions of a field K 557
13.3.5. The motion group of the plane K (y/r ) and its
representations 561
13.3.6. Evaluation of the ^} adic Bessel functions 564
13.3.7. Irreducible representations of the group of unimodular
matrices of the second order over the field K 568
13.3.8. Bessel functions of the second kind on K 572
13.3.9. Discrete series of irreducible unitary representations of
the group SL(2, K) 574
Bibliography 577
Subject Index 604
|
| any_adam_object | 1 |
| author | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_GND | (DE-588)127328122 (DE-588)115774580 |
| author_facet | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 |
| author_role | aut aut |
| author_sort | Vilenkin, Naum Ja. 1920-1991 |
| author_variant | n j v nj njv a u k au auk |
| building | Verbundindex |
| bvnumber | BV006621278 |
| ctrlnum | (OCoLC)312016209 (DE-599)BVBBV006621278 |
| format | Book |
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| id | DE-604.BV006621278 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T08:51:52Z |
| institution | BVB |
| isbn | 0792314921 0792314948 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-004231610 |
| oclc_num | 312016209 |
| open_access_boolean | |
| owner | DE-384 DE-91G DE-BY-TUM DE-739 DE-824 DE-29T DE-83 |
| owner_facet | DE-384 DE-91G DE-BY-TUM DE-739 DE-824 DE-29T DE-83 |
| physical | XVIII, 607 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Kluwer Academic Publications |
| record_format | marc |
| series | Mathematics and its applications. Soviet series |
| series2 | Mathematics and its applications. Soviet series Mathematics and its Applications / Soviet Series |
| spellingShingle | Vilenkin, Naum Ja. 1920-1991 Klimyk, Anatolij U. 1939-2008 Representation of Lie groups and special functions Mathematics and its applications. Soviet series |
| title | Representation of Lie groups and special functions |
| title_auth | Representation of Lie groups and special functions |
| title_exact_search | Representation of Lie groups and special functions |
| title_full | Representation of Lie groups and special functions 2 Class I representations, special functions, and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_fullStr | Representation of Lie groups and special functions 2 Class I representations, special functions, and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_full_unstemmed | Representation of Lie groups and special functions 2 Class I representations, special functions, and integral transforms by N. Ja. Vilenkin and A. U. Klimyk |
| title_short | Representation of Lie groups and special functions |
| title_sort | representation of lie groups and special functions class i representations special functions and integral transforms |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004231610&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005437238 (DE-604)BV004708148 |
| work_keys_str_mv | AT vilenkinnaumja representationofliegroupsandspecialfunctions2 AT klimykanatoliju representationofliegroupsandspecialfunctions2 |
Table of Contents
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Branch Library Mathematics & Informatics
| Call Number: |
0102 MAt 173f 2001 A 32780-2
Floor plan |
|---|---|
| Copy 1 | Available for loan On Shelf |