Gel'fand I. M., Graev M. I., Vilenkin N. Ya. — Generalized Functions. Volume 5. Integral Geometry and Representation Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Gel'fand I. M., Graev M. I., Vilenkin N. Ya. — Generalized Functions. Volume 5. Integral Geometry and Representation Theory

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Название: Generalized Functions. Volume 5. Integral Geometry and Representation Theory

Авторы: Gel'fand I. M., Graev M. I., Vilenkin N. Ya.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1966

Количество страниц: 467

Добавлена в каталог: 19.12.2012

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Предметный указатель

"Real axis" $X_{0}$       208 231
"Upper half plane" in four complex dimensions      310
$D_{s}$       396
$d_{\chi}$       141
$D_{\chi}$ for the real unimodular group      391
$E_{s}$       411
$E{\chi}$       150
$F_{s}$       411
$F_{\chi}$       151
$T_{\chi}(g)$       see "Representations of G"
$\mathfrak{H}$       239
$\mathfrak{H}_{K}$       239
Absolute      275
Absolute convergence of integrals on G      215
Affine complex plane      336
Affine complex space      115
Affine real space      1
Analogs of $S^{\beta}_{\alpha}$ on the $z_{1}z_{4} — z_{2}z_{3} = 1$ surface      112
Analogs of Plancherel's theorem      see "Plancherel's theorem"
Analogs of the real axis      208 231 269
Analytic continuation in the coordinates      308
Analytic representations      396
Analytic representations, invariant bilinear functional      409
Analytic representations, invariant Hermitian functionals      429
Analytic representations, operator $A_{+}$ and $A_{-}$       399
Analytic representations, reducibility on $D_{s}/E_{s}$ and $F_{-s}$       420
Analytic representations, representations of the discrete series      400
Analyticity in $\chi$ of the Fourier transform on G      231
Areas of unbounded hypersurfaces      32
Asymptotic behavior, $K(z_{1}, z_{2}; \chi)$       325
Asymptotic behavior, $\Phi(u, v; u^{'}, v^{'})$       260
Asymptotic behavior, Fourier transforms of characteristic functions      19
Asymptotic behavior, functions in $D_{\chi}$ (complex)      143
Asymptotic behavior, functions in $D_{\chi}$ (real)      392
Bilinear functional      157
Bochner's theorem      204
Bounded support on G      213
Class I      354
Closed operator      149
Closest relatives      155
Commutation relations of Lie operators      252
Commutator      252
Complex affine plane      336
Complex projective line      336
Conditionally invariant bilinear forms      175
Conditionally invariant bilinear forms for the real unimodular group      411
Cone as a homogeneous space      337
Continuity in $D_{\chi}$ of $F(\chi)$       207
Continuity in $D_{\chi}$ of $K(z_{1}, z_{2}; \chi)$       223
Continuity on G      213
Convergence in $D_{\chi}$       142
Convolution of functions on G      217
Decomposition in irreducible representations      243
Decomposition in irreducible representations with respect to compact subgroup of stability      356ff
Decomposition in irreducible representations with respect to noncompact subgroup of stability      364ff
Decomposition of representations of the Lorentz group      242 349ff
Degenerate invariant bilinear forms      174
Differentiability on G      205
Differential form, $\omega$ associated with a line on such a surface      99
Differential form, $\sigma(z)$ , associated with a quadratic surface in four complex dimensions      98
Differential operator      see "Lie operator"
Dilation of the complex plane      159
Direct integral      243
Discrete series      400 431
Dual representation      147
Dual representation for the real unimodular group      394
Element of volume on G      214
Element of volume on G on imaginary Lobachevskian space      286
Element of volume on G on the Lobachevskian space      286
Element of volume on G on the n-sphere      285
Elliptic space      274
Equivalence of representations      151 182
Equivalence of representations at integer points      153
Equivalence of representations for the real unimodular group      416
Extension of $T_{\chi}$ representations      148 395
Finite-dimensional representations of G on spaces of polynomials      140
Fourier transform on G      205
Fourier transform on horospheres of the first kind      366
Fourier transform on isotropic lines      369
Fourier transform on Lobachevskian space      359
Fourier transform on the isotropic cone      355
Fourier transform on the line      202
Generalized Beta function      51 132
Generalized hypergeometric function      51 131
Group of motions      331
Harmonic measure      289
Hermitian functional      189
Hilbert space, unitary representations on      198
Homogeneous coordinates of lines on a quadratic surface in four complex variables      96
Homogeneous space      331
Horosphere      280
Horosphere, first and second kinds      283
Horosphere, group theoretical definition      345
Horosphere, group theoretical definition, for the Lorentz group      333
Hyperbolic rotation      275
Hyperbolic space      275
Hyperboloid, Radon transform of characteristic function, n dimensions      63
Hyperboloid, Radon transform of characteristic function, three dimensions      42
Imaginary Lobachevskian motions      278
Imaginary Lobachevskian space      277
Infinitely differentiable function      205
Infinitesimal parallel displacement operators      84
Integer points      150
Integer points for the real unimodular group      396
Integral along a generator of a quadratic surface in four complex dimensions      100
Integral geometry, problem of      75
Integral transform in imaginary Lobachevskian space      304
Integral transform in imaginary Lobachevskian space, inversion formula      323
Integral transform in ordinary Lobachevskian space      290
Integral transform in ordinary Lobachevskian space, inversion formula      301
Internal operator      112
Intertwining operator      152
Intertwining operator, expressions for      181
Intertwining operator, expressions for, the real unimodular group      418
Invariant bilinear functional      157
Invariant bilinear functional, degeneracy      174
Invariant bilinear functional, existence (complex case)      158
Invariant bilinear functional, existence (real case)      424
Invariant bilinear functional, theorems concerning      167 173
Invariant Hermitian functional      189
Invariant integration on G      214
Invariant integration on G on a space of constant curvature      285
Invariant measure on G      216
Invariant subspaces      see " $E_{\chi}$ " " " "
Inverse Fourier transform      203
Inversion of a function      143 392
Inversion of the complex plane      159
Irreducibility, general      150

Irreducibility, operator      148
Irreducibility, subspace      149
Isotropic cone      337
Isotropic line      278
Kernel of Fourier transform operator      218
Kernel of Fourier transform operator, geometric interpretation of      220
Kronecker product      385
Laplacian operator for G      253
Left and right derivative operators      249
Lie derivatives or operators      248
Lie derivatives or operators, commutation relations      252
Lie derivatives or operators, Fourier transforms      255
Line complexes      77
Line generators of a quadratic surface in four complex variables      94
Little group      334
Lobachevskian motions      138 275
Lobachevskian space      275
Lobachevskian space, imaginary      277
Lorentz group      136
Mellin transform      113 221 350
Mellin transform, inverse      351
Mellin transform, Plancherel's theorem      351
Moments of a function over G      265
Motion of a space      274
Multiplier of a representation      332
Negative curvature      275
Operator irreducibility      149
Paley — Wiener theorem for the Fourier transform on G      357
Paley — Wiener theorem for the Fourier transform on the line      277
Paley — Wiener theorem for the Radon transform      15 123
Parallel isotropic lines      324
Parallel isotropic lines, distance between      326
Parallel translation of the complex plane      159
Partial equivalence of representations      184
Partial equivalence of representations for the real unimodular group      418
Partitioning method      26
Plancherel's theorem for the Fourier transform, G      237
Plancherel's theorem for the Fourier transform, imaginary Lobachevskian space      381
Plancherel's theorem for the Fourier transform, isotropic cone      356
Plancherel's theorem for the Fourier transform, line      203
Plancherel's theorem for the Fourier transform, Lobachevskian space      362
Plancherel's theorem for the Fourier transform, point pairs on the complex projective line      389
Plancherel's theorem for the Mellin transform      351
Plancherel's theorem for the radon transform, complex      121
Plancherel's theorem for the radon transform, real      12
Planes of the 1st, 2nd, and 3rd kinds in imaginary Lobachevskian space      384
Pluecker coordinates of a line      77
Point pairs on the complex projective line      337
Positive cone      135
Positive cone, interior of      310
Positive curvature      274
Principal series      156 191
Principal series for the real unimodular group      400 428
Proper Lorentz group      136
Quotient space      333
Radon transform and Fourier transform      4
Radon transform of convolutions      7 117
Radon transform of derivatives      6 117
Radon transform of entire analytic functions      129
Radon transform of generalized functions      23 124
Radon transform of generalized functions and unessential functions      24 125
Radon transform of homogeneous functions      43
Radon transform of test functions      2 116
Radon transform, homogeneity of      2 116
Radon transform, interpretation of      3
Radon transform, inverse      11 121
Radon transform, linearity of      5
Radon transform, Plancherel's theorem for      12 121
Rapidly decreasing function on a quadratic surface      112
Rapidly decreasing function on a quadratic surface on G      204 213
Realizations of $D_{\chi}$ for the real unimodular group      392
Realizations of $D_{\chi}$ on the line $z_{2} = 1$       142
Realizations of $D_{\chi}$ on the sphere $|z _{1}|^{2} + |z_{2}|^{2} = 1$       144
Regular representations of the Lorentz group, left      246
Regular representations of the Lorentz group, right      242
Regularizing integrals by analytic continuation      308
Related representations      156
Representation of a group      139
Representation of a group, dual      147
Representation of a group, unitary      140
Representations of G on $D_{\chi}$       144
Representations of G on $E_{\chi}$       150
Representations of G on $F_{\chi}$       151
Representations of G, closest relatives      156
Representations of G, equivalence      151
Representations of G, intertwining operator      152
Representations of G, principal series      156
Representations of G, supplementary series      157
Representations of G, unitary      156
Riemann surface of inequivalent representations      153
Riemann surface of inequivalent representations for the real unimodular group      398
Riemannian space      274
Right derivative      249
S (for G)      254
Scalar product      189
Scalar product for the real unimodular group      425
Space of constant curvature      274
Space of functions with vanishing moments, $F_{\chi}$       151
Space of homogeneous functions, $D_{\chi}$       141
Space of homogeneous polynomials, $E_{\chi}$       150
Spectrum of the $T_{s}$ , operators of the discrete series      438
Sphere in four complex variables      94
Subgroups of stability      334
Subgroups of stability, table      340
Subspace irreducibility      148
Subspace irreducibility for the real unimodular group      438
Subspace irreducibility of unitary representations      200
Summable function on G      205
Supplementary series      157 193
Supplementary series for the real unimodular group      400 428
Symmetry properties of Fourier transform on G      240
Table of Radon transforms      71
Table of subgroups of stability      340
Tangent space to G      247
Tangent vector      247
Trace of $T_{\chi}$ (generalized)      227
Trace of Fourier transform      226
Transitive      138 275
Unessential functions      24 125
Unimodular      133
Unimodular group, complex      133
Unimodular group, real      390ff
Unitary representations      140
Unitary representations on Hilbert space      270
Unitary representations, subspace irreducibility of      200
Unitary representations, subspace irreducibility of, for the real unimodular group      399
Universal covering space of the space of representations      153
Universal covering space of the space of representations for the real unimodular group      398
Weight of a representation      145

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