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Gelfand I.M., Gindikin S.G., Graev M.I. — Selected Topics in Integral Geometry
Gelfand I.M., Gindikin S.G., Graev M.I. — Selected Topics in Integral Geometry



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Название: Selected Topics in Integral Geometry

Авторы: Gelfand I.M., Gindikin S.G., Graev M.I.

Аннотация:

The miracle of integral geometry is that it is often possible to recover a function on a manifold just from the knowledge of its integrals over certain submanifolds. The founding example is the Radon transform, introduced at the beginning of the 20th century. Since then, many other transforms were found, and the general theory was developed. Moreover, many important practical applications were discovered. The best known, but by no means the only one, being to medical tomography.

This book is a general introduction to integral geometry, the first from this point of view for almost four decades. The authors, all leading experts in the field, represent one of the most influential schools in integral geometry. The book presents in detail basic examples of integral geometry problems, such as the Radon transform on the plane and in space, the John transform, the Minkowski-Funk transform, integral geometry on the hyperbolic plane and in the hyperbolic space, the horospherical transform and its relation to representations of $SL(2,\mathbb C)$, integral geometry on quadrics, etc. The study of these examples allows the authors to explain important general topics of integral geometry, such as the Cavalieri conditions, local and nonlocal inversion formulas, and overdetermined problems in integral geometry. Many of the results in the book were obtained by the authors in the course of their career-long work in integral geometry.



Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 190

Добавлена в каталог: 21.01.2014

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Предметный указатель
$sl(2,\mathbb{C})$      111
$SL(2,\mathbb{C})$, as a homogeneous space      111
$SL(2,\mathbb{C})$, as a hyperboloid      111
$SL(2,\mathbb{C})$, Fourier transform      131 141
$SL(2,\mathbb{C})$, Fourier transform, inversion formula      134 142
$SL(2,\mathbb{C})$, Fourier transform, Plancherel formula      135 142
$SL(2,\mathbb{C})$, Fourier transform, representation theory      137 138 143
$SL(2,\mathbb{C})$, Fourier transform, symmetry relation      133 142
$SL(2,\mathbb{C})$, Fourier transform, transform related to paraboloids      141
$SL(2,\mathbb{C})$, horosphere      128 129
$SL(2,\mathbb{C})$, horospherical function      129
$SL(2,\mathbb{C})$, horospherical function on the manifold of triples      130
$SL(2,\mathbb{C})$, horospherical function, expression      130
$SL(2,\mathbb{C})$, horospherical function, zonal      129
$SL(2,\mathbb{C})$, horospherical transform      119 120
$SL(2,\mathbb{C})$, horospherical transform, inversion formula      124
$SL(2,\mathbb{C})$, horospherical transform, symmetry relation      121
$SL(2,\mathbb{C})$, integral transform, line complex associated with horospheres      120
$SL(2,\mathbb{C})$, integral transform, related to paraboloids      126
$SL(2,\mathbb{C})$, integral transform, related to paraboloids, and an integral transform on the hyperbolic space      127
$SL(2,\mathbb{C})$, integral transform, related to paraboloids: inversion formula      127
$SL(2,\mathbb{C})$, integral transform, related to the line complex associated with horospheres: inversion formula      123
$SL(2,\mathbb{C})$, Laplace — Beltrami operator      128
$SL(2,\mathbb{C})$, manifold of horospheres      113
$SL(2,\mathbb{C})$, manifold of horospheres, associated line complex      117
$SL(2,\mathbb{C})$, manifold of horospheres, embedding in projective space      116
$SL(2,\mathbb{C})$, manifold of paraboloids      118
$SL(2,\mathbb{C})$, orbits and sections      112
$SL(2,\mathbb{C})$, paraboloid      129
$SL(2,\mathbb{C})$, zonal horospherical function      129
Abel transform      2
Admissible line complex      71 73
Admissible line complex, description      76
Admissible line complex, geometric structure      75
Admissible submanifolds of hyperplane sections in $\mathcal{L}^n$      147
Asgeirsson for the hyperbolic space      101
Asgeirsson for three-dimensional space      9
Asgeirsson on the Euclidean plane      3
Asgeirsson relations for the hyperbolic plane      85
Back front      13
Bivector      59
Bivector dual      64
Cavalieri's conditions      14
Electron microscopy      18
Forward front of the wave      13
Fourier integral on $\mathcal{L}^2$      88
Fourier integral on $\mathcal{L}^3$      103
Fourier transform in the hyperbolic space      104
Fourier transform in the hyperbolic space, inversion formula      104
Fourier transform in the hyperbolic space, Plancherel formula      105
Fourier transform in the hyperbolic space, relation to the horospherical transform      104
Fourier transform in the hyperbolic space, representation theory      106
Fourier transform in the hyperbolic space, symmetry relation      105
Fourier transform on $\mathbb{R}^2      $86
Fourier transform on S(H)      49
Fourier transform on the hyperbolic plane      88
Fourier transform on the hyperbolic plane, inversion formula      89
Fourier transform on the hyperbolic plane, Plancherel formula      91
Fourier transform on the hyperbolic plane, relation to representation theory      92
Fourier transform on the hyperbolic plane, relation to the horocycle transform      89 92
Fourier transform on the hyperbolic plane, symmetry relation      90
Fourier transform on the Schwartz space      49
Homogeneity condition      14
Horocycle      79
Horocycle functions      88
Horocycle functions, zonal      86
Horocycle transform      83
Horocycle transform, group-theoretic meaning      92
Horocycle transform, inversion formula      85 86
Horocycle waves      107
Horosphere      97 103 128 129
Horosphere oriented distance      98
Horosphere, center      98
Horosphere, parallel      98
Horospherical function      103
Horospherical function on $SL(2,\mathbb{C})$      129
Horospherical function zonal      103
Horospherical transform      98 103
Horospherical transform on $SL(2,\mathbb{R})$      128
Horospherical transform on $\mathcal{L}^3$, inversion formula      100 101
Horospherical transform, inversion formula      125
Horospherical transform, inversion formula, on $\mathcal{L}^n$      103
Horospherical waves      109
Huygens principle      13
Hyperbolic plane      79 80
Hyperbolic plane, area element      80 81
Hyperbolic plane, geodesics      62
Hyperbolic plane, horocycle      79 81
Hyperbolic plane, horocycle, center      81
Hyperbolic plane, horocycle, oriented distance      82
Hyperbolic plane, horocycle, parallel      81
Hyperbolic plane, invariant distance      80 81
Hyperbolic plane, lines      82
Hyperbolic plane, motions      79 80
Hyperbolic plane, Poincare model      79
Hyperbolic plane, Riemannian metric      80
Hyperbolic space      96
Hyperbolic space of an arbitrary dimension      102
Integral geometry for line complexes in $\mathbb{C}^3$      71
Integral geometry on the torus      21
Integral transform for spheres in $E^n$      157
Integral transform for spheres in $E^n$, inversion formula      159
Integral transform for spheres in $E^n$, operator $\kappa$      157
Integral transform of differential forms      53
Integral transform, related to completely geodesic surfaces in $\mathcal{L}^3$      102
Integral transform, related to completely geodesic surfaces in $\mathcal{L}^3$: inversion formula      102
Integral transform, related to geodesics on the hyperbolic plane      93
Integral transform, related to geodesics on the hyperbolic plane: inversion formula      96
Integral transform, related to geodesics on the hyperbolic plane: relation to the projective Radon transform      96
Integral transform, related to hyperplane sections of $\mathcal{L}^n$      146
Integral transform, related to lines on the hyperbolic plane      93
Inversion formula for a complex of lines intersecting a curve in $\mathbb{C}^3$      71
Inversion formula for the Fourier transform in the hyperbolic space      104
Inversion formula for the horocyde transform      85 86
Inversion formula for the horospherical transform on $\mathcal{L}^3$      100 101
Inversion formula for the horospherical transform on $\mathcal{L}^n$      103
Inversion formula for the Radon transform for the complex affine space      40
Inversion formula for the Radon transform for the projective space      32
Inversion formula for the Radon transform for the projective space of arbitrary dimension      38
Inversion formula for the Radon transform in three-dimensional space      8
Inversion formula for the Radon transform of arbitrary dimension      11
Inversion formula for the Radon transform on the affine plane      10
Inversion formula for the Radon transform on the Euclidean plane      2 3
Inversion formula for the Radon transform, even-dimensional case      11
Inversion formula for the Radon transform, local      11
Inversion formula for the Radon transform, nonlocal      11
Inversion formula for the Radon transform, odd-dimensional case      11
Inversion formula on $\mathcal{L}^n$      152
Inversion formula, local      38
Inversion formula, nonlocal      38
Isotropic cone      75
John transform      43 45 68
John transform in $\mathbb{C}^3$      67
John transform in $\mathbb{C}^3$, image      68
John transform in $\mathbb{C}^3$, inversion formula      68
John transform in $\mathbb{P}^3$      60 63
John transform in $\mathbb{P}^3$, image      63
John transform in $\mathbb{P}^3$, the affine John transform      61
John transform in the affine space      45
John transform in the affine space, image      46
John transform of 1-forms on $\mathbb{R}^3$      58
John transform of 1-forms on $\mathbb{R}^3$, image      58
John transform of 1-forms on $\mathbb{R}^3$, kernel      58
John transform of 2-forms on $\mathbb{R}^3$      56
John transform of 2-forms on $\mathbb{R}^3$, image      57
John transform of 3-forms on $\mathbb{R}^3$      54
John transform of 3-forms on $\mathbb{R}^3$, image      55
John transform of differential forms      53
John transform, as an intertwining operator      66
John transform, group-theoretic meaning      65 66
John transform, image      48
John transform, inversion formula      50
John transform, operator $\kappa$      47
John transform, operator $\kappa$, complex case      68
John transform, operator $\kappa$, the differential form $\kappa\varphi$      47
John transform, relation to Gauss hypergeometric function      45
Kirchhoff formula      13
Laplace — Beltrami operator on $SL(2,\mathbb{C})$      128
Laplace — Beltrami operator on $\mathcal{L}^2$      88
Laplace — Beltrami operator on $\mathcal{L}^3$      103
Line complex      71 117
Line complex, admissibility conditions      73
Line complex, admissibility conditions in $\mathbb{C}^3$      76
Line complex, critical plane      76
Line complex, critical point      76
Lorentz group ($SL(2, \mathbb{C})$)      111
Manifold of lines in $\mathbb{P}^3$      59
Manifold of lines in $\mathbb{R}^3$      47
Manifold of lines in the space      43
Manifold of lines on the plane      1
Manifold of one-dimensional subspaces      47
Manifold, homogeneous coordinates      115
Manifold, left translation      115
Method of horocycle waves      107
Method of horospheres      93
Method of plane waves      12
Minkowski — Funk transform for the three-dimensional sphere      34
Minkowski — Funk transform on two-dimensional sphere      22
Minkowski — Funk transform on two-dimensional sphere, inversion formula      24
Operator $\kappa$ for the John transform      47
Operator $\kappa$ for the John transform, analogs      52
Operator $\kappa$ for the John transform, analogs in complex case      69
Operator $\kappa$ for the John transform, complex case      68
Operator $\kappa$ in $E^n$      157
Operator $\kappa$ on $\mathcal{L}^n$      149
Operator $\kappa$ on $\mathcal{L}^n$, inversion formulas      152
Operator $\kappa$ on $\mathcal{L}^n$, local and nonlocal      151
Paraboloid      129
Plancherel formula for the Fourier transform      91
Plane waves      12
Pluecker coordinates      59
Poisson formula      12
Pullback      52
Pushdown      52
Radon transform for the complex affine space      38
Radon transform for the complex affine space of arbitrary dimension: Fourier transform      39
Radon transform for the complex affine space of arbitrary dimension: inversion formula      40
Radon transform for the projective plane      30 38
Radon transform for the projective plane, image      37
Radon transform for the projective plane, inversion formula      35
Radon transform for the projective space      30 32
Radon transform for the projective space of an arbitrary dimension      38
Radon transform for the projective space of an arbitrary dimension: inversion formula      38
Radon transform for the projective space, image      35
Radon transform for the projective space, inversion formula      32
Radon transform for the projective space, the affine Radon transform      32
Radon transform for the projective space, the Minkowski — Funk transform      32
Radon transform in three-dimensional affine space      9
Radon transform in three-dimensional space      7
Radon transform in three-dimensional space, Fourier transform      9
Radon transform in three-dimensional space, inversion formula      8
Radon transform of 1-forms on the plane      25
Radon transform of 1-forms on the plane: inversion      25
Radon transform of 2-forms in three-dimensional space      28
Radon transform of 2-forms in three-dimensional space: inversion formula      28
Radon transform of 2-forms on the plane      26
Radon transform of 2-forms on the plane: image      27
Radon transform of 3-forms in three-dimensional space      29
Radon transform of 3-forms in three-dimensional space: inversion      30
Radon transform of arbitrary dimension      10
Radon transform of arbitrary dimension, combined inversion formula      11
Radon transform of arbitrary dimension, inversion formula      11
Radon transform on the affine plane      5
Radon transform on the affine plane, inversion formula      10
Radon transform on the Euclidean plane      1
Radon transform on the Euclidean plane, image      15 18
Radon transform on the Euclidean plane, inversion formula      3
Radon transform on the Euclidean plane, Paley — Wiener theorem      15 18
Radon transform, "complex"      11
Radon transform, discrete      19 21
Radon transform, inversion formula, local      11 14
Radon transform, inversion formula, nonlocal      9 14
Radon transform, inversion formula, using the moments      17
Radon transform, Poisson formula      19
Radon transform, relation to Fourier series      21
Radon transform, relation to the Fourier transform      5
Radon transform, tomography problems      4
Radon transform, wave equation on the plane      12 13
Rapidly decreasing function on a vector bundle      47
Reconstruction of unknown directions      18
Right regular representation      138
Symmetry relation for the Fourier transform on $SL(2,\mathbb{C})$      133 142
Symmetry relation for the Fourier transform on the hyperbolic plane      90
Symmetry relation for the hyperbolic plane      86
Symmetry relation for the hyperbolic space      101
Two-dimensional Fourier series      21
Ultrahyperbolic differential equation      43
Wave equation for the hyperbolic plane      106
Wave equation for the hyperbolic space      109
Wave equation, three-dimensional      13
Wave equation, two-dimensional      12
X-ray transform      43
Zonal horocycle functions      38
Zonal horospherical function      103
Zonal horospherical function on $SL(2,\mathbb{C})$      129
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