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Generalized Functions, Volume 5: Integral Geometry and Representation Theory [Kõva köide]

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  • Formaat: Hardback, 449 pages, kõrgus x laius: 254x178 mm, kaal: 988 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470426633
  • ISBN-13: 9781470426637
Teised raamatud teemal:
Generalized Functions, Volume 5: Integral Geometry and Representation TheorySuurem pilt
  • Formaat: Hardback, 449 pages, kõrgus x laius: 254x178 mm, kaal: 988 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470426633
  • ISBN-13: 9781470426637
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470426637.html
Märksõnad:
The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gelfand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory.

The unifying idea of Volume 5 in the series is the application of the theory of generalized functions developed in earlier volumes to problems of integral geometry, to representations of Lie groups, specifically of the Lorentz group, and to harmonic analysis on corresponding homogeneous spaces. The book is written with great clarity and requires little in the way of special previous knowledge of either group representation theory or integral geometry; it is also independent of the earlier volumes in the series. The exposition starts with the definition, properties, and main results related to the classical Radon transform, passing to integral geometry in complex space, representations of the group of complex unimodular matrices of second order, and harmonic analysis on this group and on most important homogeneous spaces related to this group. The volume ends with the study of representations of the group of real unimodular matrices of order two.
Translator's Note v
Foreword vii
Chapter 1 Radon Transform of Test Functions and Generalized Functions on a Real Affine Space
1(74)
1 The Radon Transform on a Real Affine Space
1(20)
1.1 Definition of the Radon Transform
1(3)
1.2 Relation between Radon and Fourier Transforms
4(1)
1.3 Elementary Properties of the Radon Transform
5(3)
1.4 The Inverse Radon Transform
8(4)
1.5 Analog of Plancherel's Theorem for the Radon Transform
12(3)
1.6 Analog of the Paley-Wiener Theorem for the Radon Transform
15(4)
1.7 Asymptotic Behavior of Fourier Transforms of Characteristic Functions of Regions
19(2)
2 The Radon Transform of Generalized Functions
21(34)
2.1 Definition of the Radon Transform for Generalized Functions
22(3)
2.2 Radon Transform of Generalized Functions Concentrated on Points and Line Segments
25(1)
2.3 Radon Transform of (x1) λ + δ (x2 ... xn)
26(1)
2.3a Radon Transform of (x1) k + δ (x2 ... xn) for Nonnegative Integer k
27(4)
2.4 Integral of a Function over a Given Region in Terms of Integrals over Hyperplanes
31(4)
2.5 Radon Transform of the Characteristic Function of One Sheet of a Cone
35(3)
Appendix to Section 2.5
38(2)
2.6 Radon Transform of the Characteristic Function of One Sheet of a Two-Sheeted Hyperboloid
40(3)
2.7 Radon Transform of Homogeneous Functions
43(1)
2.8 Radon Transform of the Characteristic Function of an Octant
44(7)
2.9 The Generalized Hypergeometric Function
51(4)
3 Radon Transforms of Some Particular Generalized Functions
55(14)
3.1 Radon Transforms of the Generalized Functions (P + i0) λ (P -- i0) λ and Pλ. for Nondegenerate Quadratic Forms P
56(3)
Appendix to Section 3.1
59(2)
3.2 Radon Transforms of (P + c + i0) λ, (P + c - i0) λ, and (P + c) λ + for Nondegenerate Quadratic Forms
61(2)
3.3 Radon Transforms of the Characteristic Functions of Hyperboloids and Cones
63(3)
3.4 Radon Transform of a Delta Function Concentrated on a Quadratic-Surface
66(3)
4 Summary of Radon Transform Formulas
69(6)
Chapter II Integral Transforms in the Complex Domain
75(58)
1 Line Complexes in a Space of Three Complex Dimensions and Related Integral Transforms
77(17)
1.1 Pliicker Coordinates of a Line
77(1)
1.2 Line Complexes
78(2)
1.3 A Special Class of Complexes
80(2)
1.4 The Problem of Integral Geometry for a Line Complex
82(4)
1.5 The Inversion Formula. Proof of the Theorem of Section 1.4
86(3)
1.6 Examples of Complexes
89(3)
1.7 Note on Translation Operators
92(2)
2 Integral Geometry on a Quadratic Surface in a Space of Four Complex Dimensions
94(21)
2.1 Statement of the Problem
94(1)
2.2 Line Generators of Quadratic Surfaces
95(3)
2.3 Integrals of f(z) over Quadratic Surfaces and along Complex Lines
98(2)
2.4 Expression for f(z) on a Quadratic Surface in Terms of Its Integrals along Line Generators
100(3)
2.5 Derivation of the Inversion Formula
103(4)
2.6 Another Derivation of the Inversion Formula
107(4)
2.7 Rapidly Decreasing Functions on Quadratic Surfaces. The Paley-Wiener Theorem
111(4)
3 The Radon Transform in the Complex Domain
115(18)
3.1 Definition of the Radon Transform
115(2)
3.2 Representation off(z) in Terms of Its Radon Transform
117(4)
3.3 Analog of Plancherel's Theorem for the Radon Transform
121(2)
3.4 Analog of the Paley-Wiener Theorem for the Radon Transform
123(1)
3.5 Radon Transform of Generalized Functions
124(1)
3.6 Examples
125(6)
3.7 The Generalized Hypergeometric Function in the Complex Domain
131(2)
Chapter III Representations of the Group of Complex Unimodular Matrices in Two Dimensions
133(69)
1 The Group of Complex Unimodular Matrices in Two Dimensions and Some of Its Realizations
134(5)
1.1 Connection with the Proper Lorentz Group
134(3)
1.2 Connection with Lobachevskian and Other Motions
137(2)
2 Representations of the Lorentz Group Acting on Homogeneous Functions of Two Complex Variables
139(9)
2.1 Representations of Groups
139(2)
2.2 The Dx Spaces of Homogeneous Functions
141(1)
2.3 Two Useful Realizations of the Dx
142(2)
2.4 Representation of G on Dx
144(1)
2.5 The Tx(g) Operators in Other Realizations of Dx
145(2)
2.6 The Dual Representations
147(1)
3 Summary of Basic Results concerning Representations on Dx
148(9)
3.1 Irreducibility of Representations on the Dx and the Role of Integer Points
148(3)
3.2 Equivalence of Representations on the Dx and the Role of Integer Points
151(2)
3.3 The Problem of Equivalence at Integer Points
153(3)
3.4 Unitary Representations
156(1)
4 Invariant Bilinear Functionals
157(21)
4.1 Statement of the Problem and the Basic Results
152(7)
4.2 Necessary Condition for Invariance under Parallel Translation and Dilation
159(4)
4.3 Conditions for Invariance under Inversion
163(2)
4.4 Sufficiency of Conditions for the Existence of Invariant Bilinear Functionals (Nonsingular Case)
165(3)
4.5 Conditions for the Existence of Invariant Bilinear Functionals (Singular Case)
168(6)
4.6 Degeneracy of Invariant Bilinear Functionals
174(1)
4.7 Conditionally Invariant Bilinear Functionals
175(3)
5 Equivalence of Representations of G
178(11)
5.1 Intertwining Operators
178(4)
5.2 Equivalence of Two Representations
182(2)
5.3 Partially Equivalent Representations
184(5)
6 Unitary Representations of G
189(13)
6.1 Invariant Hermitian Functionals on Dx
189(1)
6.2 Positive Definite Invariant Hermitian Functionals
190(3)
6.3 Invariant Hermitian Functionals for Noninteger ρ, | ρ | > 1
193(3)
6.4 Invariant Hermitian Functionals in the Special Case of Integer n1 = n2
196(2)
6.5 Unitary Representations of G by Operators on Hilbert Space
198(2)
6.6 Subspace Irreducibility of the Unitary Representations
200(2)
Chapter IV Harmonic Analysis on the Group of Complex Unimodular Matrices in Two Dimensions
202(71)
1 Definition of the Fourier Transform on a Group. Statement of the Problems and Summary of the Results
202(14)
1.1 Fourier Transform on the Line
202(2)
Functions on G
204(1)
1.3 Fourier Transform on G
205(2)
1.4 Domain of Definition of F(x)
207(2)
1.5 Summary of the Results of
Chapter IV
209(4)
Appendix. Functions on G
213(3)
2 Properties of the Fourier Transform on G
216(61)
2.1 Simplest Properties
216(2)
2.2 Fourier Transform as Integral Operator
218(2)
2.3 Geometric Interpretation of K (z1, z2; Χ) The Functions φ (z1, z2; λ) and φ (u, v; u' v')
220(2)
2.4 Properties of K (z1, z2 ; Χ)
222(1)
2.5 Continuity of K (z1, z2; χ)
223(2)
2.6 Asymptotic Behavior of K (z1, z2; Χ)
225(1)
2.7 Trace of the Fourier Transform
226(1)
3 Inverse Fourier Transform and Plancherel's Theorem for G
227(1)
3.1 Statement of the Problem
227(3)
3.2 Expression for φ (z1, z2; λ) in Terms of K (z1, z2 ; χ)
230(2)
3.3 Expression for f(g) in Terms of (z1, z2; λ)
232(3)
3.4 Expression for f(g) in Terms of Its Fourier Transform F(Χ)
235(2)
3.5 Analog of Plancherel's Theorem for G
237(3)
3.6 Symmetry Properties of F(x)
240(2)
3.7 Fourier Integral and the Decomposition of the Regular Representation of the Lorentz Group into Irreducible Representations
242(5)
4 Differential Operators on G
247(9)
4.1 Tangent Space to G
247(1)
4.2 Lie Operators
248(2)
4.3 Relation between Left and Right Derivative Operators
250(2)
4.4 Commutation Relations for the Lie Operators
252(1)
4.5 Laplacian Operators
253(1)
4.6 Functions on G with Rapidly Decreasing Derivatives
254(1)
4.7 Fourier Transforms of Lie Operators
255(1)
5 The Paley-Wiener Theorem for the Fourier Transform on G
256(17)
5.1 Integrals of f(g) along "Line Generators"
257(1)
5.2 Behavior of φ (u, v; u', v') under Translation and Differentiation of f(g)
258(2)
5.3 Differentiability and Asymptotic Behavior of φ (u, v; u', v')
260(2)
5.4 Conditions on K(z1, z2 ; χ)
262(3)
5.5 Moments of f(g) and Their Expression in Terms of the Kernel
265(2)
5.6 The Paley-Wiener Theorem for the Fourier Transform on G
267(6)
Chapter V Integral Geometry in a Space of Constant Curvature
273(58)
1 Spaces of Constant Curvature
274(16)
1.1 Spherical and Lobachevskian Spaces
274(2)
1.2 Some Models of Lobachevskian Spaces
276(1)
1.3 Imaginary Lobachevskian Spaces
277(1)
1.3a Isotropic Lines of an Imaginary Lobachevskian Space
278(2)
1.4 Spheres and Horospheres in a Lobachevskian Space
280(2)
1.5 Spheres and Horospheres in an Imaginary Lobachevskian Space
282(3)
1.6 Invariant Integration in a Space of Constant Curvature
285(2)
1.7 Integration over a Horosphere
287(1)
1.8 Measures on the Absolute
288(2)
2 Integral Transform Associated with Horospheres in a Lobachevskian Space
290(14)
2.1 Integral Transform Associated with Horospheres
291(2)
2.2 Inversion Formula for n = 3
293(7)
2.3 Inversion Formula for Arbitrary Dimension
300(2)
2.4 Functions Depending on the Distance from a Point to a Horosphere, and Their Averages
302(2)
3 Integral Transform Associated with Horospheres in an Imaginary Lobachevskian Space
304(27)
3.1 Statement of the Problem and Preliminary Remarks
304(4)
3.2 Regularizing Integrals by Analytic Continuation in the Coordinates
308(6)
3.3 Derivation of the Inversion Formula
314(5)
3.4 Derivation of the Inversion Formula (Continued)
319(5)
3.4a Parallel Isotropic Lines
324(2)
3.5 Calculation of Φ (x, a; μ)
326(5)
Chapter VI Harmonic Analysis on Spaces Homogeneous with Respect to the Lorentz Group
331(59)
1 Homogeneous Spaces and the Associated Representations of the Lorentz Group
331(18)
1.1 Homogeneous Spaces
331(1)
1.2 Representations of the Lorentz Group Associated with Homogeneous Spaces
331(1)
1.3 The Relation between Representation Theory and Integral Geometry
332(2)
1.4 Homogeneous Spaces and Associated Subgroups of Stability
334(1)
1.5 Examples of Spaces Homogeneous with Respect to the Lorentz Group
335(4)
1.6 Group-Theoretical Definition of Horospheres
339(6)
1.7 Fourier Integral Expansions of Functions on Homogeneous Spaces
345(4)
2 Representations of the Lorentz Group Associated with the Complex Affine Plane and with the Cone, and Their Irreducible Components
349(7)
2.1 Unitary Representations of the Lorentz Group Associated with the Complex Affine Plane
349(3)
2.2 Unitary Representation of the Lorentz Group Associated with the Cone
352(4)
3 Decomposition of the Representation of the Lorentz Group Associated with Lobachevskian Space
356(8)
3.1 Representation of the Lorentz Group Associated with Lobachevskian Space
356(1)
3.2 Decomposition by the Horosphere Method
357(5)
3.3 The Analog of Plancherel's Theorem for Lobachevskian Space
362(2)
4 Decomposition of the Representation of the Lorentz Group Associated with Imaginary Lobachevskian Space
364(21)
4.1 Representation of the Lorentz Group Associated with Imaginary Lobachevskian Space
364(1)
4.2 Decomposition of the Representation Associated with Horospheres of the First Kind
365(2)
4.3 Decomposition of the Representation Associated with Isotropic Lines
367(6)
4.4 Decomposition of the Representation Associated with Imaginary Lobachevskian Space
373(8)
4.5 The Analog of Plancherel's Theorem for Imaginary Lobachevskian Space
381(2)
4.6 Integral Transform Associated with Planes in Lobachevskian Space
383(2)
5 Integral Geometry and Harmonic Analysis on the Point Pairs on the Complex Projective Line
385(5)
Chapter VII Representations of the Group of Real Unimodular Matrices in Two Dimensions
390(50)
1 Representations of the Real Unimodular Matrices in Two Dimensions Acting on Homogeneous Functions of Two Real Variables
390(5)
1.1 The Dx Spaces of Homogeneous Functions
390(2)
1.2 Two Useful Realizations of Dx
392(1)
1.3 Representation of G on Dx
392(1)
1.4 The Tx(g) Operators in Other Realizations of Dx
393(1)
1.5 The Dual Representations
394(1)
2 Summary of the Basic Results concerning Representations on Dx
395(5)
2.1 Irreducibility of Representations on Dx
395(2)
2.2 Equivalence of Representations on Dx and the Role of Integer Points
397(1)
2.3 The Problem of Equivalence at Integer Points
398(1)
2.4 Unitary Representations
399(1)
3 Invariant Bilinear Functionals
400(13)
3.1 Invariance under Translation and Dilation
401(3)
3.2 Necessary and Sufficient Conditions for the Existence of an Invariant Bilinear Functional
404(5)
3.3 Degenerate Invariant Bilinear Functionals for Analytic Representations
409(2)
3.4 Conditionally Invariant Bilinear Functionals
411(2)
4 Equivalence of Two Representations
413(11)
4.1 Intertwining Operators
413(3)
4.2 Equivalence of Two Representations
416(2)
4.3 Partially Equivalent Representations
418(5)
4.4 Other Models of Ft and F+
423(1)
5 Unitary Representations of G
424(16)
5.1 Existence of an Invariant Hermitian Functional
424(2)
5.2 Positive Definite Invariant Hermitian Functionals (Nonanalytic Representations)
426(3)
5.3 Invariant Hermitian Functionals for Analytic Representations
429(3)
5.4 Invariant Positive Definite Hermitian Functionals on the Analytic Function Spaces F+ and F-
432(2)
5.5 Unitary Representations of G by Operators on Hilbert Space
434(3)
5.6 Inequivalence of the Representations of the Discrete Series
437(1)
5.7 Subspace Irreducibility of the Unitary Representations
438(2)
Notes and References to the Literature 440(2)
Bibliography 442(3)
Index 445