Integral Geometry and Representation Theory

Translator's NoteForewordChapter I Radon Transform of Test Functions and Generalized Functions on a Real Affine Space 1. The Radon Transform on a Real Affine Space 1.1. Definition of the Radon Transform 1.2. Relation Between Radon and Fourier Transforms 1.3. Elementary Properties of the Radon Transform 1.4. The Inverse Radon Transforms 1.5. Analog of Plancherel's Theorem for the Radon Transform 1.6. Analog of the Paley-Wiener Theorem for the Radon Transform 1.7. Asymptotic Behavior of Fourier Transforms of Characteristic Functions of Regions 2. The Radon Transform of Generalized Functions 2.1. Definition of the Radon Transform for Generalized Functions 2.2. Radon Transform of Generalized Functions Concentrated on Points and Line Segments 2.3. Radon Transform of (x1)+λ δ(x2,...,xn) 2.3a. Radon Transform of (x1)+λ δ(x2,...,xn) for Nonnegative Integer k 2.4. Integral of a Function over a Given Region in Terms of Integrals over Hyperplanes 2.5. Radon Transform of the Characteristic Function of One Sheet of a Cone Appendix to Section 2.5 2.6. Radon Transform of the Characteristic Function of One Sheet of a Two-Sheeted Hyperboloid 2.7. Radon Transform of Homogeneous Functions 2.8. Radon Transform of the Characteristic Function of an Octant 2.9. The Generalized Hypergeometric Function 3. Radon Transforms of Some Particular Generalized Functions 3.1. Radon Transforms of the Generalized Functions (P + i0)λ, (P — i0)λ, and P+λ for Nondegenerate Quadratic Forms P Appendix to Section 3.1 3.2. Radon Transforms of {P + c + i0)+λ, (P + c — i0)λ, and (P + c)+λ for Nondegenerate Quadratic Forms 3.3. Radon Transforms of the Characteristic Functions of Hyperboloids and Cones 3.4. Radon Transform of a Delta Function Concentrated on a Quadratic Surface 4. Summary of Radon Transform FormulasChapter II Integral Transforms In the Complex Domain 1. Line Complexes in a Space of Three Complex Dimensions and Related Integral Transforms 1.1. Plücker Coordinates of a Line 1.2. Line Complexes 1.3. A Special Class of Complexes 1.4. The Problem of Integral Geometry for a Line Complex 1.5. The Inversion Formula. Proof of the Theorem of Section 1.4 1.6. Examples of Complexes 1.7. Note on Translation Operators 2. Integral Geometry on a Quadratic Surface in a Space of Four Complex Dimensions 2.1. Statement of the Problem 2.2. Line Generators of Quadratic Surfaces 2.3. Integrals of ƒ(z) over Quadratic Surfaces and Along Complex Lines 2.4. Expression for ƒ(z) on a Quadratic Surface in Terms of Its Integrals Along Line Generators 2.5. Derivation of the Inversion Formula 2.6. Another Derivation of the Inversion Formula 2.7. Rapidly Decreasing Functions on Quadratic Surfaces. The Paley-Wiener Theorem 3. The Radon Transform in the Complex Domain 3.1. Definition of the Radon Transform 3.2. Representation of ƒ(z) in Terms of Its Radon Transform 3.3. Analog of Plancherel's Theorem for the Radon Transform 3.4. Analog of the Paley-Wiener Theorem for the Radon Transform 3.5. Radon Transform of Generalized Functions 3.6. Examples 3.7. The Generalized Hypergeometric Function in the Complex DomainChapter III Representations of the Group of Complex Unimodular Matrices in Two Dimensions 1. The Group of Complex Unimodular Matrices in Two Dimensions and Some of Its Realizations 1.1. Connection with the Proper Lorentz Group 1.2. Connection with Lobachevskian and Other Motions 2. Representations of the Lorentz Group Acting on Homogeneous Functions of Two Complex Variables 2.1. Representations of Groups 2.2. The Dx Spaces of Homogeneous Functions 2.3. Two Useful Realizations of the Dx 2.4. Representation of G on Dx 2.5. The Tx(g) Operators in Other Realizations of Dx 2.6. The Dual Representations 3. Summary of Basic Results concerning Representations on Dx 3.1. Irreducibility of Representations on the Dx and the Role of Integer Points 3.2. Equivalence of Representations on the Dx and the Role of Integer Points 3.3. The Problem of Equivalence at Integer Points 3.4. Unitary Representations 4. Invariant Bilinear Functionals 4.1. Statement of the Problem and the Basic Results 4.2. Necessary Condition for Invariance under Parallel Translation and Dilation 4.3. Conditions for Invariance under Inversion 4.4. Sufficiency of Conditions for the Existence of Invariant Bilinear Functionals (Nonsingular Case) 4.5. Conditions for the Existence of Invariant Bilinear Functionals (Singular Case) 4.6. Degeneracy of Invariant Bilinear Functionals 4.7. Conditionally Invariant Bilinear Functionals 5. Equivalence of Representations of G 5.1. Intertwining Operators 5.2. Equivalence of Two Representations 5.3. Partially Equivalent Representations 6. Unitary Representations of G 6.1. Invariant Hermitian Functionals on Dx 6.2. Positive Definite Invariant Hermitian Functionals 6.3. Invariant Hermitian Functionals for Noninteger ρ,