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Generalized Functions, Volumes 1-6 [Kõva köide]

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  • Formaat: Hardback, 2165 pages, kõrgus x laius: 254x178 mm, kaal: 5169 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470428857
  • ISBN-13: 9781470428853
Teised raamatud teemal:
Generalized Functions, Volumes 1-6Suurem pilt
  • Formaat: Hardback, 2165 pages, kõrgus x laius: 254x178 mm, kaal: 5169 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470428857
  • ISBN-13: 9781470428853
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470428853.html
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.
Translator's Note v
Foreword to the First Russian Edition vii
Foreword to the Second Russian Edition xi
Chapter I Definition and Simplest Properties of Generalized Functions
1(139)
1 Test Functions and Generalized Functions
1(17)
1.1 Introductory Remarks
1(1)
1.2 Test Functions
2(1)
1.3 Generalized Functions
3(2)
1.4 Local Properties of Generalized Functions
5(2)
1.5 Addition and Multiplication by a Number and by a Function
7(1)
1.6 Translations, Rotations, and Other Linear Transformations on the Independent Variables
8(2)
1.7 Regularization of Divergent Integrals
10(3)
1.8 Convergence of Generalized Function Sequences
13(2)
1.9 Complex Test Functions and Generalized Functions
15(1)
1.10 Other Test-Function Spaces
16(2)
2 Differentiation and Integration of Generalized Functions
18(27)
2.1 Fundamental Definitions
18(3)
2.2 Examples for the Case of a Single Variable
21(6)
2.3 Examples for the Case of Several Variables
27(2)
2.4 Differentiation as a Continuous Operation
29(5)
2.5 Delta-Convergent Sequences
34(5)
2.6 Differential Equations for Generalized Functions
39(5)
2.7 Differentiation in S
44(1)
3 Regularization of Functions with Algebraic Singularities
45(37)
3.1 Statement of the Problem
45(3)
3.2 The Generalized Functions xλ+ and xλ-
48(2)
3.3 Even and Odd Combinations of xλ+ and xλ-
50(4)
3.4 Indefinite Integrals of xλ+, xλ-, |x|λ sgn x
54(1)
3.5 Normalization of xλ+, xλ-, |x|λ sgn x
55(4)
3.6 The Generalized Functions (x + i0)λ and (x -- i0)λ
59(2)
3.7 Canonical Regularization
61(4)
3.8 Regularization of Other Integrals
65(6)
3.9 The Generalized Function rλ
71(3)
3.10 Plane-Wave Expansion of rλ
74(4)
3.10 Homogeneous Functions
78(4)
4 Associate Functions
82(18)
4.1 Definition
82(2)
4.2 Taylor's and Laurent Series for xλ+ and xλ-
84(5)
4.3 Expansion of |x|λ and |x|λ sgn x
89(4)
4.4 The Generalized Functions (x + i0)λ and (x -- i0)λ
93(3)
4.5 Taylor's Series for (x + i0)λ and (x - i0)λ
96(2)
4.6 Expansion of rλ
98(2)
5 Convolutions of Generalized Functions
100(22)
5.1 Direct Product of Generalized Functions
100(3)
5.2 Convolutions of Generalized Functions
103(3)
5.3 Newtonian Gravitational Potential and Elementary Solutions of Differential Equations
106(3)
5.4 Poisson's Integral and Elementary Solutions of Cauchy's Problem
109(6)
5.5 Integrals and Derivatives of Higher Orders
115(7)
6 Elementary Solutions of Differential Equations with Constant Coefficients
122(18)
6.1 Elementary Solutions of Elliptic Equations
122(6)
6.2 Elementary Solutions of Regular Homogeneous Equations
128(4)
6.3 Elementary Solutions of Cauchy's Problem
132(8)
Appendix 1 Local Properties of Generalized Functions
140(7)
A1.1 Test Functions as Averages of Continuous Functions
141(1)
A1.2 Partition of Unity
142(2)
A1.3 Local Properties of Generalized Functions
144(2)
A1.4 Differentiation as a Local Operation
146(1)
Appendix 2 Generalized Functions Depending on a Parameter
147(221)
A2.1 Continuous Functions
147(1)
A2.2 Differentiable Functions
148(1)
A2.3 Analytic Functions
149(4)
Chapter II Fourier Transforms of Generalized Functions
153(56)
1 Fourier Transforms of Test Functions
153(13)
1.1 Fourier Transforms of Functions in K
153(2)
1.2 The Space Z
155(2)
1.3 The Case of Several Variables
157(1)
1.4 Functionals on Z
158(2)
1.5 Analytic Functionals
160(5)
1.6 Fourier Transforms of Functions in S
165(1)
2 Fourier Transforms of Generalized Functions. A Single Variable
166(24)
2.1 Definition
166(2)
2.2 Examples
168(2)
2.3 Fourier Transforms of xλ+, xλ-, |x|λ, and |x|λ sgn x
170(4)
2.4 Fourier Transforms of xλ+, In x+ and Similar Generalized Functions
174(8)
2.5 Fourier Transform of the Generalized Function (ax2 + bx + c)λ+
182(6)
2.6 Fourier Transforms of Analytic Functionals
188(2)
3 Fourier Transforms of Generalized Functions. Several Variables
190(10)
3.1 Definitions
190(1)
3.2 Fourier Transform of the Direct Product
191(1)
3.3 Fourier Transform of rλ
192(4)
3.4 Fourier Transform of Generalized Function with Bounded Support
196(4)
3.5 The Fourier Transform as the Limit of a Sequence of Functions
200(1)
4 Fourier Transforms and Differential Equations
200(9)
4.1 Introductory Remarks
200(1)
4.2 The Iterated Laplace Equation Δmu = f
201(1)
4.3 The Wave Equation in Space of Odd Dimension
202(2)
4.4 The Relation between the Elementary Solution of an Equation and the Corresponding Cauchy Problem
204(2)
4.5 Classical Operational Calculus
206(3)
Chapter III Particular Types of Generalized Functions
209(159)
1 Generalized Functions Concentrated on Smooth Manifolds of Lower Dimension
209(38)
1.1 Introductory Remarks on Differential Forms
214(6)
1.2 The Form ω
220(2)
1.3 The Generalized Function δ(P)
222(4)
1.4 Example: Derivation of Green's Theorem
226(2)
1.5 The Differential Forms ωk(φ) and the Generalized Functions δ(k)(P)
228(4)
1.6 Recurrence Relations for the δ(k)(P)
232(4)
1.7 Recurrence Relations for the δ(k)(aP)
236(1)
1.8 Multiplet Layers
237(2)
1.9 The Generalized Function δ(P1,..., Pk) and its Derivatives
239(8)
2 Generalized Functions Associated with Quadratic Forms
247(48)
2.1 Definition of δ1(k)(P) and δ2(k)(P)
247(6)
2.2 The Generalized Function Pλ
253(16)
2.3 The Generalized Function Pλ Associated with a Quadratic Form with Complex Coefficients
269(5)
2.4 The Generalized Functions (P + i0)λ and (P -- i0)λ
274(5)
2.5 Elementary Solutions of Linear Differential Equations
279(4)
2.6 Fourier Transforms of (P + i0)λ and (P - i0)λ
283(2)
2.7 Generalized Functions Associated with Bessel Functions
285(2)
2.8 Fourier Transforms of (c2 + P + i0)λ and (c2 + P - i0)λ
287(3)
2.9 Fourier Transforms of (c2 + P)λ+ and (c2 + P)λ-
290(1)
2.10 Fourier Transforms of (c2 + P)λ+/Γ(λ + 1) and (c2 + P)λ+/Γ(λ + 1) for Integral λ
291(4)
3 Homogeneous Functions
295(18)
3.1 Introduction
295(2)
3.2 Positive Homogeneous Functions of Several Independent Variables
297(6)
3.3 Generalized Homogeneous Functions of Degree -n
303(6)
3.4 Generalized Homogeneous Functions of Degree -n - m
309(2)
3.5 Generalized Functions of the Form rλf, where f Is a Generalized Function on the Unit Sphere
311(2)
4 Arbitrary Functions Raised to the Power λ
313(55)
4.1 Reducible Singular Points
313(2)
4.2 The Generalized Function Gλ when G = 0 Consists Entirely of First-Order Points
315(3)
4.3 The Generalized Function Gλ when G = 0 Has No Points of Order Higher Than Two
318(5)
4.4 The Generalized Function Gλ in General
323(3)
4.5 Integrals of an Infinitely Differentiable Function over a Surface Given by G = c
326(4)
Summary of Fundamental Definitions and Equations of Volume I
330(29)
Table of Fourier Transforms
359(9)
Appendix A Proof of the Completeness of the Generalized-Function Space
368(2)
Appendix B Generalized Functions of Complex Variables
370(43)
B1 Generalized Functions of a Single Complex Variable
371(16)
B1.1 The Variables z and z
371(1)
B1.2 Homogeneous Functions of a Complex Variable
372(1)
B1.3 The Homogeneous Generalized Functions zλzμ
373(4)
B1.4 The Generalized Functions z-k-1 and Its Derivatives
377(1)
B1.5 Associated Homogeneous Functions
378(1)
B1.6 Uniqueness Theorem for Homogeneous Generalized Functions
379(2)
B1.7 Fourier Transforms of Test Functions and of Generalized Functions
381(4)
B1.8 The Generalized Functions fλ(z) fμ(z), Where f(z) is a Meromorphic Function
385(2)
B2 Generalized Functions of m Complex Variables
387
B2.1 The Generalized Functions δ(P) and δ(k,l)(P)
387(3)
B2.2 The Generalized Functions GλGμ
390(1)
B2.3 Homogeneous Generalized Functions
391(2)
B2.4 Associated Homogeneous Functions
393(1)
B2.5 The Residue of a Homogeneous Function
394(2)
B2.6 Homogeneous Generalized Functions of Degree (-m,-m)
396(2)
B2.7 The Generalized Function PλPμ, Where P Is a Nondegenerate Quadratic Form
398(6)
B2.8 Elementary Solutions of Linear Differential Equations in the Complex Domain
404(2)
B2.9 The Generalized Function GλGμ (General Case)
406(5)
B2.10 Generalized Functions Corresponding to Meromorphic Functions of m Complex Variables
411
Notes and References to the Literature 413(3)
Bibliography 416(3)
Index 419(3)
Index of Particular Generalized Functions 422
Preface to the Russian Edition v
Chapter I Linear Topological Spaces
1(76)
1 Definition of a Linear Topological Space
1(10)
2 Normed Spaces. Comparability and Compatibility of Norms
11(4)
3 Countably Normed Spaces
15(17)
4 Continuous Linear Functionals and the Conjugate Space
32(9)
5 Topology in a Conjugate Space
41(12)
6 Perfect Spaces
53(7)
7 Continuous Linear Operators
60(6)
8 Union of Countably Normed Spaces
66(11)
Appendix 1 Elements, Functionals, Operators Depending on a Parameter
70(2)
Appendix 2 Differentiable Abstract Functions
72(1)
Appendix 3 Operators Depending on a Parameter
73(2)
Appendix 4 Integration of Continuous Abstract Functions with Respect to the Parameter
75(2)
Chapter II Fundamental and Generalized Functions
77(45)
1 Definition of Fundamental and Generalized Functions
77(9)
2 Topology in the Spaces K{Mp} and Z{Mp}
86(12)
3 Operations with Generalized Functions
98(11)
4 Structure of Generalized Functions
109(13)
Chapter III Fourier Transformations of Fundamental and Generalized Functions
122(44)
1 Fourier Transformations of Fundamental Functions
122(6)
2 Fourier Transforms of Generalized Functions
128(7)
3 Convolution of Generalized Functions and Its Connection to Fourier Transforms
135(19)
4 Fourier Transformation of Entire Analytic Functions
154(12)
Chapter IV Spaces of Type S
166
1 Introduction
166(3)
2 Various Modes of Defining Spaces of Type S
169(7)
3 Topological Structure of Fundamental Spaces
176(8)
4 Simplest Bounded Operations in Spaces of Type S
184(9)
5 Differential Operators
193(4)
6 Fourier Transformations
197(10)
7 Entire Analytic Functions as Elements or Multipliers in Spaces of Type S
207(18)
8 The Question of the Nontriviality of Spaces of Type S
225(12)
9 The Case of Several Independent Variables
237
Appendix 1 Generalization of Spaces of Type S
244(2)
Appendix 2 Spaces of Type W
246
Notes and References 253(4)
Bibliography 257(2)
Index 259
Translator's Note v
Preface to the Russian Edition vii
Chapter I Spaces of Type W
1(28)
1 Definitions
1(11)
2 Bounded Operators in Spaces of Type W
12(6)
3 Fourier Transforms
18(7)
4 The Case of Several Variables
25(4)
Chapter II Uniqueness Classes for the Cauchy Problem
29(76)
1 Introduction
29(3)
2 The Cauchy Problem in a Topological Vector Space
32(4)
3 The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method
36(15)
4 The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms
51(9)
5 Examples
60(4)
6 The Connection between the Reduced Order of a System and Its Characteristic Roots
64(16)
7 A Theorem of the Phragmen-Lindelof Type
80(25)
Appendix 1 Convolution Equations
90(4)
Appendix 2 Equations with Coefficients Which Depend on x
94(6)
Appendix 3 Systems with Elliptic Operators
100(5)
Chapter III Correctness Classes for the Cauchy Problem
105(60)
1 Introduction
105(6)
2 Parabolic Systems
111(15)
3 Hyperbolic Systems
126(8)
4 Systems Which Are Petrovskii-Correct
134(22)
5 On the Solutions of Incorrect Systems
156(9)
Chapter IV Generalized Eigenfunction Expansions
165
1 Introduction
165(7)
2 Differentiation of Functionals of Strongly Bounded Variation
172(4)
3 Differentiation of Functionals of Weakly Bounded Variation
176(6)
4 Existence and Completeness Theorems for the System of Eigenfunctionals
182(7)
5 Generalized Eigenfunctions of Self-Adjoint Operators
189(12)
6 The Structure of the Generalized Eigenfunctions
201(5)
7 Dynamical Systems
206
Notes and References 211(6)
Bibliography 217(4)
Index 221(150)
Translator's Note v
Foreword vii
Chapter I The Kernel Theorem. Nuclear Spaces. Rigged Hilbert Space
1(134)
1 Bilinear Functionals on Countably Normed Spaces. The Kernel Theorem
2(24)
1.1 Convex Functionals
3(4)
1.2 Bilinear Functionals
7(4)
1.3 The Structure of Bilinear Functionals on Specific Spaces (the Kernel Theorem)
11(9)
Appendix. The Spaces K, S, and 2
20(6)
2 Operators of Hilbert-Schmidt Type and Nuclear Operators
26(30)
2.1 Completely Continuous Operators
27(5)
2.2 Hilbert-Schmidt Operators
32(5)
2.3 Nuclear Operators
37(10)
2.4 The Trace Norm
47(5)
2.5 The Trace Norm and the Decomposition of an Operator into a Sum of Operators of Rank 1
52(4)
3 Nuclear Spaces. The Abstract Kernel Theorem
56(47)
3.1 Countably Hilbert Spaces
57(5)
3.2 Nuclear Spaces
62(4)
3.3 A Criterion for the Nuclearity of a Space
66(5)
3.4 Properties of Nuclear Spaces
71(2)
3.5 Bilinear Functionals on Nuclear Spaces
73(6)
3.6 Examples of Nuclear Spaces
79(7)
3.7 The Metric Order of Sets in Nuclear Spaces
86(12)
3.8 The Functional Dimension of Linear Topological Spaces
98(5)
4 Rigged Hilbert Spaces. Spectral Analysis of Self-Adjoint and Unitary Operators
103(24)
4.1 Generalized Eigenvectors
103(3)
4.2 Rigged Hilbert Spaces
106(4)
4.3 The Realization of a Hilbert Space as a Space of Functions, and Rigged Hilbert Spaces
110(4)
4.4 Direct Integrals of Hilbert Spaces, and Rigged Hilbert Spaces
114(5)
4.5 The Spectral Analysis of Operators in Rigged Hilbert Spaces
119(8)
Appendix. The Spectral Analysis of Self-Adjoint and Unitary Operators in Hilbert Space
127(1)
1 The Abstract Theorem on Spectral Decomposition
127(2)
2 Cyclic Operators
129(1)
3 The Decomposition of a Hilbert Space into a Direct Integral Corresponding to a Given Self-Adjoint Operator
130(5)
Chapter II Positive and Positive-Definite Generalized Functions
135(102)
1 Introduction
135(7)
1.1 Positivity and Positive Definiteness
136(6)
2 Positive Generalized Functions
142(9)
2.1 Positive Generalized Functions on the Space of Infinitely Differentiable Functions Having Bounded Supports
142(3)
2.2 The General Form of Positive Generalized Functions on the Space S
145(2)
2.3 Positive Generalized Functions on Some Other Spaces
147(2)
2.4 Multiplicatively Positive Generalized Functions
149(2)
3 Positive-Definite Generalized Functions. Bochner's Theorem
151(24)
3.1 Positive-Definite Generalized Functions on S
151(1)
3.2 Continuous Positive-Definite Functions
152(5)
3.3 Positive-Definite Generalized Functions on K
157(9)
3.4 Positive-Definite Generalized Functions on Z
166(1)
3.5 Translation-Invariant Positive-Definite Hermitean Bilinear Functionals
167(2)
3.6 Examples of Positive and Positive-Definite Generalized Functions
169(6)
4 Conditionally Positive-Definite Generalized Functions
175(21)
4.1 Basic Definitions
175(1)
4.2 Conditionally Positive Generalized Functions (Case of One Variable)
176(3)
4.3 Conditionally Positive Generalized Functions (Case of Several Variables)
179(9)
4.4 Conditionally Positive-Definite Generalized Functions on K
188(1)
4.5 Bilinear Functionals Connected with Conditionally Positive-Definite Generalized Functions
189(5)
Appendix
194(2)
5 Evenly Positive-Definite Generalized Functions
196(20)
5.1 Preliminary Remarks
196(2)
5.2 Evenly Positive-Definite Generalized Functions on S1\2
198(13)
5.3 Evenly Positive-Definite Generalized Functions on S1/2
211(2)
5.4 Positive-Definite Generalized Functions and Groups of Linear Transformations
213(3)
6 Evenly Positive-Definite Generalized Functions on the Space of Functions of One Variable with Bounded Supports
216(13)
6.1 Positive and Multiplicatively Positive Generalized Functions
216(3)
6.2 A Theorem on the Extension of Positive Linear Functionals
219(1)
6.3 Even Positive Generalized Functions on Z
220(6)
6.4 An Example of the Nonuniqueness of the Positive Measure Corresponding to a Positive Functional on Z+
226(3)
7 Multiplicatively Positive Linear Functionals on Topological Algebras with Involutions
229(8)
7.1 Topological Algebras with Involutions
229(3)
7.2 The Algebra of Polynomials in Two Variables
232(5)
Chapter III Generalized Random Processes
237(66)
1 Basic Concepts Connected with Generalized Random Processes
237(9)
1.1 Random Variables
237(5)
1.2 Generalized Random Processes
242(2)
1.3 Examples of Generalized Random Processes
244(1)
1.4 Operations on Generalized Random Processes
245(1)
2 Moments of Generalized Random Processes. Gaussian Processes. Characteristic Functionals
246(16)
2.1 The Mean of a Generalized Random Process
246(2)
2.2 Gaussian Processes
248(4)
2.3 The Existence of Gaussian Processes with Given Means and Correlation Functionals
252(5)
2.4 Derivatives of Generalized Gaussian Processes
257(1)
2.5 Examples of Gaussian Generalized Random Processes
257(3)
2.6 The Characteristic Functional of a Generalized Random Process
260(2)
3 Stationary Generalized Random Processes. Generalized Random Processes with Stationary nth-Order Increments
262(11)
3.1 Stationary Processes
262(1)
3.2 The Correlation Functional of a Stationary Process
263(2)
3.3 Processes with Stationary Increments
265(3)
3.4 The Fourier Transform of a Stationary Generalized Random Process
268(5)
4 Generalized Random Processes with Independent Values at Every Point
273(16)
4.1 Processes with Independent Values
273(2)
4.2 A Condition for the Positive Definiteness of the Functional exp(∫f[ φ(t)]dt)
275(4)
4.3 Processes with Independent Values and Conditionally Positive-Definite Functions
279(4)
4.4 A Connection between Processes with Independent Values at Every Point and Infinitely Divisible Distribution Laws
283(1)
4.5 Processes Connected with Functionals of the nth Order
284(1)
4.6 Processes of Generalized Poisson Type
285(1)
4.7 Correlation Functionals and Moments of Processes with Independent Values at Every Point
286(2)
4.8 Gaussian Processes with Independent Values at Every Point
288(1)
5 Generalized Random Fields
289(14)
5.1 Basic Definitions
289(1)
5.2 Homogeneous Random Fields and Fields with Homogeneous sth-Order Increments
290(2)
5.3 Isotropic Homogeneous Generalized Random Fields
292(2)
5.4 Generalized Random Fields with Homogeneous and Isotropic sth-Order Increments
294(3)
5.5 Multidimensional Generalized Random Fields
297(4)
5.6 Isotropic and Vectorial Multidimensional Random Fields
301(2)
Chapter IV Measures in Linear Topological Spaces
303
1 Basic Definitions
303(9)
1.1 Cylinder sets
303(2)
1.2 Simplest Properties of Cylinder Sets
305(2)
1.3 Cylinder Set Measures
307(2)
1.4 The Continuity Condition for Cylinder Set Measures
309(2)
1.5 Induced Cylinder Set Measures
311(1)
2 The Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Nuclear Spaces
312(23)
2.1 The Additivity of Cylinder Set measures
312(5)
2.2 A Condition for the Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Countably Hilbert Spaces
317(3)
2.3 Cylinder Sets Measures in the Adjoint Spaces of Nuclear Countably Hilbert Spaces
320(10)
2.4 The Countable Additivity of Cylinder Set Measures in Spaces Adjoint to Union Spaces of Nuclear Spaces
330(3)
2.5 A Condition for the Countable Additivity of Measures on the Cylinder Sets in a Hilbert Space
333(2)
3 Gaussian Measures in Linear Topological Spaces
335(10)
3.1 Definition of Gaussian Measures
335(4)
3.2 A Condition for the Countable Additivity of Gaussian Measures in the Conjugate Spaces of Countably Hilbert Spaces
339(6)
4 Fourier Transforms of Measures in Linear Topological Spaces
345(5)
4.1 Definition of the Fourier Transform of a Measure
345(2)
4.2 Positive-Definite Functionals on Linear Topological Spaces
347(3)
5 Quasi-Invariant Measures in Linear Topological Spaces
350
5.1 Invariant and Quasi-Invariant Measures in Finite-Dimensional Spaces
350(4)
5.2 Quasi-Invariant Measures in Linear Topological Spaces
354(5)
5.3 Quasi-Invariant Measures in Complete Metric Spaces
359(3)
5.4 Nuclear Lie Groups and Their Unitary Representations. The Commutation Relations of the Quantum Theory of Fields
362
Notes and References to the Literature 371(6)
Bibliography 377(4)
Subject Index 381(59)
Translator's Note v
Foreword vii
Chapter I 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 Plucker 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 of f(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
157(2)
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)
1.2 Functions on G
204(1)
1.3 Fourier Transform on G
205(2)
1.4 Domain of Definition of F(χ)
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(11)
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, ν u', ν')
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(20)
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 φ(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(Χ)
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, ν u', ν') under Translation and Differentiation of f(g)
258(2)
5.3 Differentiability and Asymptotic Behavior of Φ(u, ν u', ν')
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
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 F+8 and F-8
423(1)
5 Unitary Representations of G
424
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+8 and F-8
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
Notes and References to the Literature 440(2)
Bibliography 442(3)
Index 445
Chapter 1 Homogeneous Spaces with a Discrete Stability Group
1(119)
§1 Generalities
1(16)
1 Homogeneous Spaces and Their Stability Subgroups
1(1)
2 The Connection Between the Homogeneous Spaces X = Γ\G and Riemann Surfaces
2(3)
3 The Fundamental Domain of a Discrete Group Γ
5(3)
4 Discrete Groups with a Compact Fundamental Domain
8(3)
5 The Structure of a Fundamental Domain in the Lobachevskii Plane
11(6)
§2 Representations of a Group G Induced by a Discrete Subgroup
17(16)
1 Definition of Induced Representations
18(2)
2 The Operators Tφ
20(4)
3 The Discreteness of the Spectrum of the Induced · Representation in the Case of a Compact Space X = Γ\G
24(2)
4 The Trace Formula
26(4)
5 Another Form of the Trace Formula
30(3)
§3 Irreducible Unitary Representations of the Group of Real Unimodular Matrices of Order 2
33(10)
1 The Principal Series of Irreducible Unitary Representations
33(2)
2 The Supplementary Series of Representations
35(1)
3 The Discrete Series of Representations
36(1)
4 Another Realization of the Representations of the Principal and Supplementary Series
36(4)
5 The Laplace Operator Δ. The Spaces Ωs
40(3)
§4 The Duality Theorem
43(20)
1 Automorphic Forms
45(2)
2 Statement of the Duality Theorem
47(1)
3 The Laplace Operator
48(2)
4 Proof of the Duality Theorem for Representations of the Continuous Series
50(3)
5 Proof of the Duality Theorem for Representations of the Discrete Series
53(4)
6 The General Duality Theorem
57(6)
§5 The Trace Formula for the Group G of Real Unimodular Matrices of Order 2
63(24)
1 Statement of the Problem
63(2)
2 The Function h
65(2)
3 Contribution of the Hyperbolic Elements to the Trace Formula
67(3)
4 Contribution of the Elliptic Elements
70(5)
5 Contribution of the Elements e and - e to the Trace Formula
75(1)
6 The Final Trace Formula
76(1)
7 Formulae for the Multiplicities of the Representations of the Discrete Series
77(1)
8 Complete Splitting of the Trace Formula
78(1)
9 Construction of the Functions φ+n(g) and φ-n(g)
79(3)
10 The Asymptotic Formula
82(2)
11 The Trace Formula for the Case When - e Does Not Belong to Γ
84(3)
Appendix I to §5 A Theorem on Continuous Deformations of a Discrete Subgroup
87(3)
Appendix II to §5 The Trace Formula for the Group of Complex Unimodular Matrices of Order 2
90(4)
1 Irreducible Unitary Representations of G
90(1)
2 The Trace Formula for G
91(3)
3 The Asymptotic Formula
94(1)
§6 Investigation of the Spectrum of a Representation Generated by a Noncompact Space X = Γ\G (Separation of the Discrete Part of the Spectrum)
94(12)
1 Horospheres in a Homogeneous Space
95(1)
2 Statement of the Main Theorem
96(2)
3 Cylindrical Sets
98(2)
4 Reduction of the Main Theorem
100(1)
5 Proof that the Trace PkTφPk in Hok is Finite
101(5)
Appendix to
Chapter 1 Arithmetic Subgroups of the Group G of Real Unimodular Matrices of Order 2
106(14)
1 Definition of an Arithmetic Subgroup
106(1)
2 The Modular Group
107(4)
3 Some Subgroups of the Modular Group
111(4)
4 Quaternion groups
115(5)
Chapter 2 Representations of the Group of Unimodular Matrices of Order 2 with Elements from a Locally Compact Topological Field
120(122)
§1 Structure of Locally Compact Fields
123(14)
1 Classification of Locally Compact Fields
123(2)
2 The Norm in K
125(1)
3 Structure of Disconnected Fields
126(1)
4 Additive and Multiplicative Characters of K
127(2)
5 The Structure of the Subgroup A. The Functions exp x and In x
129(2)
6 Quadratic Extensions of a Disconnected Field
131(1)
7 The Multiplicative Characters signr x
132(1)
8 Circles in K (√τ)
133(1)
9 Cartesian and Polar Coordinates in K (√τ)
134(1)
10 Invariant Measures on K and in its Quadratic Extension K (√τ)
135(1)
11 Additive and Multiplicative Characters on the "Plane" K√τ
136(1)
§2 Test and Generalized Functions on a Locally Compact Disconnected Field K
137(20)
1 The Space of Test Functions
137(1)
2 Generalized Functions Concentrated at a Point
138(1)
3 Homogeneous Generalized Functions
138(3)
4 The Fourier Transform of Test Functions
141(2)
5 The Fourier Transform of Generalized Homogeneous Functions. The Gamma-Function and Beta-Function
143(2)
6 Additional Information on the Gamma-Function
145(6)
7 The Integral ∫Χ(utt) dt
151(1)
8 Functions Resembling Analytic Functions in the Upper and the Lower Half-Plane
152(1)
9 The Mellin Transform
153(2)
10 The Relation Between the Gamma-Function Connected with the Ground Field K and the Gamma-Function Connected with the Quadratic Extension K(√τ) of K
155(2)
§3 Irreducible Representations of the Group of Matrices of Order 2 with Elements from a Locally Compact Field (the Continuous Series)
157(26)
1 The Continuous Series of Unitary Representations of G
157(2)
2 Another Realization of the Representations of the Continuous Series
159(4)
3 Equivalence of Representations of the Continuous Series
163(1)
4 The Irreducibility of the Representations of the Continuous Series
163(3)
5 The Decomposition of the Representations Tπτ(g), πτ(t) = signτt, into Irreducible Representations
166(1)
6 The Quasiregular Representation of G and its Decomposition into Irreducible Representations
167(2)
7 The Supplementary Series of Irreducible Unitary Representations of G
169(2)
8 The Singular Representation of G
171(1)
9 Representations in the Spaces Dπ
172(2)
10 Spherical Functions
174(2)
11 The Operator of the Horospherical Automorphism
176(7)
§4 The Discrete Series of Irreducible Unitary Representations of G
183(15)
1 Description of the Representations of the Discrete Series
183(2)
2 Continuous Dependence of the Operators Tπ(g) on g
185(2)
3 Proof of the Relation Tπ(g1g2) = Tπ(g1)Tπ(g2)
187(2)
4 Unitariness of the Operators Tπ(g)
189(1)
5 The π-Realization of the Representations of the Discrete Series
190(2)
6 Another Realization of the Representations of the Discrete Series
192(2)
7 Equivalence of Representations of the Discrete Series
194(4)
8 Discrete Series for the Field of 2-adic Numbers
198(1)
§5 The Traces of Irreducible Representations of G
198(11)
1 Statement of the Problem
198(1)
2 The Traces of the Representations of the Continuous Series
199(2)
3 Trace of the Singular Representation
201(1)
4 Traces of the Representations of the Discrete Series
202(5)
5 Traces of the Representations of the Discrete Series for the Field of Real Numbers
207(2)
§6 The Inversion Formula and the Plancherel Formula on G
209(12)
1 Statement of the Problem
209(2)
2 The Inversion Formula for a Disconnected Field
211(5)
3 Computation of Certain Integrals
216(3)
4 Computation of the Constant c in the Inversion Formula
219(1)
5 The Inversion Formulae for Connected Fields
220(1)
Appendix to
Chapter 2
221(21)
1 Some Facts from the Theory of Operator Rings in Hilbert Space
221(3)
2 Connection Between the Unitary Representations of the Group G of all Nonsingular Matrices of Order 2 and the Subgroup of Matrices of the Form (a b 0 1)
224(3)
3 Theorem on the Complete Continuity of the Operator Tφ
227(1)
4 The Decomposition of an Irreducible Representation of G Relative to Representations of its Maximal Compact Subgroup. The Theorem on the Existence of a Trace
228(3)
5 Representations of the Unimodular Group
231(1)
6 Classification of all Irreducible Representations of G and G
232(10)
Chapter 3 Representations of Adele Groups
242(172)
§1 Adeles and Ideles
242(15)
1 The Group of Characters of the Additive Group of Rational Numbers
242(2)
2 Definition of Adeles and Ideles
244(1)
3 Another Construction of the Group of Adeles
245(1)
4 The Isomorphisms Q → A and Q* → A*
246(2)
5 The Group of Additive Characters of the Ring of Adeles A
248(3)
6 The Characters of the Group A/Q
251(1)
7 Invariant Measures in the Group of Adeles and the Group of Ideles
251(1)
8 The Function |λ|
252(1)
9 The Characters of the Group of Ideles A*
253(2)
10 The Characters of the Group A*/Q*
255(2)
Appendix to §1 On a Zeta-Function
257(1)
§2 Analysis on the Group of Adeles
258(11)
1 Schwartz-Bruhat Functions
258(1)
2 The Fourier Transform of Schwartz-Bruhat Functions
259(2)
3 The Poisson Summation Formula
261(1)
4 The Mellin Transform of Schwartz-Bruhat Functions. The Tate Formula
262(5)
5 The Space An
267(2)
Appendix to §2 Tate Rings
269(2)
§3 The Groups of Adeles GA and their Representations
271(12)
1 Definition of the Group of Adeles GA
271(1)
2 Irreducible Unitary Representations of the Group of Adeles
272(2)
3 Proof of a Theorem on Tensor Products
274(4)
4 Criteria for the Existence of a Single Linearly Independent Invariant Vector
278(3)
5 Second Theorem on Tensor Products
281(2)
§4 The Adele Group of the Group of Unimodular Matrices of Order 2
283(59)
1 Statement of the Problem and Summary of the Results
283(3)
2 The Structure of the Space X
286(1)
3 Description of the Space Ω of all Compact Horospheres of X
287(3)
4 Cylindrical Sets
290(3)
5 The Horospherical Map
293(1)
6 Investigation of the Kernel of the Horospherical Map (Discreteness of the Spectrum)
294(2)
7 The Spaces Y, Ω and E
296(3)
8 The Operation of Multiplication in the Spaces A2, Y and E
299(2)
9 Decomposition of the Representations Generated by Y and Ω into Irreducible Representations
301(5)
10 The Operator B (Definition)
306(2)
11 Properties of the Operator B
308(3)
12 Schwartz-Bruhat Functions in Ω
311(6)
13 The Fourier Transform in L2(Ω)
317(6)
14 The Operator M
323(2)
15 An Explicit Expression for M
325(3)
16 The Family M of Functions on Ω
328(7)
17 Decomposition of the Representation in H' into Irreducible Representations
335(2)
18 Connection of the Operator of the Horospherical Automorphism B with Dirichlet L-Functions
337(5)
Appendix I to §4
342(10)
1 Lemma on the Completeness of the Family Φ∞
343(4)
2 Lemma on Functions Defined on the Half-Line 0 ≤ τ < ∞ and Belonging to L2
347(5)
Appendix II to §4
352(9)
1 On the Connection Between the Homogeneous Space GQ\GA and the Homogeneous Spaces of the Group G∞
352(4)
2 The Generalized Peterson Conjecture
356(5)
§5 The Space of Horospheres
361(17)
1 Reductive Algebraic Groups
361(2)
2 The Space L2(DQZA\GA)
363(5)
3 The Operators Bs
368(3)
4 Properties of the Operators Bs
371(2)
5 Main Theorem on the Operators Bs
373(3)
6 Reduction to Rank 1
376(2)
§6 Representations Generated by the Homogeneous Space GQ\GA
1 The Homogeneous Space GQ\GA
378(1)
2 Investigation of the Spectrum of the Representation for a Compact Space GQ\GA/KA
379(2)
3 The Space of Horospheres
381(1)
4 The Horospherical Map and the Operator M
382(1)
5 An Explicit Expression for the Operator M
383(1)
6 The Structure of the Space H'
384(2)
§7 Discreteness of the Spectrum
386(21)
1 Horospheres in the Space X = GQ\GA
386(3)
2 Statement of the Main Theorem
389(1)
3 Siegel Sets on GA
390(2)
4 Regular Siegel Sets
392(3)
5 Regular Siegel Sets Connected with Π-Horospheres
395(2)
6 Reduction of the Main Theorem
397(2)
7 The p-Norm
399(1)
8 Proof of the Main Theorem
400(2)
9 Solvable Algebras and Groups. Statement of the Fundamental Lemma
402(2)
10 Proof of the Fundamental Lemma
404(3)
Appendix to §7 Functions on Regular Nilpotent Lie Groups
407(7)
1 Regular Nilpotent Algebras
407(2)
2 Regular Nilpotent Lie Groups
409(5)
Guide to the Literature 414(3)
Bibliography 417(4)
Index of Names 421(2)
Subject Index 423