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Generalized Functions, Volume 4: Applications of Harmonic Analysis [Kõva köide]

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  • Formaat: Hardback, 384 pages, kõrgus x laius: 254x178 mm, kaal: 857 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470426625
  • ISBN-13: 9781470426620
Teised raamatud teemal:
Generalized Functions, Volume 4: Applications of Harmonic AnalysisSuurem pilt
  • Formaat: Hardback, 384 pages, kõrgus x laius: 254x178 mm, kaal: 857 g
  • Sari: Chelsea Publications
  • Ilmumisaeg: 30-Apr-2016
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470426625
  • ISBN-13: 9781470426620
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470426620.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 main goal of Volume 4 is to develop the functional analysis setup for the universe of generalized functions. The main notion introduced in this volume is the notion of rigged Hilbert space (also known as the equipped Hilbert space, or Gelfand triple). Such space is, in fact, a triple of topological vector spaces $E \subset H \subset E'$, where $H$ is a Hilbert space, $E'$ is dual to $E$, and inclusions $E\subset H$ and $H\subset E'$ are nuclear operators. The book is devoted to various applications of this notion, such as the theory of positive definite generalized functions, the theory of generalized stochastic processes, and the study of measures on linear topological spaces.
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(1)
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(68)
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(21)
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(9)
Notes and References to the Literature 371(6)
Bibliography 377(4)
Subject Index 381