Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions (Ams Chelsea Publishing)

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. Gel'fand 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. Volume 2 is devoted to detailed study of generalized functions as linear functionals on appropriate spaces of smooth test functions. In Chapter 1, the authors introduce and study countable-normed linear topological spaces, laying out a general theoretical foundation for the analysis of spaces of generalized functions. The two most important classes of spaces of test functions are spaces of compactly supported functions and Schwartz spaces of rapidly decreasing functions. In Chapters 2 and 3 of the book, the authors transfer many results presented in Volume 1 to generalized functions corresponding to these more general spaces. Finally, Chapter 4 is devoted to the study of the Fourier transform; in particular, it includes appropriate versions of the Paley-Wiener theorem. Cover Title page Preface to the Russian Edition Contents Chapter I Linear Topological Spaces 1. Definition of a Linear Topological Space 2. Normed Spaces. Comparability and Compatibility of Norms 3. Countably Normed Spaces 4. Continuous Linear Functionals and the Conjugate Space 5. Topology in a Conjugate Space 6. Perfect Spaces 7. Continuous Linear Operators 8. Union of Countably Normed Spaces Appendix 1. Elements, Functionals, Operators Depending on a Parameter Appendix 2. Differentiable Abstract Functions Appendix 3. Operators Depending on a Parameter Appendix 4. Integration of Continuous Abstract Functions with Respect to the Parameter Chapter II Fundamental and Generalized Functions 1. Definition of Fundamental and Generalized Functions 2. Topology in the Spaces K{MV) and Z{MV) 3. Operations with Generalized Functions 4. Structure of Generalized Functions Chapter III Fourier Transformations of Fundamental and Generalized Functions 1. Fourier Transformations of Fundamental Functions 2. Fourier Transforms of Generalized Functions 3. Convolution of Generalized Functions and Its Connection to Fourier Transforms 4. Fourier Transformation of Entire Analytic Functions Chapter IV Spaces of Type S 1. Introduction 2. Various Modes of Defining Spaces of Type S 3. Topological Structure of Fundamental Spaces 4. Simplest Bounded Operations in Spaces of Type S 5. Differential Operators 6. Fourier Transformations 7. Entire Analytic Functions as Elements or Multipliers in Spaces of Type S 8. The Question of the Nontriviality of Spaces of Type S 9. The Case of Several Independent Variables Appendix 1. Generalization of Spaces of Type S Appendix 2. Spaces of Type W Notes and References Bibliography Index Volume 1. Properties And Operations -- Volume 2. Spaces Of Fundamental And Generalized Functions / I.m. Gel'fand, G.e. Shilov ; Translated By Morris D. Friedman, Amiel Feinstein, Christian P. Peltzer -- Volume 3. Theory Of Differential Equations / I.m. Gel'fand, G.e. Shilov ; Translated By Meinhard E. Mayer -- Volume 4. Applications Of Harmonic Analysis / I.m. Gel'fand, N. Ya. Vilenkin ; Translated By Amiel Feinstein -- Volume 5. Integral Geometry And Representation Theory / I.m. Gel'fand, M.i. Graev, N. Ya. Vilenkin ; Translated By Eugene Saletan -- Volume 6. Representation Theory And Automorphic Functions / I.m. Gel'fand, M.i. Graev, I.i. Pyatetskii-shapiro ; Translated By K.a. Hirsch. I.m. Gel'fand, G.e. Shilov ; Translated By Eugene Saletan. Originally Published In Russian In 1958. Originally Published In English As 5 Volume Set: New York : Academic Press, 1964-[1968]. Includes Bibliographical References And Index. 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. Gel2 and 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. Volume 2 volume 1. Properties and operations volume 2. Spaces of fundamental and generalized functions / I.M. Gel'fand, G.E. Shilov ; translated by Morris D. Friedman, Amiel Feinstein, and Christian P. Peltzer volume 3. Theory of differential equations / I.M. Gel'fand, G.E. Shilov ; translated by Meinhard E. Mayer volume 4. Applications of harmonic analysis / I.M. Gel'fand, N. Ya. Vilenkin ; translated by Amiel Feinstein volume 5. Integral geometry and representation theory / I.M. Gel'fand, M.I. Graev, N. Ya. Vilenkin ; translated by Eugene Saletan The six-volume collection, Generalized Functions, 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. Volume 1 is devoted to basics of the theory of generalized functions.