توابع عمومی، جلد سوم: نظریه معادلات دیفرانسیل

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. In Volume 3, applications of generalized functions to the Cauchy problem for systems of partial differential equations with constant coefficients are considered. The book includes the study of uniqueness classes of solutions of the Cauchy problem and the study of classes of functions where the Cauchy problem is well posed. The last chapter of this volume presents results related to spectral decomposition of differential operators related to generalized eigenfunctions. Cover Title page Translator's Note Preface to the Russian Edition Contents Chapter I Spaces of Type W 1. Definitions 2. Bounded Operators in Spaces of Type W 3. Fourier Transforms 4. The Case of Several Variables Chapter II Uniqueness Classes for the Cauchy Problem 1. Introduction 2. The Cauchy Problem in a Topological Vector Space 3. The Cauchy Problem for Systems of Partial Differential Equations. The Operator Method 4. The Cauchy Problem for Systems of Partial Differential Equations. The Method of Fourier Transforms 5. Examples 6. The Connection between the Reduced Order of a System and Its Characteristic Roots 7. A Theorem of the Phragmen-Lindelof Type Appendix I. Convolution Equations Appendix 2. Equations with Coefficients Which Depend on ? Appendix 3. Systems with Elliptic Operators Chapter III Correctness Classes for the Cauchy Problem 1. Introduction 2. Parabolic Systems 3. Hyperbolic Systems 4. Systems Which Are Petrovskii-Correct 5. On the Solutions of Incorrect Systems Chapter IV Generalized Eigenfunction Expansions 1. Introduction 2. Differentiation of Functionals of Strongly Bounded Variation 3. Differentiation of Functionals of Weakly Bounded Variation 4. Existence and Completeness Theorems for the System of Eigenfunctionals 5. Generalized Eigenfunctions of Self-Adjoint Operators 6. The Structure of the Generalized Eigenfunctions 7. Dynamical Systems Notes and References Bibliography 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. In Volum 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 2 is devoted to detailed study of generalized functions as linear functionals on appropriate spaces of smooth test functions. 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.