Tauberian theorems for generalized functions: V.S. Vladimorov, Yu.N. Drozzinov and B.I. Zavialov: (mathematics and its applications—soviet series. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988, 293 pp. Dfl185.00/U.S.$89.00/£57.25, ISBN 90-277-2383-4

Abstract

Notation and Definitions.- 1: Some Facts on the Theory of Distributions.- 1. Distributions and their properties.- 1. Spaces of test functions.- 2. The space of distributions D?(O).- 3. The space of distributions S?(F).- 4. Linear operations on distributions.- 5. Change of variables.- 6. L -invariant distributions.- 7. Direct product of distributions.- 8. Convolution of distributions.- 9. Convolution algebras of distributions.- 2. Integral transformations of distributions.- 1. The Fourier transform of tempered distributions.- 2. Fourier series of periodic distributions.- 3. The B -transform of distributions.- 4. Fractional derivatives (primitives).- 5. The Laplace transform of tempered distributions.- 6. The Cauchy kernel of the tube domain TC.- 7. Regular cones.- 8. Fractional derivatives (primitives) with respect to a cone.- 9. The Radon transform of distributions with compact support in an odd-dimensional space.- 3. Quasi-asymptotics of distributions.- 1. General definitions and basic properties.- 2. Automodel (regularly varying) functions.- 3. Quasi-asymptotics over one-parameter groups of transformations.- 4. The one-dimensional case. Quasi-asymptotics at infinity and at zero.- 5. The one-dimensional case. Asymptotics by translations.- 6. Quasi-asymptotics by selected variable.- 2: Many-Dimensional Tauberian Theorems.- 4. The General Tauberian theorem and its consequences.- 1. The Tauberian theorem for a family of linear transformations.- 2. The general Tauberian theorem for the dilatation group.- 3. Tauberian theorems for nonnegative measures.- 4. Tauberian theorems for holomorphic functions of bounded argument.- 5. Admissible and strictly admissible functions.- 1. Families of linear transformations under which a cone is invariant.- 2. Strictly admissible functions for a family of linear transformations.- 3. Admissible functions of a cone.- 4. Some examples of admissible functions of a cone.- 6. Comparison Tauberian theorems.- 1. Preliminary theorems.- 2. The comparison Tauberian theorems for measures and for holomorphic functions with nonnegative imaginary part.- Comments on Chapter 2.- 3: One-Dimensional Tauberian Theorems.- 7. The general Tauberian theorem and its consequences.- 1. The general Tauberian theorem and its particular cases.- 2. Quasi-asymptotics of a distribution f from S+? and a function arg f?.- 3. Tauberian theorem for distributions from the class .- 4. The decomposition theorem.- 8. Quasi-asymptotic properties of distributions at the origin.- 1. The general case.- 2. Quasi-asymptotics of distributions from H and asymptotic properties of the reproducting functions of measures.- 9. Asymptotic properties of the Fourier transform of distributions from M+.- 1. Asymptotic properties of the Fourier transform of finite measures.- 2. Asymptotic properties of the Fourier transform of distributions from M+.- 3. The Abel and Cezaro series summation with respect to an automodel weight.- 10. Quasi-asymptotic expansions.- 1. Open and closed quasi-asymptotic expansions.- 2. Quasi-asymptotic expansions and convolutions.- 4: Asymptotic Properties of Solutions of Convolutions Equations.- 11. Quasi-asymptotics of the fundamental solutions of convolution equations.- 1. Quasi-asymptotics and convolution.- 2. Quasi-asymptotics of the fundamental solutions of hyperbolic operators with constant coefficients.- 3. Quasi-asymptotics of the solutions of the Cauchy problem for the heat equation.- 12. Quasi-asymptotics of passive operators.- 1. The translationally-invariant passive operators.- 2. The fundamental solution and the Cauchy problem.- 3. Quasi-asymptotics of passive operators and their fundamental solutions.- 4. Differential operators of the passive type.- 5. Examples.- Comments on Chapter 4.- 5: Tauberian Theorems for Causal Functions.- 13. The Jost-Lehmann-Dyson representation.- 1. The Jost-Lehmann-Dyson representation in the symmetric case.- 2. Inversion of the Jost-Lehmann-Dyson representation in the symmetric case.- 3. The Jost-Lehmann-Dyson representation in the general case.- 14. Automodel asymptotics for the causal functions and singularities of their Fourier transforms on the light cone.- 1. Some preliminary results and definitions.- 2. The main theorems.- 3. On forbidden asymptotics in the Bjorken domain.- 4. Asymptotic properties of the two-point Wightman function.- Comments on Chapter 5.