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# Tauberian Theorems for Generalized Functions

## byV.s. Vladimirov, Yu.N. Drozzinov, O.I. Zavialov

### Paperback | October 2, 2011

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Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. The Scandal of Father G. K. Chesterton. 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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Title:Tauberian Theorems for Generalized FunctionsFormat:PaperbackPublished:October 2, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401077746

ISBN - 13:9789401077743

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Table of Contents

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 Björken domain.- 4. Asymptotic properties of the two-point Wightman function.- Comments on Chapter 5.