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Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces
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Authors: Sergey Ludkovsky (Dept. of Applied Mathematics, Moscow State Technical Univ., MIREA, Moscow, Russia) 
Book Description:
This book is devoted to new results of investigations of non-Archimedean functional analysis, which is becoming more important nowadays due to the development of non-Archimedean mathematical physics, particularly, quantum mechanics, quantum field theory, theory of super-strings and super-gravity (VV89, VVZ94, ADV88, Cas02, DD00, Ish84, Khr90, Lud99t, Lud03b, Mil84, Jan 98). Recently non-Archimedean analysis was found to be useful in dynamical systems, mathematical biology, mathematical psychology, cryptology and information theory. On the other hand, quantum mechanics is based on measure theory and probability theory. The results of this
book published mainly in papers [Lud02a, Lud03s2, Lud04a, Lud96c, Lud99a, Lud00a, Lud99t, Lud01f, Lud00f, Lud99s, Lud04b] have served for investigations of non-Archimedean stochastic processes (Lud0321, Lud0341, Lud0348,Lud01f, LK02). Stochastic approach in quantum-field theory is actively used and investigated especially in recent years (see, for example, and references therein (AHKMT93, AHKT84]). As it is well-known in the theory of functions great role is played by continuous functions and differentiable functions.
In classical measure theory, the analog of continuity is quasi-invariance relative to shifts and actions of linear or non-linear operators in the Banach space. Moreover, differentiability of measures is the stronger condition and there is a very large theory about it in the classical case. Apart from it, the non-Archimedean case was less studied. Since there are not any non-trivial differentiable functions from the p-adic field Qp into R or into another p-adic non-Archimedean field Qp' with p not equal to p', then instead of differentiability of measures their pseudo-differentiability is considered.
Traditional or classical mathematical analysis and functional analysis work mainly over the real and complex fields. But there are well-known many other infinite fields with non-trivial multiplicative norms since the end of the 19-th century and is called the non-Archimedean norm. Such fields and vector spaces with non-Archimedean norms are frequently called for short non-Archimedean fields and non-Archimedean normed spaces correspondingly. Therefore, mathematical analysis and functional analysis over non-Archimedean fields have developed already during a rather long period of time, but they remain substantially less elaborated in comparison with classical ones

Table of Contents:
Preface

Acknowledgement

Notation

1. Real-valued measures, pp.1
1.1 Introduction, pp. 1
1.2 Distributions and families of measures, pp. 3
1.3 Quasi-invariant measures, pp. 28
1.4 Pseudo-differentiable measures, pp. 57
1.5 Convergence of measures, pp. 67
1.6 Measures with particular properties, pp. 73
1.7 Comments, pp. 86

2. Non-Archimedean valued measures, pp. 91
2.1 Introduction, pp. 91
2.2 Non-Archimedean valued distributions, pp. 92
2.3 Quasi-invariant Ks-valued measures, pp. 109
2.4 Pseudo-differentiable Ks-valued measures, pp. 121
2.5 Convergence of Ks-valued measures, pp. 124
2.6 Measures with particular properties, pp. 128
2.7 Comments, pp. 140

3. Algebras of real measures on groups, pp. 143
3.1 Introduction, pp. 143
3.2 Algebras of measures and functions, pp. 143
3.3 Commentspp, pp. 152

4. Algebras of non-Archimedean measures on groups, pp. 153
4.1 Introduction, pp. 153
4.2 Algebras of measures and functions, pp. 153
4.3 Comments, pp. 162

A. Operators in Banach spaces, pp. 173

B. Non-Archimedean polyhedral expansions, pp. 179
B.1 Ultra-uniform spaces, pp. 179
B.2 Polyhedral expansions, pp. 184

References

Index

   Series:
      Mathematics Research Developments
   Binding: Hardcover
   Pub. Date: 2009
   Pages: 7 x 10, 198 pp
   ISBN: 978-1-60692-734-2
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
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Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces