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 nonArchimedean functional analysis, which is becoming more important nowadays due to the development of nonArchimedean mathematical physics, particularly, quantum mechanics, quantum field theory, theory of superstrings and supergravity (VV89, VVZ94, ADV88, Cas02, DD00, Ish84, Khr90, Lud99t, Lud03b, Mil84, Jan 98). Recently nonArchimedean 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 nonArchimedean stochastic processes (Lud0321, Lud0341, Lud0348,Lud01f, LK02). Stochastic approach in quantumfield theory is actively used and investigated especially in recent years (see, for example, and references therein (AHKMT93, AHKT84]). As it is wellknown in the theory of functions great role is played by continuous functions and differentiable functions.
In classical measure theory, the analog of continuity is quasiinvariance relative to shifts and actions of linear or nonlinear 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 nonArchimedean case was less studied. Since there are not any nontrivial differentiable functions from the padic field Qp into R or into another padic nonArchimedean field Qp' with p not equal to p', then instead of differentiability of measures their pseudodifferentiability is considered.
Traditional or classical mathematical analysis and functional analysis work mainly over the real and complex fields. But there are wellknown many other infinite fields with nontrivial multiplicative norms since the end of the 19th century and is called the nonArchimedean norm. Such fields and vector spaces with nonArchimedean norms are frequently called for short nonArchimedean fields and nonArchimedean normed spaces correspondingly. Therefore, mathematical analysis and functional analysis over nonArchimedean 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. Realvalued measures, pp.1
1.1 Introduction, pp. 1
1.2 Distributions and families of measures, pp. 3
1.3 Quasiinvariant measures, pp. 28
1.4 Pseudodifferentiable measures, pp. 57
1.5 Convergence of measures, pp. 67
1.6 Measures with particular properties, pp. 73
1.7 Comments, pp. 86
2. NonArchimedean valued measures, pp. 91
2.1 Introduction, pp. 91
2.2 NonArchimedean valued distributions, pp. 92
2.3 Quasiinvariant Ksvalued measures, pp. 109
2.4 Pseudodifferentiable Ksvalued measures, pp. 121
2.5 Convergence of Ksvalued 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 nonArchimedean 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. NonArchimedean polyhedral expansions, pp. 179
B.1 Ultrauniform 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: 9781606927342 
Status: AV 

Status Code 
Description 
AN 
Announcing 
FM 
Formatting 
PP 
Page Proofs 
FP 
Final Production 
EP 
Editorial Production 
PR 
At Prepress 
AP 
At Press 
AV 
Available 



