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Theory of P-Adic Distributions Linear and Nonlinear Models

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ISBN-10: 0521148561

ISBN-13: 9780521148566

Edition: 2010

Authors: S. Albeverio, A. Yu Khrennikov, V. M. Shelkovich

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Book details

List price: $93.95
Copyright year: 2010
Publisher: Cambridge University Press
Publication date: 3/18/2010
Binding: Paperback
Pages: 368
Size: 5.75" wide x 8.75" long x 0.75" tall
Weight: 1.144
Language: English

A. Yu. Khrennikov is Professor of Applied Mathematics and Director of the International Center for Mathematical Modeling in Physics, Engineering and Cognitive Sciences at V+xj_ University, Sweden.

V. M. Shelkovich is Professor in the Mathematical Department at the St Petersburg State University of Architecture and Civil Engineering, Russia.

Preface
p-adic numbers
Introduction
Archimedean and non-Archimedean normed fields
Metrics and norms on the field of rational numbers
Construction of the completion of a normed field
Construction of the field of p-adic numbers Q<sub>p</sub>
Canonical expansion of p-adic numbers
The ring of p-adic integers Z<sub>p</sub>
Non-Archimedean topology of the field Q<sub>p</sub>
Q<sub>p</sub> in connection with R
The space Q<sup>n</sup><sub>p</sub>
p-adic functions
Introduction
p-adic power series
Additive and multiplicative characters of the field Q<sub>p</sub>
p-adic integration theory
Introduction
The Haar measure and integrals
Some simple integrals
Change of variables
p-adic distributions
Introduction
Locally constant functions
The Bruhat-Schwartz test functions
The Bruhat-Schwartz distributions (generalized functions)
The direct product of distributions
The Schwartz "kernel" theorem
The convolution of distributions
The Fourier transform of test functions
The Fourier transform of distributions
Some results from p-adic L<sup>1</sup> - and L<sup>2</sup>-theories
Introduction
L<sup>1</sup>-theory
L<sup>2</sup>-theory
The theory of associated and quasi associated homogeneous p-adic distributions
Introduction
p-adic homogeneous distributions
p-adic quasi associated homogeneous distributions
The Fourier transform of p-adic quasi associated homogeneous distributions
New type of p-adic �-functions
p-adic Lizorkin spaces of test functions and distributions
Introduction
The real case of Lizorkin spaces
p-adic Lizorkin spaces
Density of the Lizorkin spaces of test functions in L<sup>p</sup>(Q<sub>n</sub><sub>p</sub>)
The theory of p-adic wavelets
Introduction
p-adic Haar type wavelet basis via the real Haar wavelet basis
p-adic multiresolution analysis (one-dimensional case)
Construction of the p-adic Haar multiresolution analysis
Description of one-dimensional 2-adic Haar wavelet bases
Description of one-dimensional p-adic Haar wavelet bases
p-adic refinable functions and multiresolution analysis
p-adic separable multidimensional MRA
Multidimensional p-adic Haar wavelet bases
One non-Haar wavelet basis in L<sup>2</sup>(Q<sub>p</sub>)
One infinite family of non-Haar wavelet bases in L<sup>2</sup>(Q<sub>p</sub>)
Multidimensional non-Haar p-adic wavelets
The p-adic Shannon-Kotelnikov theorem
p-adic Lizorkin spaces and wavelets
Pseudo-differential operators on the p-adic Lizorldn spaces
Introduction
p-adic multidimensional fractional operators
A class of pseudo-differential operators
Spectral theory of pseudo-differential operators
Pseudo-differential equations
Introduction
Simplest pseudo-differential equations
Linear evolutionary pseudo-differential equations of the first order in time
Linear evolutionary pseudo-differential equations of the second order in time
Semi-linear evolutionary pseudo-differential equations
A p-adic Schr�dinger-type operator with point interactions
Introduction
The equation D<sup>�</sup> - �I = �<sub>x</sub>
Definition of operator realizations of D<sup>�</sup> + V in L<sub>2</sub>(Q<sub>p</sub>)
Description of operator realizations
Spectral properties
The case of �-self-adjoint operator realizations
The Friedrichs extension
Two points interaction
One point interaction
Distributional asymptotics and p-adic Tauberian theorems
Introduction
Distributional asymptotics
p-adic distributional quasi-asymptotics
Tauberian theorems with respect to asymptotics
Tauberian theorems with respect to quasi-asymptotics
Asymptotics of the p-adic singular Fourier integrals
Introduction
Asymptotics of singular Fourier integrals for the real case
p-adic distributional asymptotic expansions
Asymptotics of singular Fourier integrals (�<sub>1</sub>(x) &#8801; 1)
Asymptotics of singular Fourier integrals (�<sub>1</sub>(x) &#8800; 1)
p-adic version of the Erd�lyi lemma
Nonlinear theories of p-adic generalized functions
Introduction
Nonlinear theories of distributions (the real case)
Construction of the p-adic Colombeau-Egorov algebra
Properties of Colombeau-Egorov generalized functions
Fractional operators in the Colombeau-Egorov algebra
The algebra A* of p-adic asymptotic distributions
A* as a subalgebra of the Colombeau-Egorov algebra
The theory of associated and quasi associated homogeneous real distributions
Introduction
Definitions of associated homogeneous distributions and their analysis
Symmetry of the class of distributions AH<sub>0</sub>(R)
Real quasi associated homogeneous distributions
Real multidimensional quasi associated homogeneous distributions
The Fourier transform of real quasi associated homogeneous distributions
New type of real �-functions
Two identities
Proof of a theorem on weak asymptotic expansions
One "natural" way to introduce a measure on Q<sub>p</sup>
References
Index