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Distribution Theory And Applications : Series on Concrete and Applicable Mathematics - Abdellah El Kinani

Distribution Theory And Applications

Series on Concrete and Applicable Mathematics

Hardcover Published: 8th May 2010
ISBN: 9789814304917
Number Of Pages: 202

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This book is an introductory course to the very important theory of distributions, as well as its applications in the resolution of partial differential equations (PDEs). It begins with a chapter of general interest, on the fundamental spaces (or test function spaces). The book advances and concludes with a chapter on Sobolev spaces, which are known to be very important in the resolution of PDEs.

The very basic properties of distributions are examined in detail. Several formal methods have been first used, without rigorous justifications (Dirac function, principal value of Cauchy, finite parts of Hadamard). They find their natural frame in distribution theory. It is the same for Laplace transformation which is a fundamental tool in symbolic calculations.

A detailed treatment is given to the convolution product for it is a central theme in distribution theory. Another very important instrument, covered in several chapters, is the Fourier transformation which is among the most fundamental tools in different mathematical disciplines, and also in physics. Convolution algebras, which are sufficient for the treatment of classical PDEs, are used in various applications.

The general frame for the resolution of PDEs is the theory of kernels — the first elements of which are sufficient to show the practicality of distribution theory in applications.

Comments are provided to clarify the settings and sustain calculations. This book may be used by mathematicians, physicists, engineers and graduate students.

Prefacep. vii
Introductionp. vii
Acknowledgementsp. xv
Spaces of Functionsp. 1
The space C (¿)p. 1
The space ¿m (¿)p. 2
The space ¿ (¿)p. 3
The space DKm (¿)p. 4
The space DK (¿)p. 4
The space Dm (¿)p. 5
The Topology of Dm (¿)p. 6
Density resultsp. 11
Distributionsp. 17
Definitions and characterizationsp. 17
Examplesp. 19
Embedding of Lloc1 (¿) in D' (¿)p. 22
Support of a distributionp. 23
Pasting principlep. 25
Distributions with compact supportp. 26
Operations on distributionsp. 27
Topology in D' (¿)p. 33
Continuity of differential operatorsp. 34
Tensor Productp. 37
Tensor product of functionsp. 37
Tensor product of distributionsp. 38
Properties of tensor productp. 43
Convolution of Distributionsp. 47
Regular distributions casep. 47
Convolution of distributionsp. 48
Properties of the convolution productp. 52
Convolution algebrasp. 56
Convolution equationsp. 57
Fourier Transformation in L1 (Rn)p. 67
Definitions and first propertiesp. 67
Inversion Theoremp. 71
Tempered Distributionsp. 79
The space S (Rn)p. 79
Multiplication operatorsp. 86
Fourier transformation in S (Rn)p. 89
Tempered distributionsp. 89
Convolution operatorsp. 94
Fourier transformation of tempered distributionsp. 98
Ultradistributionsp. 102
Structure of Distributionsp. 113
Local structurep. 113
Global structurep. 115
Structure of tempered distributionsp. 120
Fourier Ranges of Some Subspaces of S' (Rn)p. 125
Fourier range of L2 (Rn)p. 125
Fourier range of L1 (Rn)p. 128
Fourier range of OM(Rn)p. 128
Fourier range of D(Rn)p. 131
Fourier range of ¿'(Rn)p. 133
Laplace Transformationp. 139
Laplace transforms of functions of real variablesp. 139
Laplace transforms of distributionsp. 146
Introduction to the Theory of Kernelsp. 157
The notion of a kernelp. 157
Regular kernelsp. 159
Regularizing kernelsp. 162
Very regular kernelsp. 163
Fundamental kernels of partial differential operatorsp. 164
Hypoelliptic operatorsp. 167
Sobolev Spacesp. 175
The spaces Hm(¿), m ∈ Np. 175
The spaces H0m (¿), m ∈ Np. 180
The spaces L2(¿s), s ∈ Rp. 185
The spaces Hs (Rn), s ∈ Rp. 188
Injection theorem of Sobolevp. 190
The compact injection theoremp. 192
Bibliographyp. 197
List of Symbolsp. 199
Indexp. 201
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814304917
ISBN-10: 9814304913
Series: Series on Concrete and Applicable Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 202
Published: 8th May 2010
Country of Publication: SG
Dimensions (cm): 22.86 x 15.49  x 2.03
Weight (kg): 0.57