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E-raamat: Boundary Values And Convolution In Ultradistribution Spaces

(Wake Forest Univ, Usa), (Univ Of Rzeszow, Poland), (Univ Of Novi Sad, Serbia)
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This book provides the construction and characterization of important ultradistribution spaces and studies properties and calculations of ultradistributions such as boundedness and convolution. Integral transforms of ultradistributions are constructed and analyzed. The general theory of the representation of ultradistributions as boundary values of analytic functions is obtained and the recovery of the analytic functions as Cauchy, Fourier-Laplace, and Poisson integrals associated with the boundary value is proved.Ultradistributions are useful in applications in quantum field theory, partial differential equations, convolution equations, harmonic analysis, pseudo-differential theory, time-frequency analysis, and other areas of analysis. Thus this book is of interest to users of ultradistributions in applications as well as to research mathematicians in areas of analysis.
Preface v
1. Cones in Rn and Kernels 1
1.1 Notation
1
1.2 Cones in Rn
4
1.3 Cauchy and Poisson kernels
7
2. Ultradifferentiable Functions and Ultradistributions 13
2.1 Sequences (Mp)
13
2.2 Ultradifferential operators
17
2.3 Functions and ultradistributions of Beurling and Roumieu type
20
2.4 Fourier transform on D(*, LS) and D'(*, Ls)
26
2.5 Ultradifferentiable functions of ultrapolynomial growth
28
2.6 Tempered ultradistributions
37
2.7 Laplace transform
40
3. Boundedness 41
3.1 Boundedness in D'(*, LS)
41
3.2 Boundedness in S'*
47
4. Cauchy and Poisson Integrals 51
4.1 Cauchy and Poisson kernels as ultradifferentiable functions
51
4.2 Cauchy integral of ultradistributions
61
4.3 Poisson integral of ultradistributions
76
5. Boundary Values of Analytic Functions 81
5.1 Generalizations of Hr functions in tubes
81
5.2 Boundary values in D' ((MP), L8) for analytic functions in tubes
91
5.3 Case 2 less than r less than infinity 111
5.4 Boundary values via almost analytic extensions
118
5.5 Cases s = infinity and s = 1
129
6. Convolution of Ultradistributions 135
6.1 Introduction
135
6.2 Definitions of D'(Mp)—convolution
137
6.3 Equivalence of definitions of D'(Mp)—convolution
140
6.4 Definitions of S'(Mp) —convolution
147
6.5 Equivalence of definitions of S'(Mp) —convolution
150
6.6 Existence of D'(Mp) — and S'(Mp) — convolution
153
6.7 Compatibility conditions on supports
157
6.8 Convolution in weighted spaces
162
7. Integral Transforms of Tempered Ultradistributions 173
7.1 Introductory remarks
173
7.2 Definitions
174
7.3 Characterizations of some integral transforms
179
7.4 Laplace transform
180
7.5 Proof of equivalence of families of norms
182
7.6 Hilbert transform
186
7.6.1 One-dimensional case
187
7.6.2 Multi-dimensional case
196
7.7 Singular integral operators
199
Bibliography 205
Index 213
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