The Mellin Transformation and Fuchsian Type Partial Differential Equations

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 222 pages
This volume provides a systematic introduction to the theory of the multidimensional Mellin transformation in a distributional setting. In contrast to the classical texts on the Mellin and Laplace transformations, this work concentrates on the local properties of the Mellin transforms, i.e. on those properties of the Mellin transforms of distributions u which are preserved under multiplication of u by cut-off functions (of various types). The main part of the book is devoted to the local study of regularity of solutions to linear Fuchsian partial differential operators on a corner, which demonstrates the appearance of non-discrete asymptotic expansions (at the vertex) and of resurgence effects in the spirit of J. Ecalle. The book constitutes a part of a program to use the Mellin transformation as a link between the theory of second micro-localization, resurgence theory and the theory of the generalized Borel transformation. Chapter I contains the basic theorems and definitions of the theory of distributions and Fourier transformations which are used in the succeeding chapters. This material includes proofs which are partially transformed into exercises with hints. Chapter II presents a systematic treatment of the Mellin transform in several dimensions. Chapter III is devoted to Fuchsian-type singular differential equations. For researchers and graduate students interested in differential equations and integral transforms. This book can also be recommended as a graduate text for students of mathematics and engineering.
 

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Contents

Mellin distributions and the Mellin transformation
27
5 The spaces of Mellin distributions with support in a polyinterval
34
6 Operations of multiplication and differentiation in the space
48
7 The Mellin transformation in the space of Mellin distributions
54
8
76
10 Mellin transforms of cutoff functions continued
100
11 Important subspaces of Mellin distributions
115
12 The modified Cauchy transformation
125
Fuchsian type singular operators
139
14 Elliptic Fuchsian type partial differential equations in spaces Mſ
175
15 Fuchsian type partial differential equations in spaces with continuous
188
Appendix Generalized smooth functions and theory of resurgent
205
Bibliography
215
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