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 transformations, ie 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 1 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 2 presents a systematic treatment of
the Mellin transform in several dimensions. Chapter 3 is devoted to
Fuchsian-type singular differential equations. While aimed at
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.
General
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