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.
Part 1 Introduction: terminology and notation; elementary facts on complex topological vector spaces; a review of basic facts in the theory of distributions. Part 2 Mellin distributions and the Mellin transformation: the Fourier and the Fourier-Mellin transformations; the spaces of Mellin distributions with support in a polyinterval; operations of multiplication and differentiation in the space of Mellin distributions; the Mellin transformation in the space of Mellin distributions; the structure of Mellin distributions; Paley-Wiener type theorems for the Mellin transformation; Mellin transforms of cut-off functions (cont'd); important subspace of Mellin distributions; the modified Cauchy transformation. Part 3 Fuchsian type singular operations: Fuchsian type ordinary differential operators; elliptic Fuchsian type partial differential equations in spaces M's(w); Fuchsian type partial differential equations in space with continuous radial asymptotics. Appendix: Generalized smooth functions and theory of resurgent functions of Jean Ecalle.