## The Mellin Transformation and Fuchsian Type Partial Differential EquationsThis 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 |

215 | |

### Other editions - View all

The Mellin Transformation and Fuchsian Type Partial Differential Equations Zofia Szmydt,B. Ziemian No preview available - 2012 |

The Mellin Transformation and Fuchsian Type Partial Differential Equations Zofia Szmydt,B Ziemian No preview available - 1992 |

### Common terms and phrases

2-locally analytic function asymptotic Borel transform bounded set Cauchy Cauchy kernel Çı Cº(I Cº(R compact set conical cut-off function continuous function Corollary CŞº defined Definition denote differential operator distribution u e ends the proof equation estimation Example Exercise exists a constant extends finite fixed follows formula Fuchsian type function f Hahn–Banach theorem Hence holomorphic for Rez holomorphic function integral Lemma Let u e linear locally uniformly Mellin distribution Mellin multiplier Mellin order Mellin transform neighbourhood of zero numbers observe open set polynomial Proposition prove radius of convergence Remark resp resurgent functions satisfies seminorms sequence set ſ Show smooth function ſº solution Subsection supp Suppose tempered distribution topology uſe variable