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The Mellin Transformation and Fuchsian Type Partial Differential Equations

Author: Zofia Szmydt; Bogdan Ziemian
Publisher: Dordrecht : Springer Netherlands, 1992.
Series: Mathematics and Its Applications (East European Series), 56.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Zofia Szmydt; Bogdan Ziemian
ISBN: 9789401124249 9401124248
OCLC Number: 851388352
Description: 1 online resource (xiv, 222 pages).
Contents: I. Introduction --
§1. Terminology and notation --
§2. Elementary facts on complex topological vector spaces --
Exercise --
§3. A review of basic facts in the theory of distributions --
Exercises --
II. Mellin distributions and the Mellin transformation --
§4. The Fourier and the Fourier-Mellin transformations --
Exercises --
§5. The spaces of Mellin distributions with support in a polyinterval --
Exercises --
§6. Operations of multiplication and differentiation in the space of Mellin distributions --
Exercises --
§7. The Mellin transformation in the space of Mellin distributions --
Exercises --
§8. The structure of Mellin distributions --
Exercises --
§9. Paley-Wiener type theorems for the Mellin transformation --
Exercises --
§10. Mellin transforms of cut-off functions (continued) --
Exercises --
§11. Important subspaces of Mellin distributions --
Exercises --
§12. The modified Cauchy transformation --
Exercises --
III. Fuchsian type singular operators --
§13. Fuchsian type ordinary differential operators --
Exercises --
§14. Elliptic Fuchsian type partial differential equations in spaces % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa % dwhacqGH9aqpcaWGMbaaaa!3EE9!$$ P(x\frac{d}{{dx}})u = f $$ --
2. Case of a proper cone --
Exercise --
§15. Fuchsian type partial differential equations in spaces with continuous radial asymptotics --
Appendix. Generalized smooth functions and theory of resurgent functions of Jean Ecalle --
1. Introduction --
2. Generalized Taylor expansions --
3. Algebra of resurgent functions of Jean Ecalle --
4. Applications --
List of Symbols.
Series Title: Mathematics and Its Applications (East European Series), 56.
Responsibility: by Zofia Szmydt, Bogdan Ziemian.

Abstract:

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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