The Mellin Transformation and Fuchsian Type Partial Differential Equations

I. Introduction.- §1. Terminology and notation.- §2. Elementary facts on complex topological vector spaces.- 1. Multinormed complex vector spaces and their duals.- 2. Inductive and projective limits.- 3. Subspaces. The Hahn-Banach theorem.- Exercise.- §3. A review of basic facts in the theory of distributions.- 1. Spaces DK and (DK)1.- 2. Spaces D(A) and D’(A).- 3. Spaces S and S1.- 4. Spaces E and E1.- 5. Substitution in distributions. Homogeneous distributions.- 6. Classical order of a distribution and extendibility theorems for distributions.- 7. Convolution of distributions.- 8. Tensor product of distributions.- Exercises.- II. Mellin distributions and the Mellin transformation.- §4. The Fourier and the Fourier-Mellin transformations.- 1. The Fourier transformation in S1.- 2. The Fourier-Mellin transformation in the space of Mellin distributions with support in % MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegWuDJLgzHbIrHHhaiuqacqWFsbGufaqabeGabaaabaqcLbqacGao% 4pOBaaGcbaqcLbqacWaGaI7-q8VHRaWkaaaaaa!450D!$$R begin{array}{*{20}{c}}n + end{array} $$.- Exercises.- §5. The spaces of Mellin distributions with support in a polyinterval.- 1. Spaces Ma, ((0, t]) and M1a ((0, t]).- 2. Spaces M(?) ((0, t]) and M1(?) ((0, t]).- Exercises.- §6. Operations of multiplication and differentiation in the space of Mellin distributions.- 1. Multiplication and differentiation in Ma, M(?) and their duals.- 2. Mellin multipliers.- Exercises.- §7. The Mellin transformation in the space of Mellin distributions.- 1. The Mellin transformation in the space of Mellin distributions and its relations with the Fourier-Laplace transformation.- 2. Examples of Mellin transforms of some functions.- 3. Mellin transforms of certain cut-off functions.- 3.1. One-dimensional smooth cut-off functions.- 3.2. n-Dimensional smooth cut-off functions with a parameter.- Exercises.- §8. The structure of Mellin distributions.- 1. Characterizations of Mellin distributions.- 2. Substitution in a Mellin distribution.- 3. Mellin order of a Mellin distribution.- Exercises.- §9. Paley-Wiener type theorems for the Mellin transformation.- Exercises.- §10. Mellin transforms of cut-off functions (continued).- 1. Conical cut-off functions.- 2. The K-inequalities.- 3. The "tangent cones" ?K and related cut-off functions.- 4. Further investigation of the Mellin transform of a conical cut-off function.- Exercises.- §11. Important subspaces of Mellin distributions.- 1. Subspaces % MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI% cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa% dwhacqGH9aqpcaWGMbaaaa!3EE9!$$P(x frac{d}{{dx}})u = f $$.- 2. Subspaces SPr(s,s1 ) of Mellin distributions.- 3. Spaces M(? ?) and Zd(? ?) of distributions with continuous radial asymptotics.- Exercises.- §12. The modified Cauchy transformation.- 1. Modified Cauchy and Hilbert transformations in dimension 1.- 2. The case with parameters.- Exercises.- III. Fuchsian type singular operators.- §13. Fuchsian type ordinary differential operators.- 1. Asymptotic expansions.- 2. The equation % MathType!MTEF!2!1!+-% feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamXvP5% wqonvsaeHbmv3yPrwyGmvyUnhaiuGajugOaiadaciaaW3--Leajjad% acigaa3--nhaZPWaiaiG47p6-V

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.