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