The Mellin Transformation and Fuchsian Type Partial Differential Equations

I. Introduction.- x1. Terminology and notation.- x2. 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.- x3. 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.- x4. 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!$$ Rbegin{array}{*{20}{c}} n + end{array} $$.- Exercises.- x5. 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.- x6. 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.- x7. 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.- x8. The structure of Mellin distributions.- 1. Characterizations of Mellin distributions.- 2. Substitution in a Mellin distribution.- 3. Mellin order of a Mellin distribution.- Exercises.- x9. Paley-Wiener type theorems for the Mellin transformation.- Exercises.- x10. 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.- x11. Important subspaces of Mellin distributions.- 1. Subspaces % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa % dwhacqGH9aqpcaWGMbaaaa!3EE9!$$ P(xfrac{d}{{dx}})u = f $$.- 2. Subspaces SPr(s,s1 ) of Mellin distributions.- 3. Spaces M(? ?) and Zd(? ?) of distributions with continuous radial asymptotics.- Exercises.- x12. The modified Cauchy transformation.- 1. Modified Cauchy and Hilbert transformations in dimension 1.- 2. The case with parameters.- Exercises.- III. Fuchsian type singular operators.- x13. 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-VbaaSqaiaiG47p6-ladaciC-d9--H % caOeXafv3ySLgzGmvETj2BSbacgmGamaiGW9p0-7xYdCNamaiGW9p0 % --xkaKcabKaGaIV-O-paaaa!6A3A!$$ MI{s_{(omega )}} $$ and definition of ordinary Fuchsian type differential operators.- 3. Case of smooth coefficients.- 4. Case of analytic coefficients.- 5. Special functions as generalized analytic functions.- Exercises.- x14. Elliptic Fuchsian type partial differential equations in spaces % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI % cacaWG4bWaaSaaaeaacaWGKbaabaGaamizaiaadIhaaaGaaiykaiaa % dwhacqGH9aqpcaWGMbaaaa!3EE9!$$ P(xfrac{d}{{dx}})u = f $$.- 1. Existence and regularity of solutions on tangent cones ?K.- 2. Case of a proper cone.- Exercise.- x15. Fuchsian type partial differential equations in spaces with continuous radial asymptotics.- 1. The radial characteristic set Charg % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGamaiJeg7aHj % acacie-d9-cQcacGaGasW-maWGqbaaaa!4027!$$ alpha *P $$.- 2. Regularity of solutions in spaces M(? ?) and Zd(? ?).- 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.