The Mellin Transformation and Fuchsian Type Partial Differential Equations: Mathematics and its Applications, cartea 56

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.