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Generalized analytic functions with applications to singular ordinary and partial differential equations

Seria
Rozprawy Matematyczne tom/nr w serii: 354 wydano: 1996
Zawartość
Warianty tytułu
Abstrakty
EN
CONTENTS
Introduction.............................................................................................................................................5
I. Preliminaries.........................................................................................................................................7
   1. A review of classical results in the theory of Laplace integra............................................................7
   2. Boundary values of holomorphic functions......................................................................................10
     2.1. Distributions as boundary values of holomorphic functions.........................................................10
     2.2. Hyperfunctions in one variable....................................................................................................12
   3. Mellin analytic functionals, Mellin hyperfunctions and Mellin distributions.........................................14
   4. Laplace distributions.........................................................................................................................18
     4.1. Convolution of Laplace distributions.............................................................................................21
   5. Ecalle distributions.............................................................................................................................23
     5.1. Alien derivatives of Ecalle distributions.........................................................................................24
   6. Paley-Wiener type theorem for Mellin analytic functionals.................................................................25
     6.1. Phragmén-Lindelöf type theorems................................................................................................29
   7. The cut-off functions and their Mellin transforms...............................................................................30
   8. Modified Cauchy transformation in dimension 1.................................................................................31
II. The theory of generalized analytic functions..........................................................................................33
   9. Definition of a generalized analytic function........................................................................................34
   10. The Mellin transform of a generalized analytic function.....................................................................35
   11. Characterization of GAFs in terms of Mellin transforms.....................................................................37
   12. The Borel and Taylor transformations in the class of GAFs..............................................................40
   13. Operations on generalized analytic functions...................................................................................40
   14. Resurgent functions.........................................................................................................................44
     14.1. Alien derivatives of resurgent functions......................................................................................46
     14.2. Taylor-Fourier representation of resurgent functions..................................................................47
III. Applications to singular linear differential equations..............................................................................48
   15. Special functions as generalized analytic functions...........................................................................48
   16. Fuchsian type ODEs with generalized analytic coefficients................................................................52
   17. Fuchsian type PDEs with "constant" coefficients................................................................................58
   18. GAFs in several variables..................................................................................................................73
   19. Fuchsian type PDEs with generalized analytic coefficients................................................................78
Appendices.................................................................................................................................................84
I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84
II. Nonlinear singular differential equations.................................................................................................88
   1. The case of ordinary differential equations..........................................................................................88
   2. The case of partial differential equations.............................................................................................93
References...................................................................................................................................................94
Symbol index.................................................................................................................................................97
Subject index................................................................................................................................................99
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 354
Liczba stron
100
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLIV
Daty
wydano
1996
otrzymano
1994-12-19
Twórcy
  • Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland, ziemian@impan.gov.pl
Bibliografia
  • [A] E. Andronikoff, Intégrales de Nilsson et faisceaux constructibles, Bull. Soc. Math. France 120 (1992), 51-85.
  • [B] J. M. Bony, Second microlocalization and propagation of singularities for semilinear hyperbolic equations, Taniguchi Symp. H.E.R.T. Kataka, 1984, 11-49.
  • [Br] H. Bremermann, Distributions, Complex Variables and Fourier Transform, Addison-Wesley, 1965.
  • [C] J. B. Conway, Functions of One Complex Variable, Grad. Texts in Math. 11, Springer, 1973.
  • [E1] J. Ecalle, Les fonctions résurgentes, Publ. Math. Université de Paris-Sud, several volumes.
  • [E2] J. Ecalle, Singularités irrégulières et résurgence multiple, in: Cinq applications des fonctions résurgentes, preprint 84 T 62, Orsay 2-42.
  • [E3] J. Ecalle, Finitude des cycle-limites et accéléro-sommation de l'application de retour, preprint 90-36, Université de Paris-Sud.
  • [GS] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Academic Press, New York, 1964.
  • [Ka] A. Kaneko, Introduction to Hyperfunctions, Math. Appl., Kluwer, Dordrecht, 1988.
  • [Ko] H. Komatsu, An introduction to the theory of hyperfunctions, in: Lecture Notes in Math. 287, Springer, 1973, 1-43.
  • [L] J. Leray, Le calcul différentiel et intégral sur une variété analytique complexe, Bull. Soc. Math. France 87 (1959), 81-180.
  • [Ły1] G. Łysik, On the structure of Mellin distributions, Ann. Polon. Math. 51 (1990), 219-228.
  • [Ły2] G. Łysik, The Taylor transformation of analytic functionals with non-bounded carrier, Studia Math. 108 (1994), 159-176.
  • [Maj] H. Majima, Resurgent equations and Stokes multipliers for the generalized hypergeometric differential equation, preprint, Ochanomizu University, 1990.
  • [M] B. Malgrange, Introduction aux travaux de J. Ecalle, prépublication de l'Institut de Fourier, Université de Grénoble, 20 (1984).
  • [Mo] M. Morimoto, Analytic functionals with non-compact carrier, Tokyo J. Math. 1 (1978), 77-103.
  • [N] N. Nilsson, Some growth and ramification properties of certain multiple integrals, Ark. Mat. 5 (1965), 463-476.
  • [Pl] M. Pliś, The Mellin analytic functionals and the Laplace-Beltrami operator on the Minkowski half-plane, Studia Math. 99 (1991), 263-276.
  • [PCN] E. Pham, B. Candelpergher et C. Nosmos, Visite aux sources, prépublication de l'Université de Nice, 1989.
  • [Schl] H. Schlichtkrull, Hyperfunctions and Harmonic Analysis on Symmetric Spaces, Progr. in Math. 49, Birkhäuser, Boston, 1984.
  • [Sl] L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
  • [Sz] Z. Szmydt, Fourier Transformation and Linear Differential Equations, PWN-Polish Sci. Publ., Warszawa, and Reidel, Dordrecht, 1977.
  • [SZ] Z. Szmydt and B. Ziemian, The Mellin Transformation and Fuchsian Type PDEs, Math. Appl., Kluwer Acad. Publ., 1992.
  • [T] E. C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, London, 1949.
  • [Tou] J. C. Tougeron, Gevrey expansions and applications, preprint, University of Toronto, 1991.
  • [VDZ] V. S. Vladimirov, Yu. N. Drozhzhinov and B. I. Zav'yalov, Multidimensional Tauberian Theorems for Generalized Functions, Nauka, Moscow, 1986 (in Russian).
  • [We] A. Weinstein, The order and symbol of a distribution, Trans. Amer. Math. Soc. 241 (1978), 1-54.
  • [Wi] D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, N.J., 1946.
  • [Zie1] B. Ziemian, Taylor formula for distributions, Dissertationes Math. 264 (1988).
  • [Zie2] B. Ziemian, Elliptic corner operators in spaces with continuous radial asymptotics II, in: Partial Differential Equations, Banach Center Publ. 27, part 2, Inst. Math., Polish Acad. Sci., Warszawa, 1992, 555-580.
  • [Zie3] B. Ziemian, Continuous radial asymptotics for solutions to elliptic Fuchsian equations in 2-dimensions, in: Proc. Sympos. Microlocal Analysis and its Applications, Publ. RIMS, Kyoto Kokyuroku 750, 1991.
  • [ZieK] B. Ziemian and H. Kołakowski, Second microlocalization and the Mellin transformation, Publ. RIMS Kyoto Univ. 26 (1990), 785-801.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: 34A20, 35A20, 35C20, 46F12, 30D15, 46F15
Identyfikator YADDA
bwmeta1.element.zamlynska-50e899ca-8c56-45dd-bd25-95e339b7edec
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ISSN
0012-3862
Kolekcja
DML-PL
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