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Generalized analytic functions and a strong quasi-analyticity principle

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 Abstract: The aim of my lecture is to present some results from the theory of generalized analytic functions (GAF for short). I will pay a special attention to deriving from the theory of GAF's a kind of quasi-analyticity principle. Let me say that GAF’s behave in a simple way under basic algebraic and differential operations, and analytic change of variables. As pointed out by B. Ziemian, they form a natural subclass of distributions conormal to zero, whose symbols have a very explicit form (cf. [Zie]). Let me also say that GAF’s appear naturally as solutions to singular differential equations, and practically all special functions, after suitable change of variables, are examples of GAF's.
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  • Institute of Mathematics, Polish Academy of Sciences, P.O.B. 137, Śniadeckich 8 , 00-950 Warszawa, Poland, lysik@impan.impan.gov.pl
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Bibliografia
[Ko] H. Komatsu, Ultradistributions, I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo 20 (1973), 25-105.
[Ły 1] G. Łysik, The Taylor transformation of analytic functionals with non-bounded carrier, Studia Math. 108 (1994), 159-176.
[Ły 2] G. Łysik, Laplace ultradistributions on a half line and strong quasi-analyticity principle I, Ann. Polon. Math., to appear.
[Mo] M. Morimoto, Analytic functionals with non-compact carrier, Tokyo J. Math. 1 (1978), 77-103.
[Sz-Zie] Z. Szmydt and B. Ziemian, The Mellin Transformation and Fuchsian Type Partial Differential Equations, Kluwer, Dordrecht, 1992.
[Zie] B. Ziemian, Generalized analytic functions with applications to singular ordinary and partial differential equations, Dissertationes Math., to appear.
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DML-PL
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