Infinite dimensional holomorphy in the ring of formal power series : partial differential operators
Abstract: We study holomorphy in the ring of formal power series in an infinite number of variables. Thus we restrict our study to (infinite dimensional) holomorphy on sequence spaces and we show that we obtain a rich theory without requiring any topological structure on the domain space. We make a comprehensive PDO-study for the spaces under consideration.As a basis we establish the Martineau duality, described by the Fourier-Borel transform, between spaces of entire and of exponential type functions in an infinite number of variables. In order to study PDO:s and PDE:s in spaces in the ring of formal power series, such an established duality is a useful tool for the transpose of a differential operator become the operator of multiplication by the corresponding symbol. The second part of the thesis is devoted to applications of the Martineau duality for various PDO-related problems. The following topics are considered: Existence theorems, Approximation theorems, Fischer decompositions, Cauchy problems, PDE-preserving projectors (Kergin projector) and Pseudo-differential operators.Some of the main results are the following. We prove Malgrange type existence theorems for infinite dimensional differential operators on A, Exp and F and we show that homogenous solutions can be approximated by such solutions consisting of exponential (finitely supported) polynomials. Here A and Exp are the spaces of entire respectively exponential type functions and F is the Fischer-Fock Hilbert space (in an infinite number of variables). We extend the notions of Fischer decomposition and Fischer pair, studied by H. Shapiro, J. Aniansson etc. A Fischer pair for a space is a pair of maps (here differential operators) whose kernel respective transpose image decompose the space into a direct sum. We establish some necessary and sufficient conditions for that a given pair of maps will make up a Fischer pair. By generalizing a known result for finite dimensional domain spaces, we show the existence of non-trivial Fischer pairs for A and Exp.Moreover, we prove some infinite dimensional generalizations of results obtained by H. Shapiro. In particular we show that densely defined differential operators together with their adjoints constitute Fischer pairs for F. We show that Cauchy-and dual Cauchy problems, w.r.t differential operators, in Exp respective in Exp'=A are well-posed. We prove that the infinite dimensional Kergin operator has interpolating and PDE-preserving properties and that it is uniquely determined by these properties. A PDE-preserving projector is a projector that preserves homogeneous solutions to differential equations.
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