Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Convolution equations and harmonic analysis in spaces of entire functions

Author: D. G. Dickson
Journal: Trans. Amer. Math. Soc. 184 (1973), 373-385
MSC: Primary 30A98
MathSciNet review: 0374449
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Abstract: If H is the topological space of functions analytic in the simply connected open set $ \Omega $ of the plane with the topology of compact convergence, its dual may be identified with the space E of functions of exponential type whose Borel transforms have their singularities in $ \Omega $. For f in H and $ \phi $ in E, $ (f \ast \phi )(z) \equiv \left\langle {f,{\phi _z}} \right\rangle $ where $ {\phi _z}$ is the z-translate of $ \phi $. If $ f{\nequiv}0$ in any component of $ \Omega ,f \ast \phi = 0$ if and only if $ \phi $ is a finite linear combination of monomial-exponentials $ {z^p} \exp (\omega z)$ where $ \omega $ is a zero of f in $ \Omega $ of order at least $ p + 1$. For such f and $ \psi $ in E, $ f \ast \phi = \psi $ is solved explicitly for $ \phi $. If E is assigned its strong dual topology and $ \tau (\phi )$ is the closed linear span in E of the translates of $ \phi $, then $ \tau (\phi )$ is a finite direct sum of closed subspaces spanned by monomial-exponentials. Each closed translation invariant subspace of E is the kernel of a convolution mapping $ \phi \to f \ast \phi $; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.

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Keywords: Convolution, entire functions of exponential type, harmonic analysis
Article copyright: © Copyright 1973 American Mathematical Society