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Exponential-polynomial bases for null spaces of convolution operators in $$A^{-\infty}$$. (English) Zbl 1251.30059
Jarosz, Krzysztof (ed.), Function spaces in modern analysis. Sixth conference on function spaces, Edwardsville, IL, USA May 8–22, 2010. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5251-4/pbk). Contemporary Mathematics 547, 1-15 (2011).
Let $$D$$ be a bounded convex domain in $$\mathbb{C}$$ and $$A^{-\infty} (D)$$ the space of all holomorphic functions $$f$$ on $$D$$ such that there exists $$p>0$$ with
$\sup_{z \in D}|f(z)| \, |d_D(z)|^p < \infty,$
endowed with its natural inductive limit topology, where $$d_D(z)$$ is the distance from $$z\in D$$ to the boundary of $$D$$. Each nontrivial analytic functional $$\mu$$ on $$\mathbb{C},$$ carried by a compact convex set $$K,$$ generates a convolution operator $$\mu \ast f (z) = \big(\mu_w,f(z+w)\big)$$. Under suitable growth conditions on the Fourier-Borel transform of $$\mu$$, the convolution operator $$\mu \ast$$ acts from $$A^{-\infty} (D+K)$$ into $$A^{-\infty} (D).$$ Denote by $$\mathcal{Z}_\mu ^{-\infty} (D+K)$$ the kernel of the convolution operator $$\mu \ast,$$ endowed with the induced topology from $$A^{-\infty} (D+K).$$ The main result of the present paper is that there always exists a Schauder basis in $$\mathcal{Z}_\mu ^{-\infty} (D+K)$$ consisting of exponential-polynomial solutions of the homogeneous convolution equation $$\mu\ast f=0$$. As a consequence, it is shown that $$\mathcal{Z}_\mu ^{-\infty} (D+K)$$ can be identified with the dual space of some power series space of infinite type.
For the entire collection see [Zbl 1219.46004].

##### MSC:
 30H05 Spaces of bounded analytic functions of one complex variable 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)