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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].

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