On discrete sufficient and weakly sufficient sets in some spaces of functions of zero order. (Q1432526)

The concepts of sufficient and weakly sufficient sets for certain spaces of entire functions were introduced respectively by \textit{L. Ehrenpreis} [Fourier analysis in several complex variables (Pure and Applied Mathematics 17, Wiley, New York) (1970; Zbl 0195.10401)] and \textit{D. M. Schneider} [Trans. Am. Math. Soc. 197, 161--180 (1974; Zbl 0264.46018)]. In this paper, the authors construct discrete sufficient (weakly sufficient) sets for the spaces \(H(q,\sigma)\) defined as follows. Let \(1< q<\infty\), \(0< \sigma<\infty\). Then \(H(q,\sigma)\) is the space of all functions \(\varphi\) analytic in \(\mathbb{C}\setminus \{0\}\) such that \(|\varphi(z)|\leq C\exp(\delta \ln| z|^q)\), \(0< \delta< \sigma\), and \(C> 0\), \(\delta\) and \(C\) depending on \(\varphi\). The authors apply the result to derive integral representations with an exponential kernel, in particular exponential series representations in spaces of number sequences of finite order and type.