Isomorphism between solution spaces of convolution equations (Q2455164)

Denote by \(\Omega\) the class of all mappings from \(\mathbb{Z}\) to \(\mathbb{C}.\) For given \(\rho > 1\) and \(\alpha > 0\) it is introduced the space \(A[\rho,\alpha]=\lim_{j \to \infty}\text{proj} A_j,\) where \[ A_j=\left\{a\in\Omega:\sup_{n\in \mathbb{Z}}\frac{| a(n)| }{\exp(\alpha_j| n| ^\rho)}<\infty \right\}, \] and \(\{\alpha_j\}_{j=1}^{\infty}\) is strictly decreasing sequence such that \(\lim_{j\to \infty}\alpha_j=\alpha.\) The space \(A^{\prime}[\rho,\alpha]\) strongly dual to \(A[\rho,\alpha]\) has the form \(A^{\prime}[\rho,\alpha]=\lim_{j \to \infty}\text{ind} A_j^{\prime},\) where \[ A_j^{\prime}=\left\{b\in\Omega:\sum_{n=-\infty}^{ \infty}| b(n)| \exp(\alpha_j| n| ^\rho)<\infty\right\}. \] Let \(s\in A^{\prime}[\rho,\alpha].\) On \(A[\rho,\alpha]\) consider the convolution equation \[ (s,a(n+t))=\sum _{n\in \mathbb{Z}}s(n)a(n+t)=0,\quad t\in \mathbb{Z}. \] Denote by \(W_s\) the space of its solutions. Suppose that the characteristic function for this equation \(s_M(z)=\sum _{n\in \mathbb{Z}}s(n)z^n\), \(z\in \mathbb{C}\backslash \{0\},\) has only simple zeros \(\{z_k\}_{k\in \mathbb{N}}.\) In the paper the authors show that \(W_s\) is the closure of the linear hull of the functions \(x_k(n)=z_k^n,\) they give conditions under which the functions \(x_k(n)\) form a basis in \(W_s.\) Next, it is considered a related convolution equation on the space of entire functions of the corresponding growth and conditions are determined under which an isomorphism between solution spaces of these equations can be constructed.