Сверточные уравнения в~пространствах последовательностей с~экспоненциальным ограничением роста

A. A. Borichev

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 107-115

Résumé

We describe the solutions of convolution equations $S*x=0$ (on $\mathbb Z$ or $\mathbb Z_+$) in the spaces of sequences $X=X_{(\beta,\alpha)}=\bigcup_{\gamma\alpha}\bigcup_{\delta1/\beta}\{x:|x_n|\leqslant c\gamma^{|n|}, n0; |x_n|\leqslant c\delta^n, n\geqslant0\}$, $0\leqslant\alpha\beta\leqslant+\infty$. Every 1-invariant subspace $E$, $E\subset X$ equals $\operatorname{Ker} S$ for some $S$. After the Laplace transform $x\to\hat x$ the space $\hat E^\perp$ can be identified with $f\cdot A(K_{(\beta, \alpha)})$, where $K_{(\beta, \alpha)}=\{z:\beta|z|\alpha\}$. The space $E$ can be decomposed as $E=\operatorname{span}\{\{n^k\lambda^n\}_{n\in z}:\lambda\in\sigma\}+\{x\in X:x_k=0, k$ iff $f$ is a Weierstraas product (in $K_{(\beta, \alpha)}$) with zeros not accumulating to $|\lambda|=\beta$.