Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators

Yu. F. Korobeinik

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 49-66

Résumé

By using a general representation of nontrivial expansions of zero in absolutely representing systems of the form $\{E_\rho(\lambda_kz)\}_{k=1}^\infty$, where $\rho>0$, $E_\rho(z)=\sum\limits_{n=0}^\infty\dfrac{z^n}{\Gamma(1+\frac n\rho)}$ is the Mittag-Leffler function, and $(\lambda_k)_{k=1}^\infty$ are complex numbers, the author obtains a number of results in the theory of $\rho$-convolution operators in spaces of functions that are analytic in $\rho$-convex domains (a description of the general solution of a homogeneous $\rho$-convolution equation and of systems of such equations, a topological description of the kernel of a $\rho$-convolution operator, the construction of principal solutions, and a criterion for factorization).