Uniqueness theorems for analytic functions asymptotically representable by Dirichlet--Taylor series

M. M. Dzhrbashyan

Sbornik. Mathematics, Tome 20 (1973) no. 4, pp. 603-649

Résumé

Dirichlet–Taylor series $$ \sum^\infty_{j=1}d_je^{-\lambda_j s}s^{s_j-1} $$ are considered, where $\{d_j\}^\infty_1$ is a sequence of complex numbers, $\{\lambda_j\}^\infty_1$ is a nondecreasing sequence of positive numbers, and $s_j\geqslant1$ ($j\geqslant1$) is the number of times $\lambda_j$ occurs in the segment $\{\lambda_1,\dots,\lambda_j\}$. An “adherence principle” is established for these series. As applications of this principle, uniqueness theorems are proved for analytic functions which are asymptotically representable by partial sums of Dirichlet–Taylor series in strips. Bibliography: 16 titles.