Representation of Functions by Generalized Dirichlet Series

TLDR: In this article, it was shown that a function whose coefficients in the series are all zero must itself be zero, and conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.

Abstract: This article is concerned with the representation of functions in domains of the complex plain by series in the systems , , .In § 1 we construct systems biorthogonal to the systems , , and find the asymptotic behaviour of functions of these systems.In § 2 we determine in a natural way the coefficients of the series in the systems in question by means of the biorthogonal systems. We also find the asymptotic behaviour of the coefficients for large indices. We obtain formulae for the remainder, that is, the difference between the function and a partial sum of the corresponding series.In § 3 we prove that a function whose coefficients in the series are all zero must itself be zero. This result makes it possible, in principle, to reconstruct a function from the coefficients of its series.In § 4 we give conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.