The Closed Span of some Exponential System in Weighted Banach Spaces on the Real Line and a Moment Problem

Abstract

Let $$\{\lambda_{n}\}_{n=1}^\infty$$ be a strictly increasing sequence of positive real numbers diverging to infinity and let $$\{\mu_{n}\}_{n=1}^\infty$$ be a sequence of positive integers. Consider the exponential system $${E_{\Lambda \{ {t k}{e {{\lambda _n}t}}:k = 0,1,2,3,...,{\mu _n} - 1\} _{n = 1} \infty }}$$ Assuming the density condition $$\mathop {\lim }\limits_{t \to \infty } \frac{{\sum {_{\lambda n \leqslant {t {{\mu _n}}}}} }}{t} = d < \infty $$ and some other restrictions, we prove that every function in the closure of the linear span of EΛ in some weighted Banach spaces on the real line R is extended to an entire function represented by a Taylor–Dirichlet series $$g(z) = \sum\limits_{n = 1} \infty {(\sum\limits_{k = 0} {{\mu _n} - 1} {{c_n},{k {{z k}}}} )} {e {{\lambda _n}z}},{c_n},k \in C$$ We also consider a problem in a weighted L2(ℝ) Hilbert space as well as a moment problem on the real line.