On the completeness of the system $$\{ {t^{{\lambda _n}}}{\log ^{{m_n}}}t\} $$ in C 0(E)

Abstract

Let \(E = \bigcup\limits_{n = 1}^\infty {{I_n}} \) be the union of infinitely many disjoint closed intervals where In = [an, bn], 0 < a1 < b1 < a2 < b2 < … < bn < …, \(\mathop {\lim }\limits_{n \to \infty } \)bn = ∞. Let α(t) be a nonnegative function and \(\{ {\lambda _n}\} _{n = 1}^\infty \) a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system \(\{ {t^{{\lambda _n}}}{\log ^{{m_n}}}t\} \) in C0(E) is obtained where C0(E) is the weighted Banach space consists of complex functions continuous on E with f(t)e−α(t) vanishing at infinity.