Coefficient sequences in Hilbert and Banach space expansions

In this article we prove some theorems about the sequences of coefficients which occur for expansions relative to a basis in a Banach space, and for a certain type of basis in $L_2[-\pi,\pi]$ investigated by K. I. Babenko, namely $\{|t|^\alpha e^{int}\}_{-\infty}^\infty$, $0<\alpha<1/2$. As an application of our results, we prove that there exists no universal basis in a separable Hilbert space.