Completeness of a system of exponentials on the half-line (Q2277597)

The author investigates the completeness of the system of exponentials \[ \{\exp (-\lambda_ nt)\}^{\infty}_{n=1},\quad Re \lambda_ n>0 \] in the weighted spaces \(L^ p_{\alpha}\), i.e. the \(L^ p\)-spaces on the half-axis (0,\(\infty)\), generated by the weight \(t^{\alpha}\) \((1\leq p<\infty\), \(\alpha >-1)\). Among other things, the role of the classical \(L^ 2\)-criterion \[ \sum (Re \lambda_ n)/(1+| \lambda_ n|^ 2)<\infty \] is discussed. It is shown, in particular, that, for a given sequence of frequences \(\{\lambda_ n\}\), the completeness property may depend on p and \(\alpha\).