Approximation properties of exponential systems on the real line and the half-line

For arbitrary $a>0$ and $\alpha >1$ a class of entire functions depending on a complex parameter $\mu$ is constructed. The values of $\mu$ such that the sequence of zeros $\lambda _n$ of a function in this class generates a complete and minimal exponential system
$$ \exp \bigl (-i\lambda _nt-a|t|^\alpha \bigr) $$
in $L^p(\mathbb R)$ $(L^p(\mathbb R_+))$, $p\geqslant 2$, are described. Examples of such systems were previously known only for $\alpha=2$.