Systems with nonextendable convergence of quasipolynomials

The system $e(\Lambda)=\bigl\{(it)^ke^{i\lambda_nt}, 0\le k\le m_n-1\bigr\}_{n=1}^\infty$, where $\Lambda=\{\lambda_n\}$ is the set of zeros (of multiplicities $m_n$ ) of the Fourier transform
$$ L(z)=\int_{-a}^ae^{izt}\,d\mathscr L(t) $$
of a singular Cantor-Lebesgue measure, is examined. We prove that $e(\Lambda)$ is complete and minimal in $L_p(-a,a)$, with $p\ge1$, and that $|L(x+iy)|^2$ does not satisfy the Muckenhoupt condition on any horizontal line $\operatorname{Im}z=y\ne0$ in the complex plane. This implies that $e(\Lambda)$ does not have the property of convergence extension.