Bases of exponentials, sines and cosines in weighted spaces on a finite interval

We obtain a result concerning the basis property in a weighted space on an interval $(-a,a)$ for a system of exponentials generated by the zeros of the Fourier transform of a function with singularities at the ends of the support interval $(-a,a)$. For an arbitrary $\Delta\in\mathbb{C}$ we find a criterion for the basis property of the system $(e^{i(n+\Delta\operatorname{sign} n)t})_{n\in\mathbb{Z}}$ in a weighted space on the interval $(-\pi,\pi)$ and the systems of sines $(\sin((n+\Delta)t))_{n\in\mathbb{N}}$ and cosines $1\cup (\cos((n+\Delta)t))_{n\in\mathbb{N}}$ in a weighted space on the interval $(0,\pi)$. The weight is everywhere a finite product of polynomial functions.