Bases of Exponentials in Weighted Spaces Generated by Zeros of Functions of Sine Type

If $\omega$ is an $A_p$-weight with some additional condition and $(\lambda)$ is a separated sequence of all zeros of a sine-type function possessing a certain multiplier (in the sense of Fourier transforms) property, then the corresponding system of exponentials $(e^{i\lambda_nt})$ constitutes a basis in the weighted space $L^p((-\pi,\pi),\omega(t)\,dt)$, $1<\pi<\infty$.