Spectral analysis of biorthogonal expansions generated by Muckenhoupt weights

Any Muckenhoupt $A_2$-weight $\omega^2$ on a special curve $\mathcal{\gamma}_\rho$ ($\rho\geqslant1/2$) generates a function $y_{\rho,\omega}(\lambda,t)$, which coincides with the exponential $\exp\{i\lambda t\}$ if $\rho=1$, $\omega^2(z)\equiv1$.
In this paper the geometric approach of B. S. Pavlov is used to obtain criteria for a family of functions $\{y_{\rho,\omega}(\lambda_k,t): \lambda_k\in\Lambda\}$ to be an unconditional basis in the space $L_2(0,\sigma)$. The analytic machinery of the paper generalizes some results of M. M. Dzhrbashyan (for a power weight) for the case of an arbitrary Muckenhoupt $A_2$-weight.