Double bases from generalized Faber polynomials with complex-valued coefficients in weighted Lebesgue spaces

In the paper it is considered the generalized Faber polynomials defined inside and outside a regular curve on the complex plane. The weighted Smirnov spaces corresponding to bounded and unbounded regions are defined. It is proved that the generalized Faber polynomials forms a basis in weighted Smirnov spaces, if the weight function satisfies the Muckenhoupt condition on the regular curve. The double system of generalized Faber polynomials with complex-valued coefficients is also considered and the basis properties of such a system in weighted Lebesgue spaces over regular curves are studied.