In this work we consider \(X\text {-valued}\) Lebesgue space (so called Bochner space) \(L_p(\Gamma ;X)\) , \(1<p<+\infty \) , where \(\Gamma \) is a closed, simple Lyapunov or Radon curve on complex plane. We introduce the concept of a \(t\text {-basis}\) for \(L_p(\Gamma ;X)\) , and investigate systems of generalized Faber polynomials corresponding to the domains \(int\Gamma \) and \(ext\Gamma \) . Some operators are defined that act from the Hardy-Bochner classes \(H_{p}^{\pm }(X)\) to \(L_{p}(\Gamma ;X)\) , where X is a UMD space. We prove the invertibility of these operators and, using the results of \(t\text {-basicity}\) of parts of the exponential system for \(H_{p}^{\pm }(X)\) spaces. Additionally, we define the \(X\text {-valued}\) Smirnov classes \(E_{p}^{\pm }(X)\) corresponding to the domains \(int\Gamma \) and \(ext\Gamma \) , respectively. It is proved that Faber polynomials form a t-basis for Smirnov-Faber classes, and using this fact, the t-basisness of double the system of Faber polynomials for Bochner spaces is established. Some properties of Smirnov-Bochner classes are also studied.
