This work introduces the concept of \( t \)-basis, generated by some bilinear mapping \( t\left(\cdot ;\cdot \right) \). It is considered the vector-valued class \( L_{p} \left(X\right)=:L_{p} \left(J;X\right) \), \( 1\le p<+\infty \), where \( J=\left[-\pi ,\pi \right] \) and \( X \) is a Banach space with the UMD property, and it is proven that the classical system of exponents \( \left\{e^{int} \right\}_{n\in Z} \) forms \( t \)-basis for \( L_{p} \left(X\right) \), 1<p<+\( \infty \). Using this fact, the Hardy vector classes \( {}_{n} H_{p}^{\pm } \left(X\right) \), \( \)1<p<+\( \infty \), different from the classical ones, are defined, and an equivalent definition is given of these classes and some of their properties are studied. In addition, the concept of \( t \)-Riesz property is introduced in a system of exponentials in \( L_{p} \left(X\right) \), 1<p<+\( \infty \), and it is proved that this system has the \( t \)-Riesz property. A new method is given for establishing the Plemelj-Sokhotski formulas for \( X \)-valued Cauchy type integrals, when \( X \) has the UMD property. An abstract analogues of the "1/4-Kadets" theorem is obtained for \( L_{2} \left(H\right) \), where \( H \) is a Hilbert space.