  We study Jacobi structures on the dual bundle $A^\ast$ to a vector bundle $A$
such that the Jacobi bracket of linear functions is again linear and the Jacobi
bracket of a linear function and the constant function 1 is a basic function.
We prove that a Lie algebroid structure on $A$ and a 1-cocycle $\phi \in \Gamma
(A^\ast)$ induce a Jacobi structure on $A^\ast$ satisfying the above
conditions. Moreover, we show that this correspondence is a bijection. Finally,
we discuss some examples and applications.
