Efficient recovery of sparse signals from underdetermined linear systems has received extensive attentions in recent decade. This paper considers the stable recovery of a signal x , which is not k-sparse itself but is k-sparse in terms of a Parseval frame D , from the observations y = Ax + e via three unconstrained minimization approaches. We show that if the sensing matrix A satisfies the restricted isometry property adapted to D with δ t k < t − 1 t + 8 for any t > 1, then these three approaches can stably recover x . As a consequence, when t = 2 and 3, our result significantly improves existing best sufficient conditions. Furthermore, we theoretically characterize the recovery errors of these methods.