  Generalizing results by Halperin et al., Grivaux recently showed that any
linearly independent sequence $\{f_k\}_{k=1}^\infty$ in a separable Banach
space $X$ can be represented as a suborbit
$\{T^{\alpha(k)}\varphi\}_{k=1}^\infty$ of some bounded operator $T: X\to X.$
In general, the operator $T$ and the powers $\alpha(k)$ are not known
explicitly. In this paper we consider approximate representations
$\{f_k\}_{k=1}^\infty \approx \{T^{\alpha(k)}\varphi\}_{k=1}^\infty$ of certain
types of sequences $\{f_k\}_{k=1}^\infty.$ In contrast to the results in the
literature we are able to be very explicit about the operator $T$ and suitable
powers $\alpha(k),$ and we do not need to assume that the sequences are
linearly independent. The exact meaning of approximation is defined in a way
such that $\{T^{\alpha(k)}\varphi\}_{k=1}^\infty$ keeps essential features of
$\{f_k\}_{k=1}^\infty,$ e.g., in the setting of atomic decompositions and
Banach frames. We will present two different approaches. The first approach is
universal, in the sense that it applies in general Banach spaces; the technical
conditions are typically easy to verify in sequence spaces, but are more
complicated in function spaces. For this reason we present a second approach,
directly tailored to the setting of Banach function spaces. A number of
examples prove that the results apply in arbitrary weighted $\ell^p$-spaces and
$L^p$-spaces.
