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 \(T: X\rightarrow X.\) 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.

Halperin等人的归纳结果最近表明，Grivaux可分离的Banach空间X中的任何线性独立序列\（\ {f_k \} _ {k = 1} ^ \ infty \）都可以表示为子轨道\（\ {某些有界算子\（T：X \ rightarrow X. \）的T ^ {\ alpha（k）} \ varphi \} _ {k = 1} ^ \ infty \）通常，算子T和幂\（ \ alpha（k）\）并不明确。在本文中，我们考虑近似表示\（\ {f_k \} _ {k = 1} ^ \ infty \ approx \ {T ^ {\ alpha（k）} \ varphi \} _ {k = 1} ^ \ infty \ ）某些类型的序列\（\ {f_k \} _ {k = 1} ^ \ infty; \）与文献中的结果相比，我们能够非常明确地了解算子T和合适的幂\（\ alpha（k），\），并且我们不需要假设序列是线性独立的。逼近的确切含义是以这样的方式定义的：\（\ {T ^ {\ alpha（k）} \ varphi \} _ {k = 1} ^ \ infty \）保留\（\ {f_k \ } _ {k = 1} ^ \ infty，\）例如，在原子分解和Banach框架中。我们将介绍两种不同的方法。第一种方法是通用的，因为它适用于一般的Banach空间。技术条件通常在序列空间中易于验证，而在功能空间中则更为复杂。因此，我们提出了第二种方法，该方法直接针对Banach函数空间的设置。大量示例证明了该结果适用于任意加权\（\ ell ^ p \）-空间和\（L ^ p \）-空间。