**Background**
----------

Let $U_t$ be the (translation) $C_0$-group on $L^1(\mathbb{R})$ defined by
$$
U_t(f)(x)\triangleq f(x-t) \, (\forall t \in \mathbb{R})(m-a.e. \, x \in \mathbb{R}).
$$

The [Wiener Tauberien theorem][1] states that, for any $f \in L^1(\mathbb{R})$ if its Fourier transform $\hat{f}$ does not vanish on $\mathbb{R}$ then 
$$
cl\left(
\operatorname{span}\left\{
U_t f: \, t \in \mathbb{R}
\right\}
\right)
 = L^1(\mathbb{R}),
$$
where $cl(\cdot)$ is the closure of a set in the norm topology on $L^1(\mathbb{R})$.


----------


**Question**
----------

Is there a "truncated version" of the Wiener theorem, which gives (reasonable) sufficient condition on $f \in L^1(\mathbb{R})$ so that, for a fixed $N\in \mathbb{Z}^+$, the set
$$
cl\left(
\left\{
\sum_{i=1}^N \beta_i U_{t_i}(f):\,
t_1,\dots,t_N \in \mathbb{R},\, \beta_1,\dots,\beta_N \in \mathbb{R}
\right\}
\right)
=
L^1(\mathbb{R})?
$$

----------

The motivation for my question comes as follows:


**Classical Proof of Wiener's Theorem:**
----------

In the operator-Theoretic proof of [J. Neerven][2], can be summarized as follows:

 - Define the Banach algebra homomorphism $L^1(\mathbb{R})\rightarrow \mathcal{L}(L^1(\mathbb{R}))$ by
$$
U(f)g\triangleq \int_{-\infty}^{\infty} f(t) U(t)(g)(x) dt = f \star g,
$$
(this is the convolution operator)
 - Define $X\triangleq cl(\operatorname{span}\{U_t f: \, t \in \mathbb{R}\})$ and write $U^Y$ to be the quotient operator on $L^1(\mathbb{R})/X$,
 - Show that $f \in Sp(U^Y)$ where $Sp(U^Y)$ is the Averson spectrum of $U^Y$ defined by
$$
\left\{
\xi \in \mathbb{R}:\,
\hat{g}(\xi)=0\, \forall g \in U^Y \mbox{ with } U^Y(g)=0
\right\},
$$
 - Use the fact that $Sp(U^Y)=0$ only if $U^Y=\{0\}$ to conclude that $Y=\{0\}$ since $f \in Sp(U^Y)$ and $\hat{f}$ was assumed to have no roots.  


----------
**Edit:**
In this proof, it's clear to me that this proof does not generalize to cover my question since the *span* operator cannot if $cl\left(
\operatorname{span}\left\{
U_t f: \, t \in \mathbb{R}
\right\}
\right)$ is replaced by $cl\left\{
\sum_{i=1}^N \beta_i U_{t_i}(f):\,
t_1,\dots,t_N \in \mathbb{R},\, \beta_1,\dots,\beta_N \in \mathbb{R}
\right\}$?  However, is there another approach to proving this theorem which can be modified to get around this added requirement.  


  [1]: https://en.wikipedia.org/wiki/Wiener%27s_tauberian_theorem#The_condition_in_L1
  [2]: https://fa.its.tudelft.nl/~neerven/publications/papers/RIMUT_97.pdf