  We show that well-established methods from the theory of Banach modules and
time-frequency analysis allow to derive completeness results for the collection
of shifted and dilated version of a given (test) function in a quite general
setting. While the basic ideas show strong similarity to the arguments used in
a recent paper by V.~Katsnelson we extend his results in several directions,
both relaxing the assumptions and widening the range of applications. There is
no need for the Banach spaces considered to be embedded into $(L^2(\mathbb{R}),
||\cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to
present the results in the setting of the Euclidean spaces, because then the
Schwartz space $\mathcal{S}^{\prime}(\mathbb{R}^d)$ ($d \geq 1$) of tempered
distributions provides a well-established environment for mathematical
analysis. We also establish connections to modulation spaces and Shubin classes
$({Q}_{s}(\mathbb{R}^d), ||\cdot{}||_{Q_s})$, showing that they are special
cases of Katsnelson's setting (only) for $s \geq 0$.
