  The Kolmogorov $N$-width describes the best possible error one can achieve by
elements of an $N$-dimensional linear space. Its decay has extensively been
studied in Approximation Theory and for the solution of Partial Differential
Equations (PDEs). Particular interest has occurred within Model Order Reduction
(MOR) of parameterized PDEs e.g.\ by the Reduced Basis Method (RBM).
  While it is known that the $N$-width decays exponentially fast (and thus
admits efficient MOR) for certain problems, there are examples of the linear
transport and the wave equation, where the decay rate deteriorates to
$N^{-1/2}$. On the other hand, it is widely accepted that a smooth parameter
dependence admits a fast decay of the $N$-width. However, a detailed analysis
of the influence of properties of the data (such as regularity or slope) on the
rate of the $N$-width seems to lack.
  In this paper, we use techniques from Fourier Analysis to derive exact
representations of the $N$-width in terms of initial and boundary conditions of
the linear transport equation modeled by some function $g$ for half-wave
symmetric data. For arbitrary functions $g$, we derive bounds and prove that
these bounds are sharp. In particular, we prove that the $N$-width decays as
${c_r N^{-r}}$ for functions {with Sobolev regularity} $g\in
H^{r{-\varepsilon}}$ for all $\varepsilon>0$ even if $g\not\in H^{r}$. Our
theoretical investigations are complemented by numerical experiments which
confirm the sharpness of our bounds and give additional quantitative insight.
