  We initiate the study of the Interval Selection problem in the (streaming)
sliding window model of computation.
  In this problem, an algorithm receives a potentially infinite stream of
intervals on the line, and the objective is to maintain at every moment an
approximation to a largest possible subset of disjoint intervals among the $L$
most recent intervals, for some integer $L$.
  We give the following results:
  - In the unit-length intervals case, we give a $2$-approximation sliding
window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, and we show that any
sliding window algorithm that computes a $(2-\varepsilon)$-approximation
requires space $\Omega(L)$, for any $\varepsilon > 0$.
  - In the arbitrary-length case, we give a
$(\frac{11}{3}+\varepsilon)$-approximation sliding window algorithm with space
$\tilde{\mathrm{O}}(|OPT|)$, for any constant $\varepsilon > 0$, which
constitutes our main result.
  We also show that space $\Omega(L)$ is needed for algorithms that compute a
$(2.5-\varepsilon)$-approximation, for any $\varepsilon > 0$.
  Our main technical contribution is an improvement over the smooth histogram
technique, which consists of running independent copies of a traditional
streaming algorithm with different start times. By employing the one-pass
$2$-approximation streaming algorithm by Cabello and P\'{e}rez-Lantero [Theor.
Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the
underlying algorithm, the smooth histogram technique immediately yields a
$(4+\varepsilon)$-approximation in this setting. Our improvement is obtained by
forwarding the structure of the intervals identified in a run to the subsequent
run, which constrains the shape of an optimal solution and allows us to target
optimal intervals differently.
