  A conventional linear model for functional data involves expressing a
response variable $Y$ in terms of the explanatory function $X(t)$, via the
model: $Y=a+\int_I b(t)X(t)dt+\hbox{error}$, where $a$ is a scalar, $b$ is an
unknown function and $I=[0, \alpha]$ is a compact interval. However, in some
problems the support of $b$ or $X$, $I_1$ say, is a proper and unknown subset
of $I$, and is a quantity of particular practical interest. In this paper,
motivated by a real-data example involving particulate emissions, we develop
methods for estimating $I_1$. We give particular emphasis to the case
$I_1=[0,\theta]$, where $\theta \in(0,\alpha]$, and suggest two methods for
estimating $a$, $b$ and $\theta$ jointly; we introduce techniques for selecting
tuning parameters; and we explore properties of our methodology using both
simulation and the real-data example mentioned above. Additionally, we derive
theoretical properties of the methodology, and discuss implications of the
theory. Our theoretical arguments give particular emphasis to the problem of
identifiability.
