  This paper studies estimation of a smooth function $f(t,s)$ when we are given
functional responses of the form $f(t,\cdot)$ + error, but scientific interest
centers on the collection of functions $f(\cdot,s)$ for different $s$. The
motivation comes from studies of human brain development, in which $t$ denotes
age whereas $s$ refers to brain locations. Analogously to varying-coefficient
models, in which the mean response is linear in $t$, the "varying-smoother"
models that we consider exhibit nonlinear dependence on $t$ that varies
smoothly with $s$. We discuss three approaches to estimating varying-smoother
models: (a) methods that employ a tensor product penalty; (b) an approach based
on smoothed functional principal component scores; and (c) two-step methods
consisting of an initial smooth with respect to $t$ at each $s$, followed by a
postprocessing step. For the first approach, we derive an exact expression for
a penalty proposed by Wood, and an adaptive penalty that allows smoothness to
vary more flexibly with $s$. We also develop "pointwise degrees of freedom," a
new tool for studying the complexity of estimates of $f(\cdot,s)$ at each $s$.
The three approaches to varying-smoother models are compared in simulations and
with a diffusion tensor imaging data set.
