  Confidence bands are confidence sets for an unknown function f, containing
all functions within some sup-norm distance of an estimator. In the density
estimation, regression, and white noise models, we consider the problem of
constructing adaptive confidence bands, whose width contracts at an optimal
rate over a range of H\"older classes.
  While adaptive estimators exist, in general adaptive confidence bands do not,
and to proceed we must place further conditions on f. We discuss previous
approaches to this issue, and show it is necessary to restrict f to
fundamentally smaller classes of functions.
  We then consider the self-similar functions, whose H\"older norm is similar
at large and small scales. We show that such functions may be considered
typical functions of a given H\"older class, and that the assumption of
self-similarity is both necessary and sufficient for the construction of
adaptive bands. Finally, we show that this assumption allows us to resolve the
problem of undersmoothing, creating bands which are honest simultaneously for
functions of any H\"older norm.
