  We show that there do not exist adaptive confidence bands for curve
estimation except under very restrictive assumptions. We propose instead to
construct adaptive bands that cover a surrogate function f^\star which is close
to, but simpler than, f. The surrogate captures the significant features in f.
We establish lower bounds on the width for any confidence band for f^\star and
construct a procedure that comes within a small constant factor of attaining
the lower bound for finite-samples.
