  Suppose that $\ff \in \reals^{n}$ is a vector of $n$ error-contaminated
measurements of $n$ smooth values measured at distinct and strictly ascending
abscissae. The following projective technique is proposed for obtaining a
vector of smooth approximations to these values. Find \yy\ minimizing $\| \yy -
\ff \|_{\infty}$ subject to the constraints that the second order consecutive
divided differences of the components of \yy\ change sign at most $q$ times.
This optimization problem (which is also of general geometrical interest) does
not suffer from the disadvantage of the existence of purely local minima and
allows a solution to be constructed in $O(nq)$ operations. A new algorithm for
doing this is developed and its effectiveness is proved. Some of the results of
applying it to undulating and peaky data are presented, showing that it is
economical and can give very good results, particularly for large
densely-packed data, even when the errors are quite large.
