  This paper considers the problem of specifying a simple approximating density
function for a given data set (x_1,...,x_n). Simplicity is measured by the
number of modes but several different definitions of approximation are
introduced. The taut string method is used to control the numbers of modes and
to produce candidate approximating densities. Refinements are introduced that
improve the local adaptivity of the procedures and the method is extended to
spectral densities.
