Properties and modifications of Whittaker-Henderson graduation

First some properties of the Whittaker-Henderson graduation are obtained. Then modifications of the Whittaker-Henderson criterion are investigated in which u is determined (a) to minimize the fit function F d(u) subject to the restriction that the smoothness function S d(u) does not exceed a specified tolerance level and (b) to minimize S d(u) subject to a similar restriction on F d(u). Sufficient conditions for optimality are given and it is shown that the optimal solution can be obtained as the unique solution of a equation of a single variable.