A note on "Data smoothing by cubic spline filters"

After a previous correspondence (see ibid., vol.46, p.2790-6, Oct. 1998) was published, the author found out that several of the key results had already been reported by Unser, Aldroubi and Eden (see ibid., vol.41, p.821-33 and p.834-48, Feb. 1993), The author regrets this oversight on his part and wishes to acknowledge the prior contribution of Unser, Aldroubi, and Eden. They have shown, in fact, that the smoothing spline and the computation of the derivatives for equally spaced data can be performed by digital filtering. They have also given the corresponding filters and discussed some of their properties. In addition, recursive implementation of these filters has been proposed and their computational efficiency over the conventional matrix solutions has been demonstrated. Thus, the author's contribution is mainly in establishing some interesting properties of the smoothing filter, especially the maximum flatness property of the filter, as well as the relationship between the cutoff frequency of the filter and the smoothing control parameter. Moreover, the author's approach to derive the smoothing filter directly from the matrix solution can always be seen as an alternative to the method proposed by Unser et al. The author has also stressed that the recursive smoothing filter gives the same result as the matrix solution except for the two extremities of the data.