Multidimensional smoothing using orthogonal expansions

Adaptive smoothing approaches are particularly useful because the amount of smoothing imposed on the data is determined automatically from the statist ical characteristics of subsets of the data itself. Although efficient meth ods are available for performing adaptive smoothings on one-dimensional (1- D) data, extension of these 1-D adaptive smoothing methods to higher dimens ions is often difficult because of the lack of a theoretical edifice and pr ohibitively large computational requirements, previously, a method was deve loped that reduces the dimensions of a data function by exploiting its Four ier transform properties and thus achieves an effective multidimensional sm oothing by use of low-dimensional smoothing methods. The purpose of this le tter is to extend this work and demonstrate the possibility of achieving a higher-dimensional smoothing by applying lower-dimensional smoothing operat ions on the partial orthogonal expansion coefficients of the data function.