Effective approximation of piecewise smooth functions by their expansion into fast convergent series in terms of functions formed by eigenfunctions of Sturm-Liouville problems

Using the eigenfunctions of two Sturm-Liouville problems (with the same operator of a general form but with two different versions of boundary conditions), a method for constructing these specific basis functions is developed. The corresponding expansions of smooth and piecewise smooth functions in terms of such basis lead to fast convergent series. This result makes it possible to approximate the functions of the above class by a small number of terms. We also give brief information on a method for constructing multidimensional (in particular, two-dimensional) specific basis functions with the above properties. The proposed method is based on the ideas of the author's earlier works. However, it is, in essence, a new method that has substantially improved characteristics and is mainly oriented to the approximation of piecewise smooth functions. We also consider several important special cases.