Choice of Approximation Bases Used in Computational Functional Algorithms for Approximating Probability Densities on the Basis of Given Sample

In this paper we formulate requirements for choosing approximation bases when constructing cost-effective optimized computational (numerical) functional algorithms for approximating probability densities on the basis of a given sample, with special attention paid to the stability and approximation of the bases. It is shown that to meet the requirements and construct efficient approaches to conditional optimization of numerical schemes, the best choice is a multi-linear approximation and the corresponding special case for both kernel and projection computational algorithms for nonparametric density estimation, which is a multidimensional analogue of the frequency polygon.