Decompositions of functions defined on finite sets in ${\mathbb{R}}^d$

Abstract

A finite subset $M\subset {\mathbb{R}}^d$ is basic, if for any function $f:M\to \mathbb{R}$ there exists a collection of functions $f_1,\dots ,f_d:\mathbb{R}\to \mathbb{R}$ such that for each element $(x_1,\dots ,x_d)\in M$ we have $f(x_1,\dots ,x_d)=f_1(x_1)+\dots +f_d(x_d)$. For certain finite sets, we prove a criterion for a set to be basic, and we show that it cannot be extended to the general case. In addition, we interpret the above criterion in terms of doubly-weighted graphs and give an estimation for the number of elements in certain basic and non-basic subsets.

arXiv comment16 pages, 9 figures