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

TL;DR

This paper characterizes when finite subsets of bR^da0are basic, meaning functions on them can be decomposed into sums of single-variable functions, providing criteria, graph interpretations, and size estimates.

Contribution

It introduces a criterion for basic sets in bR^da0and explores its limitations, linking the problem to doubly-weighted graphs and estimating set sizes.

Findings

Established a criterion for basic finite sets in bR^da0

Connected the problem to doubly-weighted graph theory

Provided bounds on the size of basic and non-basic sets

Abstract

A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in M$ we have $f(x_1, \ldots, x_d) = f_1(x_1) + \ldots + f_d(x_d)$. 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.