论文标题
在$ \ mathbb {r}^d $中定义的函数的分解
Decompositions of functions defined on finite sets in $\mathbb{R}^d$
论文作者
论文摘要
有限的子集$ m \ subset \ mathbb {r}^d $是基本的，如果对于任何功能$ f \ colon m \ to \ mathbb {r} $，则存在一个函数$ f_1，\ ldots的集合，\ ldots，f_d \ colon \ colon \ colon \ colon \ colon \ mathbb {r} {r} $ \ the for enill \ ldots，x_d）\在m $中，我们有$ f（x_1，\ ldots，x_d）= f_1（x_1） + \ ldots + f_d（x_d）$。对于某些有限集，我们证明了一个基本集的标准，我们表明它不能扩展到一般情况。此外，我们根据双重加权图来解释上述标准，并估计某些基本和非基本子集中的元素数量。
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.
