2
$\begingroup$

Consider the map $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$. When is it the case that there exist functions $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $h:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\forall x,y\in\mathbb{R}^n$, $f(x,y)=g(x)+h(y)$?

Thank you!

  • 0
    A sufficient-but-not-necessary condition is given by requiring $f$ to be an additive homomorphism, or, even more stringent, a linear map. But I suspect you're looking for something a bit broader. – 2012-10-29

2 Answers 2

3

Iff the following two mappings are constant for all fixed $x,x_1,y,y_1$: $t\mapsto f(x_1,t)-f(x,t) $ $t\mapsto f(t,y_1)-f(t,y) .$

  • 0
    I was just guessing. Please check that it indeed works.. :) – 2012-10-30
1

I don't have access to the article, but A sufficient condition for additively separable functions, looks like it answers your question in a general form:

This paper presents a set of sufficient conditions under which a completely separable function on an open SāŠ‚R^N is additively separable. The new condition is that different connected elements of intersections of parallel-to-the-axes hyperplanes with the domain S are intersected by common indifference surfaces.

though I'm not sure if the restriction to a "completely separable function" is important to your question without reading the article.