Multi-affine and multi-Jensen functions and their connection with generalized polynomials

Summary.

In 1987 Lyle Ramshaw ([R]) has introduced the concept of multi-affine functions, called blossoms. This idea unified some important approaches and methods in the theory of polynomial splines. The main observation is that any polynomial $F:\mathbb{R}^p \to \mathbb{R}^q$ of degree ≤ n is the diagonalization of a (unique) symmetric n-affine function $f:(\mathbb{R}^p )^n \to \mathbb{R}^q :\;F(x) = f(x, x, \ldots ,x)$ for all $x \in \mathbb{R}^p .$

Here we investigate these ideas with the aim to get similar characterizations for generalized polynomials F : V → W, where V, W are arbitrary vector spaces over $\mathbb{Q}.$