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

摘要

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 : R~p → R~q of degree ≤ n is the diagonalization of a (unique) symmetric n-affine function f : (R~p)~n → R~q : F(x) = f(x, x, …, x) for all x ∈ 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 Q.