Multi-affine and multi-Jensen functions and their connection with generalized polynomials
In 1987 \textit{L. Ramshaw} [Comput. Aided Geom. Des. 6, No. 4, 323--358 (1989; Zbl 0705.65008)] introduced the following concept of a symmetric \(n\)-affine function \(f:(\mathbb R^p)^n \to \mathbb R^q\), i.e., \(f\) is symmetric and satisfies \[ f(tx_1+(1-t)x_1',x_2,\dots,x_n)=tf(x_1,x_2,\dots,x_n)+(1-t)f(x_1',x_2,\dots,x_n) \] for all \(x_1,x_1',x_2,\dots,x_n \in \mathbb R^p\) and all \(t \in \mathbb R\). He showed that for every polynomial \(F:\mathbb R^p \to \mathbb R^q\) of degree \(\leq n\) there is exactly one \(n\)-affine and symmetric function \(f\) such that \(F(x)=f(x,x,\dots,x)\) for all \(x\). The authors of the present paper investigate the possibility of obtaining similar results concerning generalized polynomials. A generalized polynomial \(P\) (of degree \(\leq n\)) is a function defined on a vector space \(V\) over \(\mathbb Q\) and taking values in another \(\mathbb Q\)-vector space \(W\), such that \(P\) may be written in the form \(P=\sum_{i=0}^n P_i\), with functions \(P_i\) which are homogeneous polynomials of degree \(i\), i.e., functions \(P_i:V \to W\) whose values are given by \(P_i(x)=\widehat P_i(x,x,\dots,x)\), where the \(\widehat P_i:V^i \to W\) are \(i\)-additive (i.e., additive with respect to each variable) and symmetric. The concept which substitutes in this setting the symmetric \(n\)-affine functions is that of symmetric multi-Jensen function. A function \(f:V^n \to W\) is called a symmetric multi-Jensen function if it is symmetric and is a Jensen function in the first (and so in each) variable: \[ f\Big(\frac{x_1+x_1'}{2},x_2,\cdots,x_n\Big)=\frac{1}{2}\Big(f(x_1,x_2,\cdots,x_n)+f(x_1',x_2,\cdots,x_n) \Big). \] The main result of the paper is the following: Theorem. For every symmetric \(n\)-Jensen function \(f:V^n \to W\), its diagonalization \(F\) is a generalized polynomial of degree \(\leq n\). Moreover, given any generalized polynomial \(F:V \to W\) of degree \(\leq n\), there is a unique symmetric \(n\)-Jensen function \(f:V^n \to W\) such that \(F\) is its diagonalization. This result is extended to the case of generalized polynomials over \(K\)-vector spaces, where \(K\) is a field of characteristic zero.
scientific article; zbMATH DE number 2162500