A note on certain functional determinants
Given an arbitrary set \(X\), a positive integer \(n\) and a real or complex valued function \(F\) defined on \(X^ n\), the author establishes a necessary and sufficient condition for the representation \((*)\) \(F(x_ 1,\dots,x_ n)=Det[f_ j(x_ k)]_{n\times n}\) on \(X^ n\) with a suitable \(n\)-tuple of functions \(f_ 1,\dots,f_ n\) defined on \(X\). He also proves that any functions \(f_ 1,\dots,f_ n\) occurring in the representation \((*)\) have to be linear combinations of the functions \(F(u_ 1,\dots,u_{j-1},\cdot,u_{j+1},\dots,u_ n)\), \(j\in\{1,\dots,n\}\), where \((u_ 1,\dots,u_ n)\in X^ n\) such that \(F(u_ 1,\dots,u_ n)\neq 0\). Next he describes: 1. functions \(H\) (real or complex) defined on the product \(X\times Y\) of two sets that admit a representation \(H(x,y)=\sum^ n_{j=1}f_ j(x)f_ j(y)\) on \(X\times Y\); 2. functions \(H\) (real or complex) defined on \(X\times X\) that admit a representation \(H(x,y)=\sum^ n_{j,k=1}\alpha_{jk}f_ j(x)f_ k(y)\) on \(X\times X\).
scientific article; zbMATH DE number 85815