A criterion for a function to be represented as a sum of products of factors. (English) Zbl 0643.26009

The author proves a necessary and sufficient condition for representing a function of two real variables by means of functions of one variable in the form \[ f(x,y)=\sum^{n}_{i=1}g_ i(x)h_ i(y). \] This characterization is given for functions of class \(C^{2n}\) and is expressed in terms of the determinants of matrices of the type \([\frac{\partial^{i+j}f}{\partial x^ i\partial y^ j},\quad i,j=0,1,2,...,n].\)
The present result replaces a previous analogous characterization by C. Stephanos [Rend. Circ. Mat. Palermo 18, 360-362 (1904)], which is false, as the author shows here by an example.
Reviewer: A.Salvadori


26B40 Representation and superposition of functions