D'Alembert equation for finite sum decompositions

Consider the decomposition of a function $h ( x , y )$ into a finite sum of the form

\begin{equation*} h ( x , y ) = \sum _ { k = 1 } ^ { n } f _ { k } ( x ) g _ { k } ( y ). \end{equation*}

For sufficiently smooth $h$, a necessary condition for such a decomposition involves determinants of the form

These determinants were introduced in [a6] and [a7], and a correct formulation of the sufficient condition was given in [a3]; see also [a4].

A sufficient and necessary condition for not sufficiently smooth functions $h ( x , y )$ defined on arbitrary (even discrete) sets without any regularity conditions was formulated in [a3], [a4] by introducing a new, special matrix

\begin{equation*} \left( \begin{array} { c c c c } { h ( x _ { 1 } , y _ { 1 } ) } & { h ( x _ { 1 } , y _ { 2 } ) } & { \dots } & { h ( x _ { 1 } , y _ { n } ) } \\ { h ( x _ { 2 } , y _ { 1 } ) } & { h ( x _ { 2 } , y _ { 2 } ) } & { \dots } & { h ( x _ { 2 } , y _ { n } ) } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { h ( x _ { n } , y _ { 1 } ) } & { h ( x _ { n } , y _ { 2 } ) } & { \dots } & { h ( x _ { n } , y _ { n } ) } \end{array} \right); \end{equation*}

see also [a8] and [a9].

Several authors have dealt with problems concerning decompositions of functions of several variables and similar questions, see, e.g., [a1], [a2], [a8]. However, several open problems in this area remain (as of 2000), e.g.: find a characterization of functions $h ( x , y )$ of the form

\begin{equation*} h ( x , y ) = F ( \sum _ { k = 1 } ^ { n } f _ { k } ( x ) . g _ { k } ( y ) ), \end{equation*}

see [a5].

References

[a1]	M. Čadek, J. Šimša, "Decomposable functions of several variables" Aequat. Math. , 40 (1990) pp. 8–25
[a2]	H. Gauchman, L.A. Rubel, "Sums of products of functions of $x$ times functions of $y$" Linear Alg. & Its Appl. , 125 (1989) pp. 19–63
[a3]	F. Neuman, "Factorizations of matrices and functions of two variables" Czech. Math. J. , 32 : 107 (1982) pp. 582–588
[a4]	F. Neuman, "Functions of two variables and matrices involving factorizations" C.R. Math. Rept. Acad. Sci. Canada , 3 (1981) pp. 7–11
[a5]	F. Neuman, Th. Rassias, "Functions decomposable into finite sums of products" , Constantin Catathéodory–An Internat. Tribute , II , World Sci. (1991) pp. 956–963
[a6]	C.M. Stéphanos, "Sur une categorie d'équations fonctionalles" , Math. Kongr. Heidelberg , 1905 (1904) pp. 200–201
[a7]	C.M. Stéphanos, "Sur une categorie d'équations fonctionalles" Rend. Circ. Mat. Palermo , 18 (1904) pp. 360–362
[a8]	Th.M. Rassias, J. Šimša, "Finite sum decompositions in mathematical analysis" , Wiley (1995)
[a9]	Th.M. Rassias, J. Šimša, "19 Remark" Aequat. Math. , 56 (1998) pp. 310