View source for D'Alembert equation for finite sum decompositions

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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

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300104.png"/></td> </tr></table>

These determinants were introduced in [[#References|[a6]]] and [[#References|[a7]]], and a correct formulation of the sufficient condition was given in [[#References|[a3]]]; see also [[#References|[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 [[#References|[a3]]], [[#References|[a4]]] by introducing a new, special matrix

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

see also [[#References|[a8]]] and [[#References|[a9]]].

Several authors have dealt with problems concerning decompositions of functions of several variables and similar questions, see, e.g., [[#References|[a1]]], [[#References|[a2]]], [[#References|[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 [[#References|[a5]]].

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