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Consider the decomposition of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300101.png" /> into a finite sum of the form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300102.png" /></td> </tr></table>
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For sufficiently smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300103.png" />, a necessary condition for such a decomposition involves determinants of the form
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Consider the decomposition of a function $h ( x , y )$ into a finite sum of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300104.png" /></td> </tr></table>
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\begin{equation*} h ( x , y ) = \sum _ { k = 1 } ^ { n } f _ { k } ( x ) g _ { k } ( y ). \end{equation*}
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For sufficiently smooth $h$, a necessary condition for such a decomposition involves determinants of the form
 +
 
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<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]]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300105.png" /> defined on arbitrary (even discrete) sets without any regularity conditions was formulated in [[#References|[a3]]], [[#References|[a4]]] by introducing a new, special matrix
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300106.png" /></td> </tr></table>
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\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]]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300107.png" /> of the form
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300108.png" /></td> </tr></table>
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\begin{equation*} h ( x , y ) = F ( \sum _ { k = 1 } ^ { n } f _ { k } ( x ) . g _ { k } ( y ) ), \end{equation*}
  
 
see [[#References|[a5]]].
 
see [[#References|[a5]]].
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d1300109.png" /> times functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130010/d13001010.png" />"  ''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>
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<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>

Revision as of 17:02, 1 July 2020

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
How to Cite This Entry:
D'Alembert equation for finite sum decompositions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_equation_for_finite_sum_decompositions&oldid=13809
This article was adapted from an original article by F. Neuman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article