On some functional equations

$n+1$

f_{0}\left(x\right),F_{1}\left(x\right),\ldots,f_{n}\left(x\right),

$It is$ .

$It is$ can be found $n+1$ $c_{i},i=0,1,\ldots,$ HAVE $\sum_{i=0}^{n}c_{i}f_{i}\left(x\right)=0$ , whatever it is $x\in E$ .

$It is$ $n+1$

V\binomial{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{n+1}}=\left\|f_{j-1}\left(x_{j}\right)\right\|_{i,j=1,2,\ldots,n+1}.

on the points $x\in E,i=1,2,\ldots,n+1$ $and$ $j$

V\binomial{1,x,x^{2},\ldots,x^{n}}{x_{1},x_{2},\ldots,x_{n+1}}=V\left(x_{1},x_{2},\ldots,x_{n+1}\right)

$x_{1},x_{2},\ldots,x_{n+1}$ ,

\left[x_{1},x_{2},\ldots,x_{n+1};f\right]=\frac{V\binomial{1,x,\ldots,x^{n-1},f}{x_{1},x_{2},\ldots,x_{n+1}}}{V\left(x_{1},x_{2},\ldots,x_{n+1}\right)}

$f\left(x\right)$ $x_{1},x_{2},\ldots,x_{n+1}$ .

$It is$

V\binomial{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{n+1}}=0

$x_{i}\in E,i=1,2,\ldots,n+1$ .

For $n=0$ true for $n$ and for $n+1$

f_{i}\left(x\right),i=0,1,\ldots,n-1

exist $n$ puncture $x_{i}\in E,i=1,2,\ldots,n$ ,

V\binomial{f_{0},f_{1},\ldots,f_{n-1}}{x_{1},x_{2},\ldots,x_{n}}\neq 0

But

V\binomial{f_{0},f_{1},\ldots,f^{n}}{x_{1},x_{2},\ldots,x_{n},x}=0

whatever $x\in E$ .

2. $It is$ $It is$ . $n+1$ puncture $x_{i},i=1,2,\ldots,n+1$ $It is$

V\binomial{f_{0},f_{1},\ldots,f_{n}}{x_{1},x_{2},\ldots,x_{n+1}}\neq 0.

the distinct points $x_{1},x_{2},\ldots,x_{n+1}$ $It is$ . on the crowd $It is$ .

any system of $n+1$ $x_{i},i=1,2,\ldots,n+1$ , $n+1$ $y_{i},i=1,2,\ldots,n+1$

$y$ , $x_{i},i=1,2,\ldots,n+1$ .

3. $It is$ $\left[a,b\right]$ .

how many points $x_{1},x_{2},\ldots,x_{n+1}$ in order $x_{1}<x_{2}<x_{n+1}$ find the points $x_{i}^{\prime},x_{i}^{\prime\prime},i=1,2,\ldots,n+1$ $It is$

x_{1}^{\prime}<x_{2}^{\prime}<\ldots<x_{n+1}^{\prime},\ x_{1}^{\prime\prime}<x_{2}^{\prime\prime}<\ldots<x_{n+1}^{\prime\prime}

and

V\binomial{f_{0},f_{1},\ldots,f_{n}}{x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n+1}^{\prime}}>0,\ V\binomial{f_{0},f_{1},\ldots,f_{n}}{x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{n+1}^{\prime\prime}}\leq 0.

$x_{i}=\lambda x_{i}^{\prime\prime}+\left(1-\lambda\right)x_{i}^{\prime},i=1,2,\ldots,n+1$ , $\lambda$ for $\lambda\in\left[0,1\right]$ , $\lambda=0$ ,

$\lambda=1$ So there is a $\lambda,0<\lambda<1$ such $\lambda$ $x_{i}$

\bigtriangleup_{h}\left(x;f_{0},f_{1},\ldots,f_{n}\right)=V\binomial{f_{0},f_{1},\ldots,f_{n}}{x,x+h,x+2h,\ldots,x+nh}

$x_{i}$ for anything $x,x+nh\in\left[a,b\right]$ .

for the functions $\left(n-2\right)$

	$\displaystyle f_{0}\left(x\right)$	$\displaystyle=\left(1+x\right)\left(2+x\right),f_{1}\left(x\right)=1+x,\ f_{2},-x\left(1-x\right),\ \ \ \ \ \ \ \ \ \ \ \ x\in\left[-2-1\right]$
	$\displaystyle f_{0}\left(x\right)$	$\displaystyle=f_{1}\left(x\right)=f_{2}\left(x\right)=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in\left[-1,1\right]$
	$\displaystyle f_{0}\left(x\right)$	$\displaystyle=1-x,\ f_{1}\left(x\right)=-x\left(1-x\right),\ f_{2}\left(x\right)=\left(1-x\right)\left(2-x\right),\ \ \ \ x\in\left[1,2\right]$

$\left[-2,2\right]$ $\bigtriangleup_{h}\left(x;f_{0},f_{1},f_{2}\right)=0$ , whatever it is $-2\leq x,x+2h\leq 2$ .

However, the functions $f_{0}\left(x\right),f_{1}\left(x\right),f_{2}\left(x\right)$ $\left[-2,2\right]$

5.

f_{i}\left(x\right)=x^{i}\ ,i=0,1,\ldots,n-1

1!2!,\ldots,\left(n-1\right)!h^{\frac{n\left(n-1\right)}{2}}\sum_{i=0}^{n}\left(-1\right)^{ni}\binomial{n}{i}f_{n}\left(x+ih\right).

\sum_{i=0}^{n}\left(-1\right)^{ni}\binomial{n}{i}f\left(x+ih\right)=0,\ x,x+nh\in\left[a,b\right]

$n-1$ . $f\left(x\right)$ $f\left(x\right)$ $\left[a,b\right]$ .

on the interval $\left[a,b\right]$ .

\bigtriangleup_{h}\left(f_{0},f_{1},\ldots,f_{n-1},f\right)=0,\ \ x,x+nh\in\left[a,b\right]

6.

Let's consider $n+2$ puncture $x_{1},x_{2},\ldots,x_{n+2}$ $2n+1$

$f_{0}\left(x_{1}\right)$	$f_{1}\left(x_{1}\right)$	$\cdots$	$f_{n-1}\left(x_{1}\right)$	$f\left(x_{1}\right)$	$f_{0}\left(x_{1}\right)$	$f_{1}\left(x_{1}\right)$	$\cdots$	$f_{n-1}\left(x_{1}\right)$
$f_{0}\left(x_{2}\right)$	$f_{1}\left(x_{2}\right)$	$\cdots$	$f_{n-1}\left(x_{2}\right)$	$f\left(x_{2}\right)$	$0$	$0$	$\cdots$	$0$
$f_{0}\left(x_{3}\right)$	$f_{1}\left(x_{3}\right)$	$\cdots$	$f_{n-1}\left(x_{3}\right)$	$f\left(x_{3}\right)$	$0$	$0$	$\cdots$	$0$
		$\cdots$					$\cdots$
$f_{0}\left(x_{i-1}\right)$	$f_{1}\left(x_{i-1}\right)$	$\cdots$	$f_{n-1}\left(x_{i-1}\right)$	$f\left(x_{i-1}\right)$	$0$	$0$	$\cdots$	$0$
$f_{0}\left(x_{i}\right)$	$f_{1}\left(x_{i}\right)$	$\cdots$	$f_{n-1}\left(x_{i}\right)$	$f\left(x_{i}\right)$	$f_{0}\left(x_{i}\right)$	$f_{1}\left(x_{i}\right)$	$\cdots$	$f_{n-1}\left(x_{i}\right)$
$f_{0}\left(x_{i+1}\right)$	$f_{1}\left(x_{i+1}\right)$	$\cdots$	$f_{n-1}\left(x_{i+1}\right)$	$f\left(x_{i+1}\right)$	$0$	$0$	$\cdots$	$0$
$f_{0}\left(x_{i+2}\right)$	$f_{1}\left(x_{i+2}\right)$	$\cdots$	$f_{n-1}\left(x_{i+2}\right)$	$f\left(x_{i+2}\right)$	$0$	$0$	$\cdots$	$0$
		$\cdots$					$\cdots$
$f_{0}\left(x_{n+1}\right)$	$f_{1}\left(x_{n+1}\right)$	$\cdots$	$f_{n-1}\left(x_{n+1}\right)$	$f\left(x_{n+1}\right)$	$0$	$0$	$\cdots$	$0$
$f_{0}\left(x_{n+2}\right)$	$f_{1}\left(x_{n+2}\right)$	$\cdots$	$f_{n}\left(x_{n+2}\right)$	$f\left(x_{n+2}\right)$	$f_{0}\left(x_{n+2}\right)$	$f_{1}\left(x_{n+2}\right)$	$\cdots$	$f_{n-1}\left(x_{n+2}\right)$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(x_{2}\right)$	$f_{1}\left(x_{2}\right)$	$\cdots$	$f_{n-1}\left(x_{2}\right)$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(x_{3}\right)$	$f_{1}\left(x_{3}\right)$	$\cdots$	$f_{n-1}\left(x_{3}\right)$
		$\cdots$					$\cdots$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(xli-1\right)$	$f_{1}\left(x_{i-1}\right)$	$\cdots$	$f_{n-1}\left(x_{i-1}\right)$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(x_{i+1}\right)$	$f_{??}\left(x_{i+1}\right)$	$\cdots$	$f_{n-1}\left(x_{i+1}\right)$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(x_{i+2}\right)$	$f_{1}\left(x_{i+2}\right)$	$\cdots$	$f_{n-1}\left(x_{i+2}\right)$
		$\cdots$					$\cdots$
$0$	$0$	$\cdots$	$0$	$0$	$f_{0}\left(x_{n+1}\right)$	$f_{1}\left(x_{n+1}\right)$	$\cdots$	$f_{n-1}\left(x_{n+???}\right)$

$2\leq i\leq n+1$ $i=2$ and $i=n+1$ .

$n+1$

		$\displaystyle V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{2},x_{3},\ldots,x_{n+1}}V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+2}}=$
	$\displaystyle\|$	$\displaystyle=V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{2},x_{3},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+2}}V\binom{f_{0},f_{1},\ldots,f_{n-1},f}{x_{1},x_{2},\ldots,x_{n+1}}+$
		$\displaystyle+V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{???+1}}V\binom{f_{0},f_{1},\ldots,f_{n-1},f}{x_{2},x_{3},\ldots,x_{n+2}}$

$i=2$ or $i=n+1$ .

	$\displaystyle\left(x_{n+2}-x_{1}\right)\left[x_{1},x_{2},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots,x_{n+2};f\right]$
	$\displaystyle=\left(x_{i}-x_{1}\right)\left[x_{1},x_{2},\ldots,x_{n+1};f\right]+\left(x_{n+2}-x_{i}\right)\left[x_{2},x_{3},\ldots,x_{n+2};f\right].$

7. finite $x_{1},x_{2},\ldots,x_{m}\left(m\geq n+1\right)$ $\left[a,b\right]$

V\binom{f_{0},f_{1},\ldots,f_{n-1},f_{n}}{x_{i},x_{i+1},\ldots,x_{i+n}}=0,\ i=1,2,\ldots,m-n

$f\left(x\right)$

V\binom{f_{0},f_{1},\ldots,f_{n-1,}f}{x_{1},x_{2},\ldots,x_{n+1}}=0

$x_{1},x_{2},\ldots,x_{n+1}\in\left[a,b\right]$

points $x$ $x_{r}-x_{s}$

group of $n+1$ RANGE $\left[a,b\right]$ .

$f\left(x\right)$ $\left[a,b\right]$ , any group of $n+1$

$\left[a,b\right]$ ,

f\left(x\right)=c_{0}f_{0}\left(x\right)+c_{1}f_{1}\left(x\right)+\cdots+c_{n-1}f_{n-1}\left(x\right),

where $c_{i},\ i=0,1,\ldots,n-1$ ,

for $n\$ $x_{1},x_{2},\ldots,x_{n}$ will variable $x_{n+1}=x$

1.

f_{0}\left(x\right)=\sin x,\ f,\ \left(x\right)=\cos x

$\left[a,b\right]$ where $0\leq a<b\leq\pi,b-a<\pi=\allowbreak\pi$ .

f\left(x\right)-2\cosh f\left(x+h\right)=f\left(x+2h\right)=0

$c_{0}\sin x+c_{1}\cos x$ .

2.

f_{0}\left(x\right)=1,\ f\left(x\right)=\sin x,\ f_{2}\left(x\right)=\cos x

$\left[a,b\right]$ 0 $\leq a<b\leq 2\pi,\ b-a<2\pi$ .

f\left(x\right)-f\left(x+3h\right)=\left(2\cosh+1\right)\left[f\left(x+h\right)-f\left(x+2h\right)\right]

is therefore of the form $c_{0}+c_{1}\sin x+c_{2}\cos x$ .

\sin x,\cos x,\sin 2x,\cos 2x,\ldots,\sin nx,\cos nx

or

1,\sin x,\cos x,\sin 2x,\cos 2x,\ldots,\sin nx,\cos nx.

9. $f\left(x\right)$ $\left[a,b\right]$ $\left[a,b\right]$

\left|f\left(x\right)\right|<M,x\in\left[a,b\right].

function $f\left(x\right)$ $\left[a,b\right]$

$x_{0}$ $\alpha_{1},\alpha_{2},\ldots,\alpha_{n-1},n-1$ $x_{0}$ $\left[a,b\right]$

we

V\binom{f_{0},f_{1},\ldots,f_{n-2}}{\alpha_{1},\alpha_{2},\ldots,\alpha_{n-1},x_{0}}=\lambda\neq 0.

$V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{1},x_{2},\ldots,x_{a\ ???}}$ $x_{1},x_{2},\ldots,x_{n}$ , $\varepsilon$ $\mu<\left|\lambda\right|$ , ${}^{\text{,}}$ $\delta$ $x\in\left(x_{0}-\delta,x_{0}+\delta\right),\alpha_{i}^{\prime}\in\left(\alpha_{i}+\delta,\right)i=1,2,\ldots,n-1$

	$\displaystyle\left\|V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x,\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}\right\|>\mu$
	$\displaystyle\left\|f\left(x_{0}\right)\right\|\left\|V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{0},\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}-V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x,\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}\right\|<\frac{\mu\varepsilon}{2}$
	$\displaystyle\left\|V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x,x_{0},\alpha_{1}^{\prime},\acute{\alpha}_{2},\ldots,\alpha_{i-1}^{\prime}\alpha_{i+1},\alpha_{i+2}^{\prime\prime},\ldots,\alpha_{n-1}^{\prime}}\right\|<\frac{\mu\varepsilon}{2\left(n-1\right)M},i=1,2,\ldots,n-1.$

$x\in\left(x_{0}-\delta,x_{0}+\delta\right)$ $\alpha_{i}^{\prime}$ $\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime},x_{0},x$

V\binom{f_{0},f_{1},\ldots,f_{n-1},f}{\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime},x_{0},x}=0

from which we deduce

	$\displaystyle\left[f\left(x_{0}\right)-f\left(x\right)\right]V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{0},\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}=$
	$\displaystyle=f\left(x_{0}\right)\left[V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x_{0},\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}-V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x,\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}\right]+$
	$\displaystyle+\sum_{i=1}^{n-1}\left(-1\right)^{i}f\left(\alpha_{i}^{\prime}\right)V\binom{f_{0},f_{1},\ldots,f_{n-1}}{x,x_{0},\alpha_{1}^{\prime},\alpha_{2}^{\prime},\ldots,\alpha_{i-1}^{\prime},\alpha_{i+1}^{\prime},\alpha_{i+2}^{\prime},\ldots,\alpha_{n-1}^{\prime}}$

\left|f\left(x_{0}\right)-f\left(x\right)\right|<\frac{1}{\mu}\cdot\frac{\mu\varepsilon}{2}+\frac{\left(n-1\right)M}{\mu}\cdot\frac{\mu\varepsilon}{2\left(n-1\right)M}=\varepsilon

$\left|x_{0}-x\right|<\delta$ , $f\left(x\right)$ $x_{0}$ .

on the interval $\left[a,b\right]$ , where $c_{i},i=0,1,\ldots,n-1$

$f\left(x\right)$ $\left[a,b\right]$ . $f\left(x\right)$

10. $F\left(x,y\right)$ $x$ and $y$ $f\left(x\right)g\left(x\right)$ , $x$

$y$ .

F\left(x,y\right)=\sum_{i=1}^{m}f_{i}\left(x\right)g_{i}\left(y\right).

If $F\left(x,y\right)$

\left\|\frac{\partial^{1+j}F}{\partial x^{i}\partial y^{f}}\right\|_{i^{\prime},j=0,1,\ldots,m}=0

$F\left(x,y\right)$ .

11. $F\left(x,y\right)$ on a crowd $E$ of points $\left(x,y\right)$ that $x$

$E_{x}$ and $E_{y}$ ).

We will say that $m$ degree $m_{1}>m$ .

f_{1}\left(x\right),f_{2}\left(x\right),\ldots,j

$E_{x}$

g_{1}\left(y\right),g_{2}\left(y\right),\ldots,g\left(y\right)

$E_{y}$ .

functions only $x$ only by $y$ effectively $0$ .

is of the actual degree $m$ , $r<m$ .

find a number $r$ $m$ $r$ functions $\varphi_{1}\left(x\right),\varphi_{2}\left(x\right),\ldots,\varphi_{r}$

f_{i}\left(x\right)=c_{i,1}\varphi_{1}\left(x\right)+x_{i,2}\varphi_{2}\left(x\right)+\cdots+c_{i,r}\varphi_{r}\left(x\right),\ \ i=1,2,\ldots,m

where $c_{i,j}$

F\left(x,y\right)=\sum_{i=1}^{r}\varphi_{i}\left(x\right)\psi_{i}\left(y\right)

where

\psi_{i}\left(y\right)=c_{1,i}g_{1}\left(y\right)+\cdots+c_{m,i}g_{m}\left(y\right),\ i=1,2,\ldots,r.

12

D\left(\begin{array}[c]{cc}x_{1},x_{2},\ldots,x_{m+1}&\\ &;F\\ y_{1},y_{2},\ldots,y_{m+1}&\end{array}\right)=\left\|F\left(x_{i},y_{i}\right)\right\|_{i,j=1,2,\ldots,m+1}

where $x_{1},x_{2},\ldots,x_{m+1}$ are $m+1$ $E_{x}$ and $y_{1},y_{2},\ldots,y_{m+1}m+1$ $E_{y}$ .

D\left(\begin{tabular}[c]{l}$x_{1},x_{2},\ldots,x_{m+1}$\\ \\ $y_{1},y_{2},\ldots,y_{m+1}$\end{tabular};F\right)=0

on $E$ .

Any function $F\left(x,y\right)$ $E$ $m$ .

function $F\left(x,y\right)$ $r,1\leq r\leq m,r$ puncture $x_{1},x_{2},\ldots,x\in E_{x}$ and $\ r$ puncture $y_{1},y_{2},\ldots,y_{r}\in E_{y}$

D=D\left(\begin{array}[c]{cc}x_{1},x_{2},\ldots,x_{r}&\\ &;F\\ y_{1},y_{2},\ldots,y_{r}&\end{array}\right)\neq 0

and

D\left(\begin{array}[c]{c}x_{1},x_{2},\ldots,x_{r},x\\ \\ y_{1},y_{2},\ldots,y_{r},y\end{array};F\right)=0

whatever $\left(x,y\right)\in E$ . $F\left(x,y\right)$

If $r$ effective $r$ . $A_{i,j}$ $D$ .

F\left(x,y\right)=\sum_{i=1}^{r}f_{i}\left(x\right)g_{i}\left(y\right)

where, for example

f_{i}\left(x\right)=F\left(x,y_{i}\right),g_{i}\left(y\right)=\frac{1}{D}\sum_{s=1}^{r}\left(-1\right)^{s+r-i}F\left(x_{s},y\right)A_{s,i},\ i=1,2,\ldots,r

$f_{i}\left(x\right)$ $g_{i}\left(y\right)$

If $E_{x}$ $m+1$ $E_{y}$ $m+1$ any function $F\left(x,y\right)$ $E$ $m$ .

13. A quasi-polynomial $m$ $m_{i}<m$ .

$m$ $m-1$ .

D\left(\begin{array}[c]{c}x_{1},x_{2},\ldots,x_{m}\\ \\ y_{1},y_{2},\ldots,y_{mj???}\end{array};F\right)=V\binom{f_{1},f_{2},\ldots,f_{m}}{x_{1},x_{2},\ldots,x_{m}}V\binom{g_{1},g_{2},\ldots,g_{m}}{y_{1},y_{2},\ldots,y_{m}}.

points $x_{i}$ $y_{i}$

Actual condition $m$ $E_{x}$ $E_{y}$ .

14. $E$ $R\left(a\leq x\leq b,c\leq y\leq d\right)$ .

We can consider functions $F\left(x,y\right)$

D\left(\begin{tabular}[c]{l}$x_{1},x_{2},\ldots,x_{m}$\\ \\ $y_{1},y_{2},\ldots,y_{n}$\end{tabular};F\right)\neq 0

$x_{1},x_{2},\ldots,x_{m}\in\left[a,b\right]$ $y_{1},y_{2},\ldots,y\in\left[c,d\right]$

$F\left(x,y\right)$ with $x$ $y$ $x_{1}<x_{2}<\ldots<x_{m},y_{1}<y_{2}<\ldots<y_{m}$ ,

we note that for $y_{1},i=1,2,\ldots,m$ and for $x_{i},i=1,2,\ldots,m$

D\left(\begin{array}[c]{c}x,x+h,x+2h,\ldots,x+mh\\ \\ y,y+k,y+2k,\ldots,y+mk\end{array};F\right)=0

where $x,y,h,k$ that $a\leq x,x+mh\leq b,c\leq y,y+mk\leq d$ and $m$

If the function $F\left(x,y\right)$ $x,y$ on $R$ $m$ .

$y$ and $k$ ,

D\left(\begin{tabular}[c]{l}$x_{1},x_{2},\ldots,x_{m+1}$\\ \\ $y,y+k,\ldots,y+mk$\end{tabular};F\right)=0

whatever $x_{i}\in\left[a,b\right],i=1,2,\ldots,m+1$ .

WE $f_{i}\left(x\right)=F\left(x,y+ik\right),\ i=1,2,\ldots,m,\ f\left(x\right)=F\left(x,y\right)$ . the points $x_{1},x_{2},\ldots,x_{m+1}$ and we put $f_{i}\left(y\right)=F\left(x_{i},y\right),\ i=1,2,\ldots,m,f\left(y\right)=F\left(x,y\right)$ ,

D\left(\begin{array}[c]{c}x_{1},x_{2},\ldots,x_{m+1}\\ \\ y_{1},y_{2},\ldots,y_{m+1}\end{array};F\right)=0

whatever $x_{i}\in\left[a,b\right],y_{i}\in\left[c,d\right],i=1,2,\ldots,m+1$ .

15. symmetrical $x$ and $y$ .

f_{i}\left(x\right)=a_{i,1}g_{1}\left(x\right)+a_{i,2}g_{2}\left(x\right)+\cdots+a_{i,m}g_{m}\left(x\right),\ i=1,2,\ldots,m

where $a_{i,j}$ $a_{i,j}=a_{j,i}$ .

linear set $e$ (so that $E_{x}=E_{y}=e$ ) $m$

(with $a\,ij=a\,j,i$ )

independent on $e$ . $m$ $x_{1},x_{2},\ldots,x_{m}\in e$

V^{\ast}=V\binom{g_{1},g_{1},\ldots,g_{m}}{x_{1},x_{2},\ldots,x_{m}}\neq 0.

allows us to write

\sum_{i=1}^{m}f_{i}\left(x\right)g_{i}\left(x_{k}\right)=\sum_{i=12}^{m}f_{i}\left(x_{k}\right)g_{i}\left(x\right),\ k=1,2,\ldots,m

$f_{i}\left(x\right)$ we show that $a_{i,j}=a_{j,i}$ .

\sum_{i\neq j}^{1,2,\ldots,m}\left(a_{i,j}-a_{j,i}\right)\left[g_{i}\left(x\right)g_{i}\left(y\right)-g_{i}\left(x\right)g_{j}\left(y\right)\right]=0.

From him $x$ and $y$ $x_{x}$ and $x_{l}$ $\binom{m}{2}$ $\binom{m}{2}$ $a_{i,j}-a_{j,i}$ .

of the determinant $V^{\ast}$ . is a power of $V^{\ast}$ ( $V^{\ast m+1}$ ) $.\$

It results that $a_{i,j}-a_{j,i}=0,\ i,j=1,2,\ldots,m$ ,

factor $a_{i,j}$ $0$ effectively $m$ , with the determinant $\left\|a_{i,j}\right\|$ of $0$ .

effective degree $r<m$ amounts of $r$ only by $x$ $y$ .

of two variables $x,y$ , $\varphi_{1}\left(x\right),\varphi_{2}\left(x\right),\ldots,\varphi_{r}\left(x\right),\varphi_{1}\left(y\right),\varphi_{2}\left(y\right),\ldots,\varphi_{r}\left(y\right)$ , where $r$ $\varphi_{i}\left(x\right)$

On some functional equations

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