On mean value theorems for continuous functions

Tiberiu Popoviciu
(original), Mathematical Analysis, mean value results, paper

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T. Popoviciu
Institutul de Calcul

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Folytonos függyények középértéktételeiröl

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T. Popoviciu, Folytonos függyények középértéktételeiröl, A Magyar Tudomanyos Académia III osztályának közle – ményeiböl, 4 (1954) no. 3, pp. 353-356 (in Hungarian).

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Note: republished in English in 2004: T. Popoviciu, On the mean-value theorem for continuous functions, East J. Approximations, 10 (2004) no. 3, pp. 379-382  (translated by Z. Finta) [MR2076896]

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[MR0065600, Zbl 0059.04502]

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1954 b -Popoviciu- Hungarian Academy of Sciences - On the averages of continuous functions
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ON THE MEAN VALUE THEOREMS OF CONTINUOUS FUNCTIONS
TIBERIU POPOVICIU
Presented at the public class meeting held on June 18, 1954

  1. Some f ( x ) f ( x ) f(x)f(x)f(x)The divided differences of a function can be interpreted with the following recursion formula:
[ x 1 , x 2 , … , x n + 1 ; f ] = [ x 2 , x 3 , … , x n + 1 ; f ] − [ x 1 , x 2 , … , x n ; f ] x n + 1 − x 1 , [ x 1 ; f ] = f ( x 1 ) . x 1 , x 2 , … , x n + 1 ; f = x 2 , x 3 , … , x n + 1 ; f − x 1 , x 2 , … , x n ; f x n + 1 − x 1 , x 1 ; f = f x 1 . {:[[x_(1),x_(2),dots,x_(n+1);f]=([x_(2),x_(3),dots,x_(n+1);f]-[x_(1),x_(2),dots,x_(n);f])/(x_(n+1)-x_(1))","],[[x_(1);f]=f(x_(1)).]:}\begin{gathered} {\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{\left[x_{2}, x_{3}, \ldots, x_{n+1} ; f\right]-\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right]}{x_{n+1}-x_{1}},} \\ {\left[x_{1} ; f\right]=f\left(x_{1}\right) .} \end{gathered}[x1,x2,…,xn+1;f]=[x2,x3,…,xn+1;f]−[x1,x2,…,xn;f]xn+1−x1,[x1;f]=f(x1).
At that time
(1) [ x 1 , x 2 , … , x n + 1 ; f ] (1) x 1 , x 2 , … , x n + 1 ; f {:(1)[x_(1),x_(2),dots,x_(n+1);f]:}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right] \tag{1} \end{equation*}(1)[x1,x2,…,xn+1;f]
n n nnn-ed-order divided difference and the x 1 , x 2 , … , x n + 1 x 1 , x 2 , … , x n + 1 x_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}x1,x2,…,xn+1points are its nodes. In this definition, we assume that the points of division are distinct and that they are ordered in increasing order:
(2) x 1 < x 2 < ⋯ < x n + 1 . (2) x 1 < x 2 < ⋯ < x n + 1 . {:(2)x_(1) < x_(2) < cdots < x_(n+1).:}\begin{equation*} x_{1}<x_{2}<\cdots<x_{n+1} . \tag{2} \end{equation*}(2)x1<x2<⋯<xn+1.
Starting from the case of different nodes, with border crossings and the f ( x ) f ( x ) f(x)f(x)f(x)By introducing successive derivatives of the function, we can move to the case of partially coincident nodes. In this sense, for example, [ x , x , … , x ; f ] n [ x , x , … , x ; f ] n [x,x,dots,x;f]n[x, x, \ldots, x ; f] n[x,x,…,x;f]n-order divided difference 1 n ! f ( n ) ( x ) 1 n ! f ( n ) ( x ) (1)/(n!)f^((n))(x)\frac{1}{n!} f^{(n)}(x)1n!f(n)(x)is equal to , where f ( n ) ( x ) f ( n ) ( x ) f^((n))(x)f^{(n)}(x)f(n)(x)the f ( x ) f ( x ) f(x)f(x)f(x)nth derivative of a function. Divided differences can therefore be considered as generalizations of derivatives.
However, since Stieltjes we have known that if f ( x ) f ( x ) f(x)f(x)f(x)nth derivative of x 0 x 0 x_(0)x_{0}x0is continuous at a point, then the divided difference (1) 1 n ! f ( n ) ( x 0 ) 1 n ! f ( n ) x 0 (1)/(n!)f^((n))(x_(0))\frac{1}{n!} f^{(n)}\left(x_{0}\right)1n!f(n)(x0)to when x 1 , x 2 , … , x n + 1 → x 0 x 1 , x 2 , … , x n + 1 → x 0 x_(1),x_(2),dots,x_(n+1)rarrx_(0)x_{1}, x_{2}, \ldots, x_{n+1} \rightarrow x_{0}x1,x2,…,xn+1→x0.
Ha f ( n ) ( x ) f ( n ) ( x ) f^((n))(x)f^{(n)}(x)f(n)(x)exists in the smallest interval containing the nodes, then
(3) [ x 1 , x 2 , … , x n + 1 ; f ] = f ( n ) ( x ) n ! (3) x 1 , x 2 , … , x n + 1 ; f = f ( n ) ( x ) n ! {:(3)[x_(1),x_(2),dots,x_(n+1);f]=(f^((n))(xi))/(n!):}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{f^{(n)}(\xi)}{n!} \tag{3} \end{equation*}(3)[x1,x2,…,xn+1;f]=f(n)(x)n!
where x ∈ ( x 1 , x n + 1 ) x ∈ x 1 , x n + 1 xi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right)x∈(x1,xn+1); this is it n = 1 n = 1 n=1n=1n=1generalization of the classical Lagrange formula corresponding to this case.
In the following we present some general properties of formula (3).
2. The theory of divided differences contains a host of completely elementary formulas, which provide various relations between the function and its derivatives by passing through the limit. We will use one of these in the following.
Let it be
(4)
x 1 < x 2 < ⋯ < x m ( m ≧ n + 1 ) x 1 < x 2 < ⋯ < x m ( m ≧ n + 1 ) x_(1) < x_(2) < cdots < x_(m)quad(m >= n+1)x_{1}<x_{2}<\cdots<x_{m} \quad(m \geqq n+1)x1<x2<⋯<xm(m≧n+1)
m m mmma series of different points and introduce the
(5) D i j [ f ] = [ x i , x i + 1 , … , x i + j ; f ] ( i = 1 , 2 , … , m − j , j = 0 , 1 , … , m − 1 ) (5) D i j [ f ] = x i , x i + 1 , … , x i + j ; f ( i = 1 , 2 , … , m − j , j = 0 , 1 , … , m − 1 ) {:[(5)D_(i)^(j)[f]=[x_(i),x_(i+1),dots,x_(i+j);f]],[(i=1","2","dots","m-j","quad j=0","1","dots","m-1)]:}\begin{gather*} D_{i}^{j}[f]=\left[x_{i}, x_{i+1}, \ldots, x_{i+j} ; f\right] \tag{5}\\ (i=1,2, \ldots, m-j, \quad j=0,1, \ldots, m-1) \end{gather*}(5)Dij[f]=[xi,xi+1,…,xi+j;f](i=1,2,…,m−j,j=0,1,…,m−1)
mark. Then 1 = i 1 < i 2 < ⋯ < i n + 1 = m 1 = i 1 < i 2 < ⋯ < i n + 1 = m 1=i_(1) < i_(2) < cdots < i_(n+1)=m1=i_{1}<i_{2}<\cdots<i_{n+1}=m1=i1<i2<⋯<in+1=min case of
(6) [ x i 1 , x i 2 , … , x i n + 1 ; f ] = ∑ i = 1 m − n A i D i n [ f ] , (6) x i 1 , x i 2 , … , x i n + 1 ; f = ∑ i = 1 m − n   A i D i n [ f ] , {:(6)[x_(i_(1)),x_(i_(2)),dots,x_(i_(n+1));f]=sum_(i=1)^(m-n)A_(i)D_(i)^(n)[f]",":}\begin{equation*} \left[x_{i_{1}}, x_{i_{2}}, \ldots, x_{i_{n+1}} ; f\right]=\sum_{i=1}^{m-n} A_{i} D_{i}^{n}[f], \tag{6} \end{equation*}(6)[xi1,xi2,…,xin+1;f]=∑i=1m−nAiDin[f],
where the A i A i A_(i)A_{i}Aicoefficients do not depend on f ( x ) f ( x ) f(x)f(x)f(x)from function and
∑ i = 1 m − n A i = 1 , A i > 0 ( i = 1 , 2 , … , m − n ) . ∑ i = 1 m − n   A i = 1 , A i > 0 ( i = 1 , 2 , … , m − n ) . sum_(i=1)^(m-n)A_(i)=1,quadA_(i) > 0quad(i=1,2,dots,m-n).\sum_{i=1}^{m-n} A_{i}=1, \quad A_{i}>0 \quad(i=1,2, \ldots, m-n) .∑i=1m−nAi=1,Ai>0(i=1,2,…,m−n).
Formula (6) shows that any point chosen from (4) n + 1 n + 1 n+1n+1n+1node-specific divided difference is the series (4) n + 1 n + 1 n+1n+1n+1is the weighted arithmetic mean of the divided differences of consecutive points. This mean value theorem serves as the basis for the following theorems.
3. First of all, it can be shown that the following generalization of formula (3) is:
  1. Item If the f ( x ) f ( x ) f(x)f(x)f(x)function is continuous in an interval containing the nodes (2), then we can find such ξ ∈ ( x 1 , x n + 1 ) ξ ∈ x 1 , x n + 1 xi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right)x∈(x1,xn+1)place to ξ ξ xi\xixit is everywhere around n + 1 n + 1 n+1n+1n+1different and ξ ξ xi\xixsplit in two by x 1 ′ , x 2 ′ , … , x n + 1 ′ x 1 ′ , x 2 ′ , … , x n + 1 ′ x_(1)^('),x_(2)^('),dots,x_(n+1)^(')x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime}x1′,x2′,…,xn+1′points to which the
[ x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f ] = [ x 1 , x 2 , … , x n + 1 ; f ] x 1 ′ , x 2 ′ , … , x n + 1 ′ ; f = x 1 , x 2 , … , x n + 1 ; f [x_(1)^('),x_(2)^('),dots,x_(n+1)^(');f]=[x_(1),x_(2),dots,x_(n+1);f]\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime} ; f\right]=\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right][x1′,x2′,…,xn+1′;f]=[x1,x2,…,xn+1;f]
equality is fulfilled.
The x i ′ x i ′ x_(i)^(')x_{i}^{\prime}xi′points, then we say that the ξ ξ xi\xixplace separates them if ξ ξ xi\xixthe x i ′ x i ′ x_(i)^(')x_{i}^{\prime}xi′falls inside the smallest interval containing the points.
Regarding Theorem 1, we can make the following remark.
If the nodes (2) are equidistant:
then
where
x i = c + ( i − 1 ) h ( i = 1 , 2 , … , n + 1 ) , [ c , c + h , … , c + n h ; f ] = 1 n ! h n λ h n f ( c ) , x i = c + ( i − 1 ) h ( i = 1 , 2 , … , n + 1 ) , [ c , c + h , … , c + n h ; f ] = 1 n ! h n λ h n f ( c ) , {:[x_(i)=c+(i-1)h quad(i=1","2","dots","n+1)","],[[c","c+h","dots","c+nh;f]=(1)/(n!h^(n))lambda_(h)^(n)f(c)","]:}\begin{gathered} x_{i}=c+(i-1) h \quad(i=1,2, \ldots, n+1), \\ {[c, c+h, \ldots, c+n h ; f]=\frac{1}{n!h^{n}} \lambda_{h}^{n} f(c),} \end{gathered}xi=c+(i−1)h(i=1,2,…,n+1),[c,c+h,…,c+nh;f]=1n!hnlhnf(c),
A h n f ( c ) = ∑ i = 1 n ( − 1 ) n − i ( n i ) f ( c + i h ) A h n f ( c ) = ∑ i = 1 n   ( − 1 ) n − i ( n i ) f ( c + i h ) A_(h)^(n)f(c)=sum_(i=1)^(n)(-1)^(n-i)((n)/(i))f(c+ih)A_{h}^{n} f(c)=\sum_{i=1}^{n}(-1)^{n-i}\binom{n}{i} f(c+i h)Ahnf(c)=∑i=1n(−1)n−i(ni)f(c+ih)
It can be shown that if f ( x ) f ( x ) f(x)f(x)f(x)is continuous in an interval containing the nodes (2), then there are in this interval such c , … , c + n h c , … , c + n h c,dots,c+nhc, \ldots, c+n hc,…,c+nhpoints that
(7) [ x 1 , x 2 , … , x n + 1 ; f ] = 1 n ! h n A h n f ( c ) . (7) x 1 , x 2 , … , x n + 1 ; f = 1 n ! h n A h n f ( c ) . {:(7)[x_(1),x_(2),dots,x_(n+1);f]=(1)/(n!h^(n))A_(h)^(n)f(c).:}\begin{equation*} \left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{1}{n!h^{n}} A_{h}^{n} f(c) . \tag{7} \end{equation*}(7)[x1,x2,…,xn+1;f]=1n!hnAhnf(c).
This leads to the following theorem of S. N. Bernstein and D. Rajkov:
Ha f ( x ) f ( x ) f(x)f(x)f(x)is continuous in an interval containing the nodes (2), then there is a ξ ∈ ( x 1 , x n + 1 ) ξ ∈ x 1 , x n + 1 xi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right)x∈(x1,xn+1)just so that everything ε > 0 ε > 0 epsi > 0\varepsilon>0e>0must include at least one c c cccand one h ≠ 0 , ξ − ε < c < ξ < c + n h < ξ + ε h ≠ 0 , ξ − ε < c < ξ < c + n h < ξ + ε h!=0,xi-epsi < c < xi < c+nh < xi+epsih \neq 0, \xi-\varepsilon<c<\xi<c+n h<\xi+\varepsilonh≠0,x−e<c<x<c+nh<x+e, for which (7) holds.
4. However, an even more general extension of formula (3) can be given. This is based on a corresponding extension of the concept of divided difference.
Consider points (4) if m m mmmis large enough, and we use the notations (5). Let us form a nonnegative integer k k kkknext to the
U i 1 , i 2 , … , i n + 1 n , k [ f ] = ‖ D i j k [ x k ] D i j k [ x k + 1 ] … D i j k [ x k + n − 1 ] D i j k [ f ] ‖ ( j = 1 , 2 , … , n + 1 ) U i 1 , i 2 , … , i n + 1 n , k [ f ] = D i j k x k D i j k x k + 1 … D i j k x k + n − 1 D i j k [ f ] ( j = 1 , 2 , … , n + 1 ) U_(i_(1),i_(2),dots,i_(n+1))^(n,k)[f]=||D_(i_(j))^(k)[x^(k)]D_(i_(j))^(k)[x^(k+1)]dotsD_(i_(j))^(k)[x^(k+n-1)]D_(i_(j))^(k)[f]||(j=1,2,dots,n+1)U_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, k}[f]=\left\|D_{i_{j}}^{k}\left[x^{k}\right] D_{i_{j}}^{k}\left[x^{k+1}\right] \ldots D_{i_{j}}^{k}\left[x^{k+n-1}\right] D_{i_{j}}^{k}[f]\right\|(j=1,2, \ldots, n+1)INi1,i2,…,in+1n,k[f]=‖Dijk[xk]Dijk[xk+1]…Dijk[xk+n−1]Dijk[f]‖(j=1,2,…,n+1) ( n + 1 n + 1 n+1n+1n+1)-order determinant.
At that time
(8) V i 1 , i 2 , … , i n + 1 n , k = U i 1 , i 2 , … , i n + 1 n , k [ x l + n ] (8) V i 1 , i 2 , … , i n + 1 n , k = U i 1 , i 2 , … , i n + 1 n , k x l + n {:(8)V_(i_(1),i_(2),dots,i_(n+1))^(n,k)=U_(i_(1),i_(2),dots,i_(n+1))^(n,k)[x^(l+n)]:}\begin{equation*} V_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, k}=U_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, k}\left[x^{l+n}\right] \tag{8} \end{equation*}(8)Ini1,i2,…,in+1n,k=INi1,i2,…,in+1n,k[xl+n]
generalization of the Vandermonde determinant,
(9) D i 1 , i 2 , … , i n + 1 n , i [ f ] = U i 1 , … , i n + 1 n , k [ f ] / V i 1 , … , i n + 1 n , k (9) D i 1 , i 2 , … , i n + 1 n , i [ f ] = U i 1 , … , i n + 1 n , k [ f ] / V i 1 , … , i n + 1 n , k {:(9)D_(i_(1),i_(2),dots,i_(n+1))^(n,i)[f]=U_(i_(1),dots,i_(n+1))^(n,k)[f]//V_(i_(1),dots,i_(n+1))^(n,k):}\begin{equation*} D_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, i}[f]=U_{i_{1}, \ldots, i_{n+1}}^{n, k}[f] / V_{i_{1}, \ldots, i_{n+1}}^{n, k} \tag{9} \end{equation*}(9)Di1,i2,…,in+1n,i[f]=INi1,…,in+1n,k[f]/Ini1,…,in+1n,k
is a type of repeated divided differences.
Expression (9) naturally only makes sense if the determinant (8) is nonzero. For this, it is necessary that i j i j i_(j)i_{j}ijindices should all be different, when it can be assumed that
1 = i 1 < i 2 < ⋯ < i n + 1 = m − k . 1 = i 1 < i 2 < ⋯ < i n + 1 = m − k . 1=i_(1) < i_(2) < cdots < i_(n+1)=m-k.1=i_{1}<i_{2}<\cdots<i_{n+1}=m-k .1=i1<i2<⋯<in+1=m−k.
It can then be shown that
U i 1 , i 2 , … , i n + 1 n , k [ f ] = ∑ i = 1 m − k − n B i D i n + k [ f ] , U i 1 , i 2 , … , i n + 1 n , k [ f ] = ∑ i = 1 m − k − n   B i D i n + k [ f ] , U_(i_(1),i_(2),dots,i_(n+1))^(n,k)[f]=sum_(i=1)^(m-k-n)B_(i)D_(i)^(n+k)[f],U_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, k}[f]=\sum_{i=1}^{m-k-n} B_{i} D_{i}^{n+k}[f],INi1,i2,…,in+1n,k[f]=∑i=1m−k−nBiDin+k[f],
where the B i B i B_(i)B_{i}Bicoefficients do not depend on f ( x ) f ( x ) f(x)f(x)f(x)function and are positive. It follows that
V i 1 , i 2 , … , i n + 1 n , k = ∑ i = 1 m − l − n B i > 0 . V i 1 , i 2 , … , i n + 1 n , k = ∑ i = 1 m − l − n   B i > 0 . V_(i_(1),i_(2),dots,i_(n+1))^(n,k)=sum_(i=1)^(m-l-n)B_(i) > 0.V_{i_{1}, i_{2}, \ldots, i_{n+1}}^{n, k}=\sum_{i=1}^{m-l-n} B_{i}>0 .Ini1,i2,…,in+1n,k=∑i=1m−l−nBi>0.
It follows that the repeated divided difference (9) is
D i n + k [ f ] ( i = 1 , 2 , … , m − k − n ) D i n + k [ f ] ( i = 1 , 2 , … , m − k − n ) D_(i)^(n+k)[f]quad(i=1,2,dots,m-k-n)D_{i}^{n+k}[f] \quad(i=1,2, \ldots, m-k-n)Din+k[f](i=1,2,…,m−k−n)
arithmetic mean of the divided differences.
5. However, it can also be shown conversely that at least in the case when the (4) nodes are equidistant, any ( n + k n + k n+kn+kn+kThe )-order divided difference is the arithmetic mean of repeated divided differences. This finally gives the following general theorem:
Theorem 2. If f ( x ) f ( x ) f(x)f(x)f(x)is continuous in an interval containing the nodes (2) and if k k kkkis a natural number not greater than n, then there exist such ξ 0 , ξ 1 , … , ξ n − k ∈ ( x 1 , x n + 1 ) , ξ 0 < ξ 1 < ⋯ < ξ n − k ξ 0 , ξ 1 , … , ξ n − k ∈ x 1 , x n + 1 , ξ 0 < ξ 1 < ⋯ < ξ n − k xi_(0),xi_(1),dots,xi_(n-k)in(x_(1),x_(n+1)),xi_(0) < xi_(1) < cdots < xi_(n-k)\xi_{0}, \xi_{1}, \ldots, \xi_{n-k} \in\left(x_{1}, x_{n+1}\right), \xi_{0}<\xi_{1}<\cdots<\xi_{n-k}x0,x1,…,xn−k∈(x1,xn+1),x0<x1<⋯<xn−kpoints that all of these have in their environment ( n − k + 1 ) ( k + 1 ) ( n − k + 1 ) ( k + 1 ) (n-k+1)(k+1)(n-k+1)(k+1)(n−k+1)(k+1)number
x 1 ′ < x 2 ′ < ⋯ < x ( n − k + 1 ) ′ ( k + 1 ) x 1 ′ < x 2 ′ < ⋯ < x ( n − k + 1 ) ′ ( k + 1 ) x_(1)^(') < x_(2)^(') < cdots < x_((n-k+1))^(')(k+1)x_{1}^{\prime}<x_{2}^{\prime}<\cdots<x_{(n-k+1)}^{\prime}(k+1)x1′<x2′<⋯<x(n−k+1)′(k+1)
point, that
x j ( k + 1 ) + 1 ′ , x j ( k + 1 ) + 2 ′ , … , x ( j + 1 ) ( k + 1 ) ′ x j ( k + 1 ) + 1 ′ , x j ( k + 1 ) + 2 ′ , … , x ( j + 1 ) ( k + 1 ) ′ x_(j(k+1)+1)^('),x_(j(k+1)+2)^('),dots,x_((j+1)(k+1))^(')x_{j(k+1)+1}^{\prime}, x_{j(k+1)+2}^{\prime}, \ldots, x_{(j+1)(k+1)}^{\prime}xj(k+1)+1′,xj(k+1)+2′,…,x(j+1)(k+1)′
points for ξ j ξ j xi_(j)\xi_{j}xjspace divides it into two ( j = 0 , 1 , … , n − k ) ( j = 0 , 1 , … , n − k ) (j=0,1,dots,n-k)(j=0,1, \ldots, n-k)(j=0,1,…,n−k), and also
D 1 , ( k + 1 ) + 1 , 2 ( k + 1 ) + 1 , … , ( n − k ) ( k + 1 ) + 1 n − k , k [ f ] = [ x 1 , x 2 , … , x n + 1 ; f ] , D 1 , ( k + 1 ) + 1 , 2 ( k + 1 ) + 1 , … , ( n − k ) ( k + 1 ) + 1 n − k , k [ f ] = x 1 , x 2 , … , x n + 1 ; f , D_(1,(k+1)+1,2(k+1)+1,dots,(n-k)(k+1)+1)^(n-k,k)[f]=[x_(1),x_(2),dots,x_(n+1);f],D_{1,(k+1)+1,2(k+1)+1, \ldots,(n-k)(k+1)+1}^{n-k, k}[f]=\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right],D1,(k+1)+1,2(k+1)+1,…,(n−k)(k+1)+1n−k,k[f]=[x1,x2,…,xn+1;f],
where the repeated divided difference applies to the sequence of points (10).
It follows specifically from Theorem 2 that if f ( x ) f ( x ) f(x)f(x)f(x)is continuous in an interval containing the nodes (2) and if the f ( k ) ( x ) f ( k ) ( x ) f^((k))(x)f^{(k)}(x)f(k)(x)derivative exists ( x 1 , x n + 1 x 1 , x n + 1 x_(1),x_(n+1)x_{1}, x_{n+1}x1,xn+1) is found in this interval n − k + 1 n − k + 1 n-k+1n-k+1n−k+1different number ξ 0 , ξ 1 , … , ξ n − k ξ 0 , ξ 1 , … , ξ n − k xi_(0),xi_(1),dots,xi_(n-k)\xi_{0}, \xi_{1}, \ldots, \xi_{n-k}x0,x1,…,xn−kexactly so that
[ x 1 , x 2 , … , x n + 1 ; f ] = 1 n ( n − 1 ) … ( n − k + 1 ) [ ξ 0 , ξ 1 , … , ξ n − k ; f ( k ) ] . x 1 , x 2 , … , x n + 1 ; f = 1 n ( n − 1 ) … ( n − k + 1 ) ξ 0 , ξ 1 , … , ξ n − k ; f ( k ) . [x_(1),x_(2),dots,x_(n+1);f]=(1)/(n(n-1)dots(n-k+1))[xi_(0),xi_(1),dots,xi_(n-k);f^((k))].\left[x_{1}, x_{2}, \ldots, x_{n+1} ; f\right]=\frac{1}{n(n-1) \ldots(n-k+1)}\left[\xi_{0}, \xi_{1}, \ldots, \xi_{n-k} ; f^{(k)}\right] .[x1,x2,…,xn+1;f]=1n(n−1)…(n−k+1)[x0,x1,…,xn−k;f(k)].
  1. The importance of the mean value theorems listed is that they make it possible to carry out certain investigations related to the remainder of some approximate formulas of analysis, namely interpolation formulas and numerical differentiation and integration formulas. The remainder of such formulas is always
A [ ξ 1 ′ , ξ 2 , … , ξ n + 1 ; f ] + B [ ξ 1 ′ , ξ 2 ′ , … , ξ n + 1 ′ ; f ] A ξ 1 ′ , ξ 2 , … , ξ n + 1 ; f + B ξ 1 ′ , ξ 2 ′ , … , ξ n + 1 ′ ; f A[xi_(1)^('),xi_(2),dots,xi_(n+1);f]+B[xi_(1)^('),xi_(2)^('),dots,xi_(n+1)^(');f]A\left[\xi_{1}^{\prime}, \xi_{2}, \ldots, \xi_{n+1} ; f\right]+B\left[\xi_{1}^{\prime}, \xi_{2}^{\prime}, \ldots, \xi_{n+1}^{\prime} ; f\right]A[x1′,x2,…,xn+1;f]+B[x1′,x2′,…,xn+1′;f]
shaped, where on the one hand the ξ i ξ i xi_(i)\xi_{i}xi, on the other hand, the ξ i ′ ξ i ′ xi_(i)^(')\xi_{i}^{\prime}xi′nodes are different and the A , B A , B A,BA, BA,Bcoefficients do not depend on f ( x ) f ( x ) f(x)f(x)f(x)function. In some special cases, in addition, the A A AAAand B B BBBone of the coefficients disappears; I have already dealt with these cases before.

TABLE OF CONTENTS

SCIENTIFIC PUBLICATIONS

György Hajós: Report on the work and tasks of the Department .....
Comments by István Olda Kovács and László Kalmár ..... 303
György Alexits's reply ..... 309
Pál Gombás: The relationship between the statistical atomic model and wave mechanics ..... 315
Comments by Tibor Hoffmann, Imre Fényes and Laszló Kalmár ..... 317
Pál Gombás's reply ..... 323
326 Sándor Szalay: Studies on the adsorption of high atomic weight cations on humus colloids
Comment by Elemér Szadeczky-Kardos ..... 327
340L. Csakalov: On the two factor series in the theory of algebraic equations
Tiberiu Popoviciu: On the mean value theorems of continuous functions ..... 343
Pal Turán: On the roots of the Riemannian zeta function ..... 353
Alfréd Rényi: A new axiomatic construction of probability calculus ..... 357
Contribution by Ákos Császár ..... 369
István Vincze: On the mathematical statistical methods of quality control in mass production ..... 427
..... 429
Contributions by Karoly Sarkadi and Tibor Tallián ..... 442
1954

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