Abstract
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T. Popoviciu
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algoritmul lui Euclid; cel mai mare divizor comun; numărul împărțirilor; șirul lui Fibonacci; baza sistemului de numerație
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T. Popoviciu, Asupra aplicării algoritmului lui Euclid pentru aflarea c.m.m.d.c. a două numere. Studii și cercetări științifice (Cluj), tom. IV, nr. 1-2, pag. 59-63 (1953)
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Studii şi cercetări Stiintifice (Cluj)
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Editura Academiei Republicii Populare Române
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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
- Somef(x)f(x)The divided differences of a function can be interpreted with the following recursion formula:
{:[[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}
At that time
{:(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*}
nn-ed-order divided difference and thex_(1),x_(2),dots,x_(n+1)x_{1}, x_{2}, \ldots, x_{n+1}points 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) < cdots < x_(n+1).:}\begin{equation*}
x_{1}<x_{2}<\cdots<x_{n+1} . \tag{2}
\end{equation*}
Starting from the case of different nodes, with border crossings and thef(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,dots,x;f]n[x, x, \ldots, x ; f] n-order divided difference(1)/(n!)f^((n))(x)\frac{1}{n!} f^{(n)}(x)is equal to , wheref^((n))(x)f^{(n)}(x)thef(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 iff(x)f(x)nth derivative ofx_(0)x_{0}is continuous at a point, then the divided difference (1)(1)/(n!)f^((n))(x_(0))\frac{1}{n!} f^{(n)}\left(x_{0}\right)to whenx_(1),x_(2),dots,x_(n+1)rarrx_(0)x_{1}, x_{2}, \ldots, x_{n+1} \rightarrow x_{0}.
Haf^((n))(x)f^{(n)}(x)exists in the smallest interval containing the nodes, then
{:(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*}
wherexi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right); this is itn=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.
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)
(4)
x_(1) < x_(2) < cdots < x_(m)quad(m >= n+1)x_{1}<x_{2}<\cdots<x_{m} \quad(m \geqq n+1)
mma series of different points and introduce the
{:[(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*}
mark. Then1=i_(1) < i_(2) < cdots < i_(n+1)=m1=i_{1}<i_{2}<\cdots<i_{n+1}=min case of
{:(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*}
where theA_(i)A_{i}coefficients do not depend onf(x)f(x)from function and
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) .
Formula (6) shows that any point chosen from (4)n+1n+1node-specific divided difference is the series (4)n+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:
3. First of all, it can be shown that the following generalization of formula (3) is:
- Item If thef(x)f(x)function is continuous in an interval containing the nodes (2), then we can find suchxi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right)place toxi\xiit is everywhere aroundn+1n+1different andxi\xisplit in two byx_(1)^('),x_(2)^('),dots,x_(n+1)^(')x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n+1}^{\prime}points to which the
[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]
equality is fulfilled.
Thex_(i)^(')x_{i}^{\prime}points, then we say that thexi\xiplace separates them ifxi\xithex_(i)^(')x_{i}^{\prime}falls inside the smallest interval containing the points.
Thex_(i)^(')x_{i}^{\prime}points, then we say that thexi\xiplace separates them ifxi\xithex_(i)^(')x_{i}^{\prime}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
If the nodes (2) are equidistant:
then
where
{:[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}
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)
It can be shown that iff(x)f(x)is continuous in an interval containing the nodes (2), then there are in this interval suchc,dots,c+nhc, \ldots, c+n hpoints that
{:(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*}
This leads to the following theorem of S. N. Bernstein and D. Rajkov:
Haf(x)f(x)is continuous in an interval containing the nodes (2), then there is axi in(x_(1),x_(n+1))\xi \in\left(x_{1}, x_{n+1}\right)just so that everythingepsi > 0\varepsilon>0must include at least oneccand oneh!=0,xi-epsi < c < xi < c+nh < xi+epsih \neq 0, \xi-\varepsilon<c<\xi<c+n h<\xi+\varepsilon, 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.
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) ifmmis large enough, and we use the notations (5). Let us form a nonnegative integerkknext to the
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) ( n+1n+1)-order determinant.
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) ( n+1n+1)-order determinant.
At that time
{:(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*}
generalization of the Vandermonde determinant,
{:(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*}
is a type of repeated divided differences.
Expression (9) naturally only makes sense if the determinant (8) is nonzero. For this, it is necessary thati_(j)i_{j}indices should all be different, when it can be assumed that
Expression (9) naturally only makes sense if the determinant (8) is nonzero. For this, it is necessary thati_(j)i_{j}indices should all be different, when it can be assumed that
1=i_(1) < i_(2) < cdots < i_(n+1)=m-k.1=i_{1}<i_{2}<\cdots<i_{n+1}=m-k .
It can then be shown that
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],
where theB_(i)B_{i}coefficients do not depend onf(x)f(x)function and are positive. It follows that
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 .
It follows that the repeated divided difference (9) is
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)
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+kn+kThe )-order divided difference is the arithmetic mean of repeated divided differences. This finally gives the following general theorem:
Theorem 2. Iff(x)f(x)is continuous in an interval containing the nodes (2) and ifkkis a natural number not greater than n, then there exist suchxi_(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}points that all of these have in their environment(n-k+1)(k+1)(n-k+1)(k+1)number
5. However, it can also be shown conversely that at least in the case when the (4) nodes are equidistant, any (n+kn+kThe )-order divided difference is the arithmetic mean of repeated divided differences. This finally gives the following general theorem:
Theorem 2. Iff(x)f(x)is continuous in an interval containing the nodes (2) and ifkkis a natural number not greater than n, then there exist suchxi_(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}points that all of these have in their environment(n-k+1)(k+1)(n-k+1)(k+1)number
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)
point, that
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}
points forxi_(j)\xi_{j}space divides it into two(j=0,1,dots,n-k)(j=0,1, \ldots, n-k), and also
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],
where the repeated divided difference applies to the sequence of points (10).
It follows specifically from Theorem 2 that iff(x)f(x)is continuous in an interval containing the nodes (2) and if thef^((k))(x)f^{(k)}(x)derivative exists (x_(1),x_(n+1)x_{1}, x_{n+1}) is found in this intervaln-k+1n-k+1different numberxi_(0),xi_(1),dots,xi_(n-k)\xi_{0}, \xi_{1}, \ldots, \xi_{n-k}exactly so that
It follows specifically from Theorem 2 that iff(x)f(x)is continuous in an interval containing the nodes (2) and if thef^((k))(x)f^{(k)}(x)derivative exists (x_(1),x_(n+1)x_{1}, x_{n+1}) is found in this intervaln-k+1n-k+1different numberxi_(0),xi_(1),dots,xi_(n-k)\xi_{0}, \xi_{1}, \ldots, \xi_{n-k}exactly so that
[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] .
- 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[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]
shaped, where on the one hand thexi_(i)\xi_{i}, on the other hand, thexi_(i)^(')\xi_{i}^{\prime}nodes are different and theA,BA, Bcoefficients do not depend onf(x)f(x)function. In some special cases, in addition, theAAandBBone of the coefficients disappears; I have already dealt with these cases before.
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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
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Contributions by Karoly Sarkadi and Tibor Tallián ..... 442