Remarks on a mean value formula for the generalized divided differences

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

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T. Popoviciu
Institutul de Calcul

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T. Popoviciu, Remarques sur une formule de la moyenne des différences divisées généralisées, Mathematica (Cluj), 2(25) (1960) no. 2, pp. 323-324 (in French)

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1960 h -Popoviciu- Mathematica – Remarques sur une formule de la moyenne des differences divisees generalisees

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Mathematica Cluj

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Published by the Romanian Academy  Publishing House

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1222-9016

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2601-744X

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1960 h -Popoviciu- Mathematica - Remarks on a formula for the mean of divided differences ge
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REMARKS ON A FORMULA FOR THE MEAN OF GENERALIZED DIVIDED DIFFERENCES

byTIBERIU POPOVICIUin Cluj

I propose to give a proof of Theorem 8 of my previous work [1]. I ask the reader to refer, for the notions, notations and numbering of the formulas, lemmas, theorems, etc., which will be used here, to my cited work [1].
Theorem 8 was deduced from Lemma 2, but it also follows from the following lemma:
L e m m e 2 ∗ L e m m e 2 ∗ Lemma 2^(**)\mathrm{L} \mathrm{e} \mathrm{m} \mathrm{m} \mathrm{e} 2^{*}Lemme2∗. Under the previous assumptions, we can find n + 2 n + 2 n+2n+2n+2distinct points x i ′ , i = 1 , 2 , … , n + 2 x i ′ , i = 1 , 2 , … , n + 2 x_(i)^('),i=1,2,dots,n+2x_{i}^{\prime}, i=1,2, \ldots, n+2xi′,i=1,2,…,n+2, so that: 1 ∘ 1 ∘ 1^(@)1^{\circ}1∘they are all included in the smallest term interval containing the points x i , 2 ∘ x i , 2 ∘ x_(i),2^(@)x_{i}, 2^{\circ}xi,2∘equality (41) is verified.
The previous hypotheses, which are discussed in the statement, are that the functions (18) and the functions (19) are continuous and form regular systems (I) of order k k kkkon the interval E E EEE, that among the points x i x i x_(i)x_{i}xiof E E EEE, not all combined, the same point is repeated at most k k kkktimes and that the functions / / ////be defined and continues on E E EEE.
We can assume n ≧ 1 n ≧ 1 n >= 1n \geqq 1n≧1.
Let us therefore consider the divided difference [ x 1 , x 2 , … , x n + 2 ; f ] = C x 1 , x 2 , … , x n + 2 ; f = C [x_(1),x_(2),dots,x_(n+2);f]=C\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]=C[x1,x2,…,xn+2;f]=Cand suppose that z 1 , z 2 , … , z p z 1 , z 2 , … , z p z_(1),z_(2),dots,z_(p)z_{1}, z_{2}, \ldots, z_{p}z1,z2,…,zplet the distinct nodes having the orders respectively k 1 , k 2 , … , k p k 1 , k 2 , … , k p k_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}k1,k2,…,kpof multiplicities, 1 ≦ k i ≦ k , i = 1 , 2 , … , p ( p > 1 ) 1 ≦ k i ≦ k , i = 1 , 2 , … , p ( p > 1 ) 1 <= k_(i) <= k,i=1,2,dots,p(p > 1)1 \leqq k_{i} \leqq k, i=1,2, \ldots, p(p>1)1≦ki≦k,i=1,2,…,p(p>1). We can assume z 1 < z 2 < … < z p z 1 < z 2 < … < z p z_(1) < z_(2) < dots < z_(p)z_{1}<z_{2}<\ldots<z_{p}z1<z2<…<zp. Let's add to the nodes x i x i x_(i)x_{i}xiAgain n ( p − 1 ) n ( p − 1 ) n(p-1)n(p-1)n(p−1)distinct points, different from x i x i x_(i)x_{i}xi, and of which exactly n n nnnare included in each of the open intervals ( z i , z i + 1 ) i = 1 , 2 , … , p − 1 z i , z i + 1 i = 1 , 2 , … , p − 1 (z_(i),z_(i+1))i=1,2,dots,p-1\left(z_{i}, z_{i+1}\right) i=1,2, \ldots, p-1(zi,zi+1)i=1,2,…,p−1. We thus have in total n p + 2 n p + 2 np+2n p+2np+2points (distinct or not) that we will designate by y i , i = 1 , 2 , … , n p + 2 y i , i = 1 , 2 , … , n p + 2 y_(i),i=1,2,dots,np+2y_{i}, i=1,2, \ldots, n p+2yi,i=1,2,…,np+2, assuming y 1 ≦ y 2 ≦ … ≦ y n p + 2 y 1 ≦ y 2 ≦ … ≦ y n p + 2 y_(1) <= y_(2) <= dots <= y_(np+2)y_{1} \leqq y_{2} \leqq \ldots \leqq y_{n p+2}y1≦y2≦…≦ynp+2.
If we pose L 0 = 0 , L I = k 1 + k 2 + … + k I + I n , I = 1 , 2 , … , p − 1 L 0 = 0 , L I = k 1 + k 2 + … + k I + I n , I = 1 , 2 , … , p − 1 l_(0)=0,l_(j)=k_(1)+k_(2)+dots+k_(j)+jn,j=1,2,dots,p-1l_{0}=0, l_{j}=k_{1}+k_{2}+\ldots+k_{j}+jn, j=1,2, \ldots, p-1L0=0,LI=k1+k2+…+kI+In,I=1,2,…,p−1, We have y L I − 1 + r = z I , r = 1 , 2 , … , k I , I = 1 , 2 , … , p y L I − 1 + r = z I , r = 1 , 2 , … , k I , I = 1 , 2 , … , p y_(l_(j-1)+r)=z_(j),r=1,2,dots,k_(j),j=1,2,dots,py_{l_{j-1}+r}=z_{j}, r=1.2, \ldots, k_{j}, j=1.2, \ldots, pyLI−1+r=zI,r=1,2,…,kI,I=1,2,…,pAnd
( α ) y L I − 1 + k I < y L I − 1 + k I + 1 < ⋯ < y L I − 1 + k I + n < y L I + 1 I = 1 , 2 , … , p − 1 ( α ) y L I − 1 + k I < y L I − 1 + k I + 1 < ⋯ < y L I − 1 + k I + n < y L I + 1 I = 1 , 2 , … , p − 1 {:[(alpha")"y_(l_(j-1)+k_(j)) < y_(l_(j-1)+k_(j)+1) < cdots < y_(l_(j-1)+k_(j)+n) < y_(l_(j)+1)],[j=1","2","dots","p-1]:}\begin{gather*} y_{l_{j-1}+k_{j}}<y_{l_{j-1}+k_{j}+1}<\cdots<y_{l_{j-1}+k_{j}+n}<y_{l_{j}+1} \tag{$\alpha$}\\ j=1,2, \ldots, p-1 \end{gather*}(α)yLI−1+kI<yLI−1+kI+1<⋯<yLI−1+kI+n<yLI+1I=1,2,…,p−1
But, by Theorem 6, the divided difference [ x 1 , x 2 , … , x n + 2 ; f ] x 1 , x 2 , … , x n + 2 ; f [x_(1),x_(2),dots,x_(n+2);f]\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right][x1,x2,…,xn+2;f]is a generalized arithmetic mean (with suitable positive weights) of the divided differences
( β ) [ y i , y i + 1 , … , y i + n + 2 ; f ] , i = 1 , 2 , … , n ( p − 1 ) + 1 ( β ) y i , y i + 1 , … , y i + n + 2 ; f , i = 1 , 2 , … , n ( p − 1 ) + 1 {:(beta)"[y_(i),y_(i+1),dots,y_(i+n+2);f]","quad i=1","2","dots","n(p-1)+1:}\begin{equation*} \left[y_{i}, y_{i+1}, \ldots, y_{i+n+2}; f\right], \quad i=1,2, \ldots, n(p-1)+1 \tag{$\beta$} \end{equation*}(β)[yi,yi+1,…,yi+n+2;f],i=1,2,…,n(p−1)+1
These divided differences enjoy the property that their nodes are always included in the smallest closed interval containing the points x i x i x_(i)x_{i}xiand that at most one of these nodes is multiple, with an order of multiplicity ≦ k ≦ k <= k\leqq k≦k, the others being simple (the divided differences ( β ) ( β ) (beta)(\beta)(β)are therefore well defined).
If the differences divided ( β ) ( β ) (beta)(\beta)(β)are not all equal, we can find (at least) one whose value is < C < C < C<C<Cand (at least) one whose value is > C > C > C>C>C. The existence of distinct nodes x L ′ x L ′ x_(l)^(')x_{l}^{\prime}xL′verifying equality (41) is then established as in case 1 of nr. 18.
If the divided differences ( β β beta\betaβ) are all equal, then their common value is C C CCCand the lemma 2 ∗ 2 ∗ 2^(**)2^{*}2∗results in taking, for example, x i ′ = y k + i − 1 x i ′ = y k + i − 1 x_(i)^(')=y_(k+i-1)x_{i}^{\prime}=y_{k+i-1}xi′=yk+i−1, i = 1 , 2 , … , n + 2 i = 1 , 2 , … , n + 2 i=1,2,dots,n+2i=1,2, \ldots, n+2i=1,2,…,n+2, which are good n + 2 n + 2 n+2n+2n+2distinct points of the smallest closed interval containing the nodes x i x i x_(i)x_{i}xi.
Lemma 2* is therefore proven.
It is easy to see that Lemmas 2 and 2 ∗ 2 ∗ 2^(**)2^{*}2∗are equivalent. In this way, the few typographical and transcription errors in the manuscript that crept into the proof of Lemma 2* are eliminated.

BIBLIOGRAPHY

[1] Popoviciu T., On the remainder in certain linear approximation formulas of analysis. Mathematica, 1 (24), 95-142 (1960).
Received on 26. V. 1960.
*) These errors, from p. 116, easy to see must be corrected as follows:
line 8 k 1 + k p > 2 instead of k 1 + k 2 > 2 , . 9 k p > 1 , . k 1 > 1 . . 10 k 1 > 1 . , k 2 > 1 . . 11 k 1 = k p = 1 . , k 1 = k 2 = 1  line  8 k 1 + k p > 2  instead of  k 1 + k 2 > 2 , . 9 k p > 1 , . k 1 > 1 . . 10 k 1 > 1 . , k 2 > 1 . . 11 k 1 = k p = 1 . , k 1 = k 2 = 1 {:[" line ",8,k_(1)+k_(p) > 2," instead of ",k_(1)+k_(2) > 2],[",".,9,k_(p) > 1,",".,k_(1) > 1],[..,10,k_(1) > 1,.",",k_(2) > 1],[..,11,k_(1)=k_(p)=1,.",",k_(1)=k_(2)=1]:}\begin{array}{rrrcr} \text { line } & 8 & k_{1}+k_{p}>2 & \text { instead of } & k_{1}+k_{2}>2 \\ , . & 9 & k_{p}>1 & , . & k_{1}>1 \\ . . & 10 & k_{1}>1 & ., & k_{2}>1 \\ . . & 11 & k_{1}=k_{p}=1 & ., & k_{1}=k_{2}=1 \end{array} line 8k1+kp>2 instead of k1+k2>2,.9kp>1,.k1>1..10k1>1.,k2>1..11k1=kp=1.,k1=k2=1
1960

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