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Remarks on a mean value formula for the generalized divided differences
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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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Mathematica Cluj
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REMARKS ON A FORMULA FOR THE MEAN OF GENERALIZED DIVIDED DIFFERENCES
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:
Lemma 2^(**)\mathrm{L} \mathrm{e} \mathrm{m} \mathrm{m} \mathrm{e} 2^{*}. Under the previous assumptions, we can findn+2n+2distinct pointsx_(i)^('),i=1,2,dots,n+2x_{i}^{\prime}, i=1,2, \ldots, n+2, so that:1^(@)1^{\circ}they are all included in the smallest term interval containing the pointsx_(i),2^(@)x_{i}, 2^{\circ}equality (41) is verified.
Lemma 2^(**)\mathrm{L} \mathrm{e} \mathrm{m} \mathrm{m} \mathrm{e} 2^{*}. Under the previous assumptions, we can findn+2n+2distinct pointsx_(i)^('),i=1,2,dots,n+2x_{i}^{\prime}, i=1,2, \ldots, n+2, so that:1^(@)1^{\circ}they are all included in the smallest term interval containing the pointsx_(i),2^(@)x_{i}, 2^{\circ}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 orderkkon the intervalEE, that among the pointsx_(i)x_{i}ofEE, not all combined, the same point is repeated at mostkktimes and that the functions///be defined and continues onEE.
We can assumen >= 1n \geqq 1.
Let us therefore consider the divided difference[x_(1),x_(2),dots,x_(n+2);f]=C\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]=Cand suppose thatz_(1),z_(2),dots,z_(p)z_{1}, z_{2}, \ldots, z_{p}let the distinct nodes having the orders respectivelyk_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}of multiplicities,1 <= k_(i) <= k,i=1,2,dots,p(p > 1)1 \leqq k_{i} \leqq k, i=1,2, \ldots, p(p>1). We can assumez_(1) < z_(2) < dots < z_(p)z_{1}<z_{2}<\ldots<z_{p}. Let's add to the nodesx_(i)x_{i}Againn(p-1)n(p-1)distinct points, different fromx_(i)x_{i}, and of which exactlynnare included in each of the open intervals(z_(i),z_(i+1))i=1,2,dots,p-1\left(z_{i}, z_{i+1}\right) i=1,2, \ldots, p-1. We thus have in totalnp+2n p+2points (distinct or not) that we will designate byy_(i),i=1,2,dots,np+2y_{i}, i=1,2, \ldots, n p+2, assumingy_(1) <= y_(2) <= dots <= y_(np+2)y_{1} \leqq y_{2} \leqq \ldots \leqq y_{n p+2}.
Let us therefore consider the divided difference[x_(1),x_(2),dots,x_(n+2);f]=C\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]=Cand suppose thatz_(1),z_(2),dots,z_(p)z_{1}, z_{2}, \ldots, z_{p}let the distinct nodes having the orders respectivelyk_(1),k_(2),dots,k_(p)k_{1}, k_{2}, \ldots, k_{p}of multiplicities,1 <= k_(i) <= k,i=1,2,dots,p(p > 1)1 \leqq k_{i} \leqq k, i=1,2, \ldots, p(p>1). We can assumez_(1) < z_(2) < dots < z_(p)z_{1}<z_{2}<\ldots<z_{p}. Let's add to the nodesx_(i)x_{i}Againn(p-1)n(p-1)distinct points, different fromx_(i)x_{i}, and of which exactlynnare included in each of the open intervals(z_(i),z_(i+1))i=1,2,dots,p-1\left(z_{i}, z_{i+1}\right) i=1,2, \ldots, p-1. We thus have in totalnp+2n p+2points (distinct or not) that we will designate byy_(i),i=1,2,dots,np+2y_{i}, i=1,2, \ldots, n p+2, assumingy_(1) <= y_(2) <= dots <= y_(np+2)y_{1} \leqq y_{2} \leqq \ldots \leqq y_{n p+2}.
If we posel_(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-1, We havey_(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, pAnd
{:[(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*}
But, by Theorem 6, the divided difference[x_(1),x_(2),dots,x_(n+2);f]\left[x_{1}, x_{2}, \ldots, x_{n+2}; f\right]is a generalized arithmetic mean (with suitable positive weights) of the divided differences
{:(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*}
These divided differences enjoy the property that their nodes are always included in the smallest closed interval containing the pointsx_(i)x_{i}and that at most one of these nodes is multiple, with an order of multiplicity<= k\leqq k, the others being simple (the divided differences(beta)(\beta)are therefore well defined).
If the differences divided(beta)(\beta)are not all equal, we can find (at least) one whose value is< C<Cand (at least) one whose value is> C>C. The existence of distinct nodesx_(l)^(')x_{l}^{\prime}verifying equality (41) is then established as in case 1 of nr. 18.
If the divided differences (beta\beta) are all equal, then their common value isCCand the lemma2^(**)2^{*}results in taking, for example,x_(i)^(')=y_(k+i-1)x_{i}^{\prime}=y_{k+i-1},i=1,2,dots,n+2i=1,2, \ldots, n+2, which are goodn+2n+2distinct points of the smallest closed interval containing the nodesx_(i)x_{i}.
Lemma 2* is therefore proven.
It is easy to see that Lemmas 2 and2^(**)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.
It is easy to see that Lemmas 2 and2^(**)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:
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]:}\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}