On the Remainder in Certain Quadrature Formulas

Tiberiu Popoviciu
(original), Approximation Theory, paper, quadrature, remainder estimation

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Tiberiu Popoviciu
Institutul de Calcul

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T. Popoviciu, Sur le reste de certaines formules de quadrature, Aequationes Math., 2 (1968) no. 1, pp. 128-129 (short communication) (in French)

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1969 a -Popoviciu- Aeq. Math. – Sur le reste de certaines formules de quadrature – short communication

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Aequationes Math.

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http://doi.org/10.1007/BF01833507

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1969 a -Popoviciu- Aeq. Math. - On the remainder of certain quadrature formulas - short communicati
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On the rest of some quadrature formulas**)

Tiberiu Popoviciu

The rest R [ f ] R [ f ] R[f]R[f]R[f]of the quadrature formula
(1) ∫ has b f ( x ) d x = ∑ α = 1 n HAS α f ( x α ) + R [ f ] (1) ∫ has b   f ( x ) d x = ∑ α = 1 n   HAS α f x α + R [ f ] {:(1)int_(a)^(b)f(x)dx=sum_(alpha=1)^(n)A_(alpha)f(x_(alpha))+R[f]:}\begin{equation*} \int_{a}^{b} f(x) dx=\sum_{\alpha=1}^{n} A_{\alpha} f\left(x_{\alpha}\right)+R[f] \tag{1} \end{equation*}(1)∫hasbf(x)dx=∑α=1nHASαf(xα)+R[f]
where the nodes x α x α x_(alpha)x_{\alpha}xαof the real axis are distinct and the HAS α HAS α A_(alpha)A_{\alpha}HASαare given real constants, is of the form
(2) R [ f ] = HAS [ ξ 1 , ξ 2 , … , ξ m + 2 ; f ] + B [ ξ 1 ′ , ξ 2 ′ , … , ξ m + 2 ′ ; f ] (2) R [ f ] = HAS ξ 1 , ξ 2 , … , ξ m + 2 ; f + B ξ 1 ′ , ξ 2 ′ , … , ξ m + 2 ′ ; f {:(2)R[f]=A[xi_(1),xi_(2),dots,xi_(m+2);f]+B[xi_(1)^('),xi_(2)^('),dots,xi_(m+2)^(');f]:}\begin{equation*} R[f]=A\left[\xi_{1}, \xi_{2}, \ldots, \xi_{m+2}; f\right]+B\left[\xi_{1}^{\prime}, \xi_{2}^{\prime}, \ldots, \xi_{m+2}^{\prime}; f\right] \tag{2} \end{equation*}(2)R[f]=HAS[ξ1,ξ2,…,ξm+2;f]+B[ξ1′,ξ2′,…,ξm+2′;f]
The function f f fffis assumed to be continuous, m m mmmis the degree of accuracy of formula (1), the points ξ α ξ α xi_(alpha)\xi_{\alpha}ξαon the one hand and the points ξ α ′ ξ α ′ xi_(alpha)^(')\xi_{\alpha}^{\prime}ξα′on the other hand, are distinct but generally depend on the function f f fff. The constants HAS , B HAS , B A,BA, BHAS,Bare independent of the function f f fff. Finally [ y 1 , y 2 , … , y r ; f ] y 1 , y 2 , … , y r ; f [y_(1),y_(2),dots,y_(r);f]\left[y_{1}, y_{2}, \ldots, y_{r}; f\right][y1,y2,…,yr;f]denotes the divided difference, of order r − 1 r − 1 r-1r-1r−1, of the function f f fffon the knots y 1 , y 2 , … , y r y 1 , y 2 , … , y r y_(1),y_(2),dots,y_(r)y_{1}, y_{2}, \ldots, y_{r}y1,y2,…,yr.
When in (2) we can take B = 0 B = 0 B=0B=0B=0the rest R [ f ] R [ f ] R[f]R[f]R[f](or the quadrature formula (1)) is said to be of the simple form. This latter notion is closely linked to the theory of higher-order convex functions [1].
In the present work we show how, under well-specified hypotheses, in the case where the remainder is not of the simple form, we can re-establish a sort of simplicity by a suitable generalization of the notion of divided difference and of the corresponding convexity.
[1] Popoviciu, Tiberiu, Mathematica 1 (24), 95-142 1959.

Endomorphism ring in the Galoisschen Theory *)

Wolfgang Krull

Es sei N N NNNein beliebiger Körper. Jedem c ∈ N c ∈ N c in Nc \in Nc∈Nwerde der durch ,, c ′ has = c ⋅ has c ′ has = c ⋅ has c^(')a=c*ac^{\prime} a=c \cdot ac′has=c⋅hasfor all has ∈ N ′ ′ has ∈ N ′ ′ a inN^('')a \in N^{\prime \prime}has∈N′′festgelegte Endomorphismus c ′ c ′ c^(')c^{\prime}c′the additive group N N NNNzugeordnet. Dann gilt: Allgemeinster Unabhängigkeitssatz: Sindh HAS 1 , … HAS n HAS 1 , … HAS n A_(1),dotsA_(n)A_{1}, \ldots A_{n}HAS1,…HASnIt is possible to change the Automorphism of the Körpers (and the Endomorphism of the Additive Group) N N NNN, so folgt aus, ( ∑ i = 1 n c i ′ HAS i ) α = 0 ∑ i = 1 n   c i ′ HAS i α = 0 (sum_(i=1)^(n)c_(i)^(')A_(i))alpha=0\left(\sum_{i=1}^{n} c_{i}^{\prime} A_{i}\right) \alpha=0(∑i=1nci′HASi)α=0for all α ∈ N ′ ′ α ∈ N ′ ′ alpha inN^('')\alpha \in N^{\prime \prime}α∈N′′stets c 1 ′ = ⋯ = c n ′ = 0 c 1 ′ = ⋯ = c n ′ = 0 c_(1)^(')=cdots=c_(n)^(')=0c_{1}^{\prime}=\cdots=c_{n}^{\prime}=0c1′=⋯=cn′=0. - Der Beweis arbeitet mit der Vandermondeschen Determinante; It is a construction and a fully-fledged element. - This is what I'm doing N N NNNseparabel und normal über K K KKKwith the Galois group G = { HAS 1 , … , HAS n } G = HAS 1 , … , HAS n G={A_(1),dots,A_(n)}G=\left\{A_{1}, \ldots, A_{n}\right\}G={HAS1,…,HASn}. It's a lovely thing to do. L L LLLzwischen N N NNNand K K KKKwerde L ′ = { c ′ ∣ c ∈ L } L ′ = c ′ ∣ c ∈ L L^(')={c^(')∣c in L}L^{\prime}=\left\{c^{\prime} \mid c \in L\right\}L′={c′∣c∈L}gesetzt. Nach Artin betrachtet et man schon in der elementaren Galoisschen Theorie den Ring K G ′ K G ′ K_(G)^(')K_{G}^{\prime}KG′go Endomorphisms { ∑ i = 1 n has i ′ HAS i ∣ has i ∈ K } ∑ i = 1 n   has i ′ HAS i ∣ has i ∈ K {sum_(i=1)^(n)a_(i)^(')A_(i)∣a_(i)in K}\left\{\sum_{i=1}^{n} a_{i}^{\prime} A_{i} \mid a_{i} \in K\right\}{∑i=1nhasi′HASi∣hasi∈K}, der zum Gruppenring von G G GGGüber K K KKKisomorph ist. Angesichts des allgemeinsten Unabhängigkeitssatzes erscheint es angebracht, neben K G ′ K G ′ K_(G)^(')K_{G}^{\prime}KG′gleich auch den Ring N G ′ N G ′ N_(G)^(')N_{G}^{\prime}NG′go Endomorphisms { ∑ i = 1 n c i ′ HAS i ∣ c ′ ∈ N } ∑ i = 1 n   c i ′ HAS i ∣ c ′ ∈ N {sum_(i=1)^(n)c_(i)^(')A_(i)∣c^(')in N}\left\{\sum_{i=1}^{n} c_{i}^{\prime} A_{i} \mid c^{\prime} \in N\right\}{∑i=1nci′HASi∣c′∈N}einzuführen. N G ′ N G ′ N_(G)^(')N_{G}^{\prime}NG′ist a Vektorraum der Dimension n 2 n 2 n^(2)n^{2}n2über K ′ K ′ K^(')K^{\prime}K′, aber nicht zum Gruppenring von G G GGGüber N N NNNisomorph, weil ia HAS i c ′ = ( HAS i c ) ′ HAS i ≠ c ′ HAS i HAS i c ′ = HAS i c ′ HAS i ≠ c ′ HAS i A_(i)c^(')=(A_(i)c)^(')A_(i)!=c^(')A_(i)A_{i} c^{\prime}=\left(A_{i} c\right)^{\prime} A_{i} \neq c^{\prime} A_{i}HASic′=(HASic)′HASi≠c′HASi. Definiert man für den Körper L L LLLdas Rechtsideal R G , L ′ R G , L ′ R_(G,L)^(')R_{G, L}^{\prime}RG,L′from N G ′ N G ′ N_(G)^(')N_{G}^{\prime}NG′through R G , L ′ = { ϕ ∣ ϕ c ∈ L R G , L ′ = { ϕ ∣ ϕ c ∈ L R_(G,L)^(')={phi∣phi c in L\mathrm{R}_{G, L}^{\prime}=\{\phi \mid \phi c \in LRG,L′={ϕ∣ϕc∈Lfor all c ∈ N } c ∈ N } c in N}c \in N\}c∈N}, so gelten die Sätze: Es ist R G , L ′ R G , L ′ R_(G,L)^(')R_{G, L}^{\prime}RG,L′gleich of the Komplexprodukt L ′ ⋅ R G , K ′ L ′ ⋅ R G , K ′ L^(')*R_(G,K)^(')L^{\prime} \cdot R_{G, K}^{\prime}L′⋅RG,K′erzeugten additiven
  1. *) Received December 12, 1967
    **) Received on 12.12.1967
  2. *) Entries on 13.12.1967

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