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), http://doi.org/10.1007/BF01833507

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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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Note: this note announces the paper T. Popoviciu, Sur le reste de certaines formules de quadrature, Aequationes Math., 2 (1969) nos. 2-3, pp. 265-268 (in French) http://doi.org/10.1007/BF01817710

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

Tiberiu Popoviciu

The remainder 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 remainder 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.

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