On the form of the remainder in certain quadrature formulas

Tiberiu Popoviciu
(proceedings), Approximation Theory, paper, quadrature

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Tiberiu Popoviciu (Institutul de Calcul)

Tiberiu Popoviciu (1906-1975)

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Sur la forme du reste de certaines formules de quadrature

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T. Popoviciu, Sur la forme du reste de certaines formules de quadrature, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), pp. 365-370. Akadémiai Kiadó, Budapest, 1972 (in French)

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1972 c -Popoviciu- Proceedings – Sur la forme du reste de certaines formules de quadrature

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Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969)

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Akadémiai Kiadó, Budapest

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1972 c -Popoviciu- Proceedings - On the form of the remainder of certain quadrature formulas
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ON THE FORM OF THE REMAINDER OF CERTAIN QUADRATURE FORMULAS

T. POPOVICIU (CLUJ)

  1. In many approximation formulas of analysis, the remainder R [ f ] R [ f ] R[f]R[f]R[f]is a linear (additive and homogeneous) functional defined on a linear set S S SSSfunctions f f fffcontinuous real numbers defined on the same interval I I III(of non-zero length) of the real axis.
If the whole S S SSScontains all polynomials and if the equalities
(1) R [ 1 ] = R [ x ] = … = R [ x m ] = 0 (1) R [ 1 ] = R [ x ] = … = R x m = 0 {:(1)R[1]=R[x]=dots=R[x^(m)]=0:}\begin{equation*} R[1]=R[x]=\ldots=R\left[x^{m}\right]=0 \tag{1} \end{equation*}(1)R[1]=R[x]=…=R[xm]=0
as well as inequality
(2) R [ x m + 1 ] ≠ 0 (2) R x m + 1 ≠ 0 {:(2)R[x^(m+1)]!=0:}\begin{equation*} R\left[x^{m+1}\right] \neq 0 \tag{2} \end{equation*}(2)R[xm+1]≠0
are verified for a certain integer m ≥ − 1 m ≥ − 1 m >= -1m \geq-1m≥−1, we say that the linear functional R [ f ] R [ f ] R[f]R[f]R[f]is the degree of accuracy m m mmm(or that m m mmm(is its degree of accuracy). This number m m mmmIf it exists, it is, moreover, clearly defined. If m = − 1 m = − 1 m=-1m=-1m=−1Relations (1), (2) must be replaced by the single inequality R [ 1 ] ≠ 0 R [ 1 ] ≠ 0 R[1]!=0R[1] \neq 0R[1]≠0.
Let us consider, in particular, the quadrature formula
(3) ∫ has b f ( x ) d V ( x ) = ∑ ν = 0 n c ν f ( x ν ) + R [ f ] (3) ∫ has b   f ( x ) d V ( x ) = ∑ ν = 0 n   c ν f x ν + R [ f ] {:(3)int_(a)^(b)f(x)dV(x)=sum_(nu=0)^(n)c_(nu)f(x_(nu))+R[f]:}\begin{equation*} \int_{a}^{b} f(x) d V(x)=\sum_{\nu=0}^{n} c_{\nu} f\left(x_{\nu}\right)+R[f] \tag{3} \end{equation*}(3)∫hasbf(x)dV(x)=∑ν=0ncνf(xν)+R[f]
Or has , b ( has < b ) has , b ( has < b ) a,b(a < b)a, b(a<b)has,b(has<b)are finished, V ( x ) V ( x ) V(x)V(x)V(x)is a function with bounded variation on [ has , b ] , c ν , v = 0 , 1 , … , n [ has , b ] , c ν , v = 0 , 1 , … , n [a,b],c_(nu),v=0,1,dots,n[a, b], c_{\nu}, v=0,1, \ldots, n[has,b],cν,v=0,1,…,nindependent coefficients of the function f f fffand the knots x ν , ν = 0 , 1 , … , n x ν , ν = 0 , 1 , … , n x_(nu),nu=0,1,dots,nx_{\nu}, \nu=0,1, \ldots, nxν,ν=0,1,…,nare distinct and included within the interval [ has , b ] [ has , b ] [a,b][a, b][has,b].
The rest R [ f ] R [ f ] R[f]R[f]R[f]of formula (3) is a linear functional defined on the set ( S = S = S=S=S=) C [ a , b ] C [ a , b ] C[a,b]C[a, b]C[has,b]of all functions f f fffcontinuous over the interval ( I = ) [ a , b ] ( I = ) [ a , b ] (I=)[a,b](I=)[a, b](I=)[has,b]In this case, all polynomials belong to S S SSSand the rest R [ f ] R [ f ] R[f]R[f]R[f]a, in general, a well-defined degree of accuracy.
2. The remainder of formula (3) can be put into various forms which allow it to be delimited under suitable assumptions made about the function f f fff.
Without restricting the generality, one can assume V ( a ) = 0 V ( a ) = 0 V(a)=0V(a)=0V(has)=0And ( x − 1 = x − 1 = x_(-1)=x_{-1}=x−1=) a < x 0 < x 1 < … < x n < b ( = x n + 1 ) a < x 0 < x 1 < … < x n < b = x n + 1 a < x_(0) < x_(1) < dots < x_(n) < b(=x_(n+1))a<x_{0}<x_{1}<\ldots<x_{n}<b\left(=x_{n+1}\right)has<x0<x1<…<xn<b(=xn+1).
Then, by specifying certain results of G. Kowalewski [1] and some results obtained by various authors, R. v. Mises, in his memoir on quadrature formulas [2], showed that, if m m mmmis a natural number and the remainder R [ f ] R [ f ] R[f]R[f]R[f]is the degree of accuracy m m mmm, we can write
(4) R [ f ] = ∫ a b φ ( x ) f ( m + 1 ) ( x ) d x (4) R [ f ] = ∫ a b   φ ( x ) f ( m + 1 ) ( x ) d x {:(4)R[f]=int_(a)^(b)varphi(x)f^((m+1))(x)dx:}\begin{equation*} R[f]=\int_{a}^{b} \varphi(x) f^{(m+1)}(x) d x \tag{4} \end{equation*}(4)R[f]=∫hasbφ(x)f(m+1)(x)dx
Or φ φ varphi\varphiφis a continuous function on [ a , b ] [ a , b ] [a,b][a, b][has,b]and assuming that the function f f fffhas a continuous derivative of order m + 1 m + 1 m+1m+1m+1over this interval.
The function φ φ varphi\varphiφis given by the formula
φ ( x ) = 1 m ! [ − ∫ a x ( t − x ) m d V ( t ) + ∑ μ = 0 v c μ ( x μ − x ) m ] φ ( x ) = 1 m ! − ∫ a x   ( t − x ) m d V ( t ) + ∑ μ = 0 v   c μ x μ − x m varphi(x)=(1)/(m!)[-int_(a)^(x)(t-x)^(m)dV(t)+sum_(mu=0)^(v)c_(mu)(x_(mu)-x)^(m)]\varphi(x)=\frac{1}{m!}\left[-\int_{a}^{x}(t-x)^{m} d V(t)+\sum_{\mu=0}^{v} c_{\mu}\left(x_{\mu}-x\right)^{m}\right]φ(x)=1m![−∫hasx(t−x)mdV(t)+∑μ=0vcμ(xμ−x)m]
For x ∈ [ x ν , x ν + 1 ] , ν = − 1 , 0 , … , n x ∈ x ν , x ν + 1 , ν = − 1 , 0 , … , n x in[x_(nu),x_(nu+1)],nu=-1,0,dots,nx \in\left[x_{\nu}, x_{\nu+1}\right], \nu=-1,0, \ldots, nx∈[xν,xν+1],ν=−1,0,…,n
This function is segment polynomial. This is what I used to call an "elementary function" (of order m m mmm) [3] and what is now called a "spline" function.
When the function φ φ varphi\varphiφdoes not change sign, from (4) we deduce, by applying the first formula of the mean of integral calculus
(5) R [ f ] = K ⋅ f ( m + 1 ) ( ξ ) ( m + 1 ) ! où a < ξ < b (5) R [ f ] = K ⋅ f ( m + 1 ) ( ξ ) ( m + 1 ) !  où  a < ξ < b {:(5)R[f]=K*(f^((m+1))(xi))/((m+1)!)quad" où "quad a < xi < b:}\begin{equation*} R[f]=K \cdot \frac{f^{(m+1)}(\xi)}{(m+1)!} \quad \text { où } \quad a<\xi<b \tag{5} \end{equation*}(5)R[f]=K⋅f(m+1)(ξ)(m+1)! Or has<ξ<b
with
K = R [ x m + 1 ] = ( m + 1 ) ! ∫ a b φ ( x ) d x ( ≠ 0 ) K = R x m + 1 = ( m + 1 ) ! ∫ a b   φ ( x ) d x ( ≠ 0 ) K=R[x^(m+1)]=(m+1)!int_(a)^(b)varphi(x)dx quad(!=0)K=R\left[x^{m+1}\right]=(m+1)!\int_{a}^{b} \varphi(x) d x \quad(\neq 0)K=R[xm+1]=(m+1)!∫hasbφ(x)dx(≠0)
  1. The same representation (4) of the remainder is valid for approximation formulas more general than the quadrature formula (3). For example, for quadrature formulas containing on the right-hand side also a certain number of the values ​​at the nodes of some of the derivatives of the function f f fffSuch formulas, with the expression of the form (4) of the remainder, can be found in the cited work of R. v. Mises [2].
Let us also recall the research of E. Rémèz who studied [6, 7] the representation, in the form (4) or in a more general form using a Stieltjes integral, of a linear functional R [ f ] R [ f ] R[f]R[f]R[f]degree of accuracy m m mmmand satisfying certain continuity conditions. These formulas are obtained by applying the well-known theorem of F. Riesz on the representation in the form of an integral of a linear functional.
We will not dwell on these questions here.
4. Another representation of the linear functional R [ f ] R [ f ] R[f]R[f]R[f]is obtained in the case of its "simplicity".
The linear functional R [ f ] R [ f ] R[f]R[f]R[f]defined on the set S S SSS, the structure of which was specified in n ∘ 1 n ∘ 1 n^(@)1\mathrm{n}^{\circ} 1n∘1and having a degree of accuracy m m mmm, is said to be of simple form if one can find m + 2 m + 2 m+2m+2m+2distinct points ξ ν , v = 1 , 2 … , m + 2 ξ ν , v = 1 , 2 … , m + 2 xi_(nu),v=1,2dots,m+2\xi_{\nu}, v=1,2 \ldots, m+2ξν,v=1,2…,m+2of the interval I I IIIdefining the elements of S S SSS, generally dependent on
the function f f fffand a constant K K KKKindependent of the function f f fffsuch as one might have
(6) R [ f ] = K [ ξ 1 , ξ 2 , … , ξ m + 2 ; f ] . (6) R [ f ] = K ξ 1 , ξ 2 , … , ξ m + 2 ; f . {:(6)R[f]=K[xi_(1),xi_(2),dots,xi_(m+2);f].:}\begin{equation*} R[f]=K\left[\xi_{1}, \xi_{2}, \ldots, \xi_{m+2} ; f\right] . \tag{6} \end{equation*}(6)R[f]=K[ξ1,ξ2,…,ξm+2;f].
Here [ ξ 1 , ξ 2 , … , ξ m + 2 ; f ] ξ 1 , ξ 2 , … , ξ m + 2 ; f [xi_(1),xi_(2),dots,xi_(m+2);f]\left[\xi_{1}, \xi_{2}, \ldots, \xi_{m+2} ; f\right][ξ1,ξ2,…,ξm+2;f]denotes the difference divided (of order m + 1 m + 1 m+1m+1m+1) of the function f f fffon the (distinct) nodes ξ 1 , ξ 2 , … , ξ n + 2 ξ 1 , ξ 2 , … , ξ n + 2 xi_(1),xi_(2),dots,xi_(n+2)\xi_{1}, \xi_{2}, \ldots, \xi_{n+2}ξ1,ξ2,…,ξn+2Moreover, when m ⩾ 0 m ⩾ 0 m >= 0m \geqslant 0m⩾0We can choose the nodes ξ ν ξ ν xi_(nu)\xi_{\nu}ξνwithin the interval I I III.
The constant K K KKKis equal to R [ x m + 1 ] R x m + 1 R[x^(m+1)]R\left[x^{m+1}\right]R[xm+1]and if, in addition, m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0and the derivative ( m + 1 ) ième f ( m + 1 ) ( x ) ( m + 1 ) ième  f ( m + 1 ) ( x ) (m+1)^("ième ")f^((m+1))(x)(m+1)^{\text {ième }} f^{(m+1)}(x)(m+1)th f(m+1)(x)exists within the interval I I III, we have formula (5).
Note that if the derivative ( m + 1 ) ième f ( m + 1 ) ( x ) ( m + 1 ) ième  f ( m + 1 ) ( x ) (m+1)^("ième ")f^((m+1))(x)(m+1)^{\text {ième }} f^{(m+1)}(x)(m+1)th f(m+1)(x)is bounded, therefore if
| f ( m + 1 ) ( x ) | ≦ ( m + 1 ) ! M f ( m + 1 ) ( x ) ≦ ( m + 1 ) ! M |f^((m+1))(x)| <= (m+1)!M\left|f^{(m+1)}(x)\right| \leqq(m+1)!M|f(m+1)(x)|≦(m+1)!M
Or M M MMMis a real number independent of x x xxx, we deduce from (5) the delimitation
| R [ f ] | ≦ | K | M . | R [ f ] | ≦ | K | M . |R[f]| <= |K|M.|R[f]| \leqq|K| M .|R[f]|≦|K|M.
This same delimitation is valid, if R [ f ] R [ f ] R[f]R[f]R[f]is of the simple form and M M MMMis the supremum of the absolute value of the divided difference of order m + 1 m + 1 m+1m+1m+1of the function f f fffSo that M M MMMto be finished, it is not necessary that the derivative ( m + 1 ) ième f ( m + 1 ) ( x ) ( m + 1 ) ième  f ( m + 1 ) ( x ) (m+1)^("ième ")f^((m+1))(x)(m+1)^{\text {ième }} f^{(m+1)}(x)(m+1)th f(m+1)(x)exists. It is enough that f f fffhas a derivative m ième f ( m ) ( x ) m ième  f ( m ) ( x ) m^("ième ")f^((m))(x)m^{\text {ième }} f^{(m)}(x)mth f(m)(x)satisfying an ordinary Lipschitz condition.
5. It follows from the preceding analysis that if the quadrature formula (3) has the remainder R [ f ] R [ f ] R[f]R[f]R[f]degree of accuracy m ( ≥ 1 ) m ( ≥ 1 ) m( >= 1)m(\geq 1)m(≥1)and if the function f f fffhas a derivative of order m + 1 m + 1 m+1m+1m+1on the interval [ a , b ] [ a , b ] [a,b][a, b][has,b], formula (5), with K = R [ x m + 1 ] K = R x m + 1 K=R[x^(m+1)]K=R\left[x^{m+1}\right]K=R[xm+1], takes place, or if the "core" φ φ varphi\varphiφin (4) does not change sign or if the rest R [ f ] R [ f ] R[f]R[f]R[f]is of the simple form. To show the close link between the two properties which in this way ensure the existence of formula (5) we will demonstrate the
Theorem. For the linear functional (4) defined on the set of continuous functions having a derivative ( m + 1 ) ième ( m + 1 ) ième  (m+1)^("ième ")(m+1)^{\text {ième }}(m+1)th continues over the bounded interval [ a , b ] [ a , b ] [a,b][a, b][has,b], or degree of accuracy m ≥ 1 m ≥ 1 m >= 1m \geq 1m≥1and in its simple form, it is necessary and sufficient that one has
(7) ∫ a b φ ( x ) d x ≠ 0 (7) ∫ a b   φ ( x ) d x ≠ 0 {:(7)int_(a)^(b)varphi(x)dx!=0:}\begin{equation*} \int_{a}^{b} \varphi(x) d x \neq 0 \tag{7} \end{equation*}(7)∫hasbφ(x)dx≠0
and that the function φ φ varphi\varphiφassumed to be continuous, does not change sign over the interval [ a , b ] [ a , b ] [a,b][a, b][has,b]6.
Before proceeding to the demonstration of our theorem, we must recall some preliminary properties.
We always assume that the set S S SSScontains all polynomials and is defined as at n 0 1 n 0 1 n^(0)1\mathrm{n}^{0} 1n01.
Lemma 1. For the linear functional R [ f ] R [ f ] R[f]R[f]R[f], defined on S S SSSand degree of accuracy m m mmmeither in its simple form, it is necessary and sufficient that one has R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for any function f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of order m m mmm(over the interval I I III).
For the concept and properties of convex functions (non-concave, non-convex, concave) of order m m mmmFor the proof of Lemma 1, you can consult my previous work. The function is said to be convex (non-concave, non-convex, concave) of order m ( ≥ − 1 ) m ( ≥ − 1 ) m( >= -1)m(\geq-1)m(≥−1)if all its differences
divided by order m + 1 m + 1 m+1m+1m+1, on distinct nodes (of the interval I I III), are positive (non-negative, non-positive, negative). The reader may consult, in particular, my memoirs in "Mathematica" [4, 5] where various criteria for simplicity can also be found.
Note that if R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0for any function f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of order m m mmmthe number R [ f ] R [ f ] R[f]R[f]R[f]For f f fffconvex of order m m mmmis or is always positive m m mmmthe sum f 1 , f 2 f 1 , f 2 f_(1),f_(2)f_{1}, f_{2}f1,f2or always negative... Suppose, we have two convex functions of order m m mmmbelonging as such we aid 1 1 > 1 1 > _(1)_(1) >{ }_{1}{ }_{1}>11>, R [ f 2 ] < 0 R f 2 < 0 R[f_(2)] < 0R\left[f_{2}\right]<0R[f2]<0The function f ( x ) = f 1 ( x ) − R [ f 1 ] R [ f 2 ] f 2 ( x ) f ( x ) = f 1 ( x ) − R f 1 R f 2 f 2 ( x ) f(x)=f_(1)(x)-(R[f_(1)])/(R[f_(2)])f_(2)(x)f(x)=f_{1}(x)-\frac{R\left[f_{1}\right]}{R\left[f_{2}\right]} f_{2}(x)f(x)=f1(x)−R[f1]R[f2]f2(x)belongs to S S SSS, is convex of order m m mmmand checks the equality R [ f ] = 0 R [ f ] = 0 R[f]=0R[f]=0R[f]=0This contradicts the hypothesis. The result is...
Lemma 2. If R [ f ] R [ f ] R[f]R[f]R[f]is a linear functional defined on S S SSSdegree of accuracy m m mmmand if we can find two functions f 1 , f 2 ∈ S f 1 , f 2 ∈ S f_(1),f_(2)in Sf_{1}, f_{2} \in Sf1,f2∈Sconvex of order m m mmmsuch as one might have R [ f 1 ] R [ f 2 ] < 0 R f 1 R f 2 < 0 R[f_(1)]R[f_(2)] < 0R\left[f_{1}\right] R\left[f_{2}\right]<0R[f1]R[f2]<0, so it is not of simple form. Let's also demonstrate the
Lemma 3. R [ f ] R [ f ] R[f]R[f]R[f]being a linear functional defined on S S SSSdegree of accuracy m m mmm, if f ∈ S f ∈ S f in Sf \in Sf∈Sis a non-concave function of order m m mmmsuch as R [ f ] ≠ 0 R [ f ] ≠ 0 R[f]!=0R[f] \neq 0R[f]≠0, then we can find a function f ∗ ∈ S f ∗ ∈ S f^(**)in Sf^{*} \in Sf*∈Sconvex of order m m mmmsuch that one has R [ f ] R [ f ∗ ] > 0 R [ f ] R f ∗ > 0 R[f]R[f^(**)] > 0R[f] R\left[f^{*}\right]>0R[f]R[f*]>0.
∼ ∼ ∼\sim~Indeed
, if ε ε epsi\varepsilonεis a positive constant, the function f ∗ ( x ) = f ( x ) + ε x m + 1 f ∗ ( x ) = f ( x ) + ε x m + 1 f^(**)(x)=f(x)+epsix^(m+1)f^{*}(x)=f(x)+\varepsilon x^{m+1}f*(x)=f(x)+εxm+1is convex of order m m mmmand we have
R [ f ∗ ] = R [ f ] + ε R [ x m + 1 ] = R [ f ] { 1 + ε R [ x m + 1 ] R [ f ] } R f ∗ = R [ f ] + ε R x m + 1 = R [ f ] 1 + ε R x m + 1 R [ f ] R[f^(**)]=R[f]+epsi R[x^(m+1)]=R[f]{1+epsi(R[x^(m+1)])/(R[f])}R\left[f^{*}\right]=R[f]+\varepsilon R\left[x^{m+1}\right]=R[f]\left\{1+\varepsilon \frac{R\left[x^{m+1}\right]}{R[f]}\right\}R[f*]=R[f]+εR[xm+1]=R[f]{1+εR[xm+1]R[f]}
All you have to do is take ε < | R [ f ] R [ x m + 1 ] | ε < R [ f ] R x m + 1 epsi < |(R[f])/(R[x^(m+1)])|\varepsilon<\left|\frac{R[f]}{R\left[x^{m+1}\right]}\right|ε<|R[f]R[xm+1]|to deduce the property stated in
Lemma 3. Lemma 3.
Finally, we have
Lemma 4. R [ f ] R [ f ] R[f]R[f]R[f]being a linear functional defined on S S SSSdegree of accuracy m m mmm, if we can find the functions f 1 , f 2 ∈ S f 1 , f 2 ∈ S f_(1),f_(2)in Sf_{1}, f_{2} \in Sf1,f2∈Snon-concave of order m m mmmsuch as one has R [ f 1 ] R [ f 2 ] < 0 R f 1 R f 2 < 0 R[f_(1)]R[f_(2)] < 0R\left[f_{1}\right] R\left[f_{2}\right]<0R[f1]R[f2]<0, so it is not of the simple form.
Indeed, f 1 ∗ , f 2 ∗ ∈ S f 1 ∗ , f 2 ∗ ∈ S f_(1)^(**),f_(2)^(**)in Sf_{1}^{*}, f_{2}^{*} \in Sf1*,f2*∈Sconvex functions of order m m mmm, such as R [ f 1 ] R [ f 1 ∗ ] > 0 , R [ f 2 ] R [ f 2 ∗ ] > 0 R f 1 R f 1 ∗ > 0 , R f 2 R f 2 ∗ > 0 R[f_(1)]R[f_(1)^(**)] > 0,R[f_(2)]R[f_(2)^(**)] > 0R\left[f_{1}\right] R\left[f_{1}^{*}\right]>0, R\left[f_{2}\right] R\left[f_{2}^{*}\right]>0R[f1]R[f1*]>0,R[f2]R[f2*]>0, which do indeed exist according to Lemma 3. It then follows R [ f ∗ ] R [ f 2 ∗ ] < 0 R f ∗ R f 2 ∗ < 0 R[f^(**)]R[f_(2)^(**)] < 0R\left[f^{*}\right] R\left[f_{2}^{*}\right]<0R[f*]R[f2*]<0and then apply Lemma
2.7. After this digression, we can proceed to the proof of our theorem stated in n ∘ 5 n ∘ 5 n^(@)5\mathrm{n}^{\circ} 5n∘5(7 )
results from
The condition is necessary, the relation (1) results in equality
(8) R [ x m + 1 ] = ( m + 1 ) ! ∫ a b φ ( x ) d x . (8) R x m + 1 = ( m + 1 ) ! ∫ a b   φ ( x ) d x . {:(8)R[x^(m+1)]=(m+1)!int_(a)^(b)varphi(x)dx.:}\begin{equation*} R\left[x^{m+1}\right]=(m+1)!\int_{a}^{b} \varphi(x) d x . \tag{8} \end{equation*}(8)R[xm+1]=(m+1)!∫hasbφ(x)dx.
The function φ φ varphi\varphiφis not identically zero on [ a , b ] [ a , b ] [a,b][a, b][has,b]So then x 0 ∈ [ a , b ] x 0 ∈ [ a , b ] x_(0)in[a,b]x_{0} \in[a, b]x0∈[has,b]such as φ ( x 0 ) ≠ 0 φ x 0 ≠ 0 varphi(x_(0))!=0\varphi\left(x_{0}\right) \neq 0φ(x0)≠0As a result of the continuity of φ φ varphi\varphiφthere exists a subinterval [ c , d ] [ c , d ] [c,d][c, d][c,d]of [ a , b ] [ a , b ] [a,b][a, b][has,b], of non-zero length (therefore a ≦ c < d ≦ b a ≦ c < d ≦ b a <= c < d <= ba \leqq c<d \leqq bhas≦c<d≦b), such as sg φ ( x ) = φ ( x ) = varphi(x)=\varphi(x)=φ(x)=
= sg φ ( x 0 ) = sg φ x 0 =sg varphi(x_(0))=\operatorname{sg} \varphi\left(x_{0}\right)=sgφ(x0)For x ∈ [ c , d ] x ∈ [ c , d ] x in[c,d]x \in[c, d]x∈[c,d]Let us now consider the functions
(9) φ k , λ = ( | x − λ | + x − λ 2 ) k − 1 (9) φ k , λ = | x − λ | + x − λ 2 k − 1 {:(9)varphi_(k,lambda)=((|x-lambda|+x-lambda)/(2))^(k-1):}\begin{equation*} \varphi_{k, \lambda}=\left(\frac{|x-\lambda|+x-\lambda}{2}\right)^{k-1} \tag{9} \end{equation*}(9)φk,λ=(|x−λ|+x−λ2)k−1
Or k k kkkis an integer > 1 > 1 > 1>1>1We then have
φ k , λ ( i ) = ( k − 1 ) ! ( k − i − 1 ) ! φ k − i , λ , i = 0 , 1 , … , k − 2 . φ k , λ ( i ) = ( k − 1 ) ! ( k − i − 1 ) ! φ k − i , λ , i = 0 , 1 , … , k − 2 . varphi_(k,lambda)^((i))=((k-1)!)/((k-i-1)!)varphi_(k-i,lambda),quad i=0,1,dots,k-2.\varphi_{k, \lambda}^{(i)}=\frac{(k-1)!}{(k-i-1)!} \varphi_{k-i, \lambda}, \quad i=0,1, \ldots, k-2 .φk,λ(i)=(k−1)!(k−i−1)!φk−i,λ,i=0,1,…,k−2.
The function
(10) f = 2 ( m + 1 ) ! [ φ m + 3 , c − 2 φ m + 3 , c + d 2 + φ m + 3 , d ] (10) f = 2 ( m + 1 ) ! φ m + 3 , c − 2 φ m + 3 , c + d 2 + φ m + 3 , d {:(10)f=(2)/((m+1)!)[varphi_(m+3,c)-2varphi_(m+3,(c+d)/(2))+varphi_(m+3,d)]:}\begin{equation*} f=\frac{2}{(m+1)!}\left[\varphi_{m+3, c}-2 \varphi_{m+3, \frac{c+d}{2}}+\varphi_{m+3, d}\right] \tag{10} \end{equation*}(10)f=2(m+1)![φm+3,c−2φm+3,c+d2+φm+3,d]
is non-concave of order m m mmmand has a derivative ( m + 1 ) ième ( m + 1 ) ième  (m+1)^("ième ")(m+1)^{\text {ième }}(m+1)th continuous which is equal to | x − c | + | x − d | − 2 | x − c + d 2 | | x − c | + | x − d | − 2 x − c + d 2 |x-c|+|x-d|-2|x-(c+d)/(2)||x-c|+|x-d|-2\left|x-\frac{c+d}{2}\right||x−c|+|x−d|−2|x−c+d2|, therefore positive on the open interval ( c , d ) ( c , d ) (c,d)(c, d)(c,d)and zero outside this interval. For function (10) we have
R [ f ] = ∫ c d φ ( x ) f ( m + 1 ) ( x ) d x R [ f ] = ∫ c d   φ ( x ) f ( m + 1 ) ( x ) d x R[f]=int_(c)^(d)varphi(x)f^((m+1))(x)dxR[f]=\int_{c}^{d} \varphi(x) f^{(m+1)}(x) d xR[f]=∫cdφ(x)f(m+1)(x)dx
from which it follows that sg R [ f ] = sg φ ( x 0 ) sg R [ f ] = sg φ x 0 sg R[f]=sg varphi(x_(0))\operatorname{sg} R[f]=\operatorname{sg} \varphi\left(x_{0}\right)sgR[f]=sgφ(x0).
Fig. 1
If the function φ φ varphi\varphiφchanges sign on [ a , b ] [ a , b ] [a,b][a, b][has,b]we can find two points x 1 , x 2 ∈ [ a , b ] x 1 , x 2 ∈ [ a , b ] x_(1),x_(2)in[a,b]x_{1}, x_{2} \in[a, b]x1,x2∈[has,b]such as φ ( x 1 ) φ ( x 2 ) < 0 φ x 1 φ x 2 < 0 varphi(x_(1))varphi(x_(2)) < 0\varphi\left(x_{1}\right) \varphi\left(x_{2}\right)<0φ(x1)φ(x2)<0, therefore also two functions f 1 , f 2 f 1 , f 2 f_(1),f_(2)f_{1}, f_{2}f1,f2nonconcave of order m m mmmsuch as sg R [ f 1 ] = sg φ ( x 1 ) sg R f 1 = sg φ x 1 sg R[f_(1)]=sg varphi(x_(1))\operatorname{sg} R\left[f_{1}\right]=\operatorname{sg} \varphi\left(x_{1}\right)sgR[f1]=sgφ(x1), sg R [ f 2 ] = sg φ ( x 2 ) sg R f 2 = sg φ x 2 sg R[f_(2)]=sg varphi(x_(2))\operatorname{sg} R\left[f_{2}\right]=\operatorname{sg} \varphi\left(x_{2}\right)sgR[f2]=sgφ(x2)It follows that R [ f 1 ] R [ f 2 ] < 0 R f 1 R f 2 < 0 R[f_(1)]R[f_(2)] < 0\mathrm{R}\left[f_{1}\right] R\left[f_{2}\right]<0R[f1]R[f2]<0. As a consequence of Lemma 4, the linear functional R [ f ] R [ f ] R[f]R[f]R[f]Therefore, it is not of the simple form.
In the theorem, the invariance of the sign of the function φ φ varphi\varphiφis therefore also necessary.
The condition is sufficient. The sufficiency of condition (7) also follows from formula (8). Let [ c , d c , d c,dc, dc,da non-zero-length subinterval of [ a , b ] ( a ≦ c < d ≦ b ) [ a , b ] ( a ≦ c < d ≦ b ) [a,b](a <= c < d <= b)[a, b](a \leqq c<d \leqq b)[has,b](has≦c<d≦b)on which φ φ varphi\varphiφis positive. Then f f fffa convex function of order m m mmmhaving a derivative ( m + 1 ) ième ( m + 1 ) ième  (m+1)^("ième ")(m+1)^{\text {ième }}(m+1)th continue on [ a , b ] [ a , b ] [a,b][a, b][has,b]We then have f ( m + 1 ) ( x ) ≧ 0 f ( m + 1 ) ( x ) ≧ 0 f^((m+1))(x) >= 0f^{(m+1)}(x) \geqq 0f(m+1)(x)≧0For x ∈ [ a , b ] x ∈ [ a , b ] x in[a,b]x \in[a, b]x∈[has,b]. But f ( m + 1 ) f ( m + 1 ) f^((m+1))f^{(m+1)}f(m+1)cannot be identically zero on any non-zero subinterval of length [ a , b ] [ a , b ] [a,b][a, b][has,b]because otherwise f f fffwould only be non-concave and not convex of order m m mmmon [ a , b ] [ a , b ] [a,b][a, b][has,b]As a result, there exists a [ c ′ , d ′ ] ⊆ [ c , d ] c ′ , d ′ ⊆ [ c , d ] [c^('),d^(')]sube[c,d]\left[c^{\prime}, d^{\prime}\right] \subseteq[c, d][c′,d′]⊆[c,d], of non-zero length
( c ≦ c ′ < d ′ ≦ d c ≦ c ′ < d ′ ≦ d c <= c^(') < d^(') <= dc \leqq c^{\prime}<d^{\prime} \leqq dc≦c′<d′≦d), on which f ( m + 1 ) ( x ) > 0 f ( m + 1 ) ( x ) > 0 f^((m+1))(x) > 0f^{(m+1)}(x)>0f(m+1)(x)>0We then have
R [ f ] = ∫ a b φ ( x ) f ( m + 1 ) ( x ) d x ≥ ∫ c ′ d ′ φ ( x ) f ( m + 1 ) ( x ) d x > 0 R [ f ] = ∫ a b   φ ( x ) f ( m + 1 ) ( x ) d x ≥ ∫ c ′ d ′   φ ( x ) f ( m + 1 ) ( x ) d x > 0 R[f]=int_(a)^(b)varphi(x)f^((m+1))(x)dx >= int_(c^('))^(d^('))varphi(x)f^((m+1))(x)dx > 0R[f]=\int_{a}^{b} \varphi(x) f^{(m+1)}(x) d x \geq \int_{c^{\prime}}^{d^{\prime}} \varphi(x) f^{(m+1)}(x) d x>0R[f]=∫hasbφ(x)f(m+1)(x)dx≥∫c′d′φ(x)f(m+1)(x)dx>0
SO R [ f ] > 0 R [ f ] > 0 R[f] > 0R[f]>0R[f]>0When
the function f f fffis non-positive, we demonstrate, in the same way that R [ f ] < 0 R [ f ] < 0 R[f] < 0R[f]<0R[f]<0, for any convex function of order m m mmmon [ a , b ] [ a , b ] [a,b][a, b][has,b].
The linear functional (4) is indeed of the simple form.
The theorem stated in n ∘ 5 n ∘ 5 n^(@)5\mathrm{n}^{\circ} 5n∘5is thus demonstrated.
8. R. v. Mises studied [ 2 ] [ 2 ] [2][2][2]the representation of the remains R [ f ] R [ f ] R[f]R[f]R[f]quadrature formulas also in the more general form of a Stieltjes integral
R [ f ] = ∫ a b f ( m + 1 ) ( x ) d α ( x ) R [ f ] = ∫ a b   f ( m + 1 ) ( x ) d α ( x ) R[f]=int_(a)^(b)f^((m+1))(x)d alpha(x)R[f]=\int_{a}^{b} f^{(m+1)}(x) d \alpha(x)R[f]=∫hasbf(m+1)(x)dα(x)
Or α ( x ) α ( x ) alpha(x)\alpha(x)α(x)is a function with bounded variation on the bounded interval [ a , b ] [ a , b ] [a,b][a, b][has,b]We can also study the simplicity of a linear functional of this form. We will return to this question in another work.

Bibliography

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[3] T. Popoviciu, Notes sur les fonctions convexes d'ordre supérieur (IX). Bull. Math. Soc. Roumaine des sciences, 43 (1942) 84-141.
[4] T. Popoviciu, Sur le reste dans certains formules ligneaux d'approximation de l'analyse, Mathematica 1 (24) (1959) 95-142.
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1972

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