On certain mean value formulas

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

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Tiberiu Popoviciu
(Institutul de Calcul)

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Asupra unor formule de medie

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T. Popoviciu, Asupra unor formule de medie, Rev. Anal. Numer. Teoria Aproximaţiei, 1 (1972), pp. 97-107

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1972 T. Popoviciu, On some average formulas, Rev. Anal. Numer. Approximation Theory
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JOURNAL OF NUMERICAL ANALYSIS AND APPROXIMATION THEORY Volume 1, Fascicola 1, 1972, pp. 97-107

ON SOME AVERAGE FORMULAS*ofTIBERIU POPOVICIU(Cluj)

  1. The classical formula of finite growth
(1) f ( x 2 ) − f ( x 1 ) = ( x 2 − x 1 ) f ′ ( ξ ) (1) f x 2 − f x 1 = x 2 − x 1 f ′ ( ξ ) {:(1)f(x_(2))-f(x_(1))=(x_(2)-x_(1))f^(')(xi):}\begin{equation*} f\left(x_{2}\right)-f\left(x_{1}\right)=\left(x_{2}-x_{1}\right) f^{\prime}(\xi) \tag{1} \end{equation*}(1)f(x2)−f(x1)=(x2−x1)f′(ξ)
occurs if f f fffis a continuous function on the bounded and closed interval [ x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2] ( x 1 < x 2 ) x 1 < x 2 (x_(1) < x_(2))\left(x_{1}<x_{2}\right)(x1<x2), differentiable on the open interval ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2) and ξ ξ xi\xiξis a convenient point of the latter interval. The point ξ ξ xi\xiξdepends on the function f f fffbut the only indication that can be given about it, in general, is that it belongs to the interval ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2). In fact, whatever it is c ∈ ( x 1 , x 2 ) c ∈ x 1 , x 2 c in(x_(1),x_(2))c \in\left(x_{1}, x_{2}\right)c∈(x1,x2), we can easily construct a function f f fffwhich meets the conditions imposed above for the validity of formula (1) and for which c c cccis the only possible value of ξ ξ xi\xiξHowever, in the case when the function f f fffbelongs to a particular set of functions, the position of the point ξ ξ xi\xiξit can, in certain cases, be made more precise by the existence of such a point in a certain particular subset of ( x 1 , x 2 x 1 , x 2 x_(1),x_(2)x_{1}, x_{2}x1,x2). In the following we will examine such problems for average formulas that generalize formula (1) of finite increments. 臨
2. Let us consider a real linear (hence additive and homogeneous) functional R ( f ) R ( f ) R(f)R(f)R(f), defined on a linear set S S SSSconsisting of real and continuous functions f f fff, defined over a given interval and and and\mathbf{I}and(of non-zero length) of the real axis. We will always assume that S S SSScontains all polynomials. The set S S SSSmay coincide with the set of all continuous functions f : and → R f : and → R f:IrarrRf: \mathbf{I} \rightarrow \mathbf{R}f:and→R, but it can be even more restricted. In the following, when necessary, we will specify the set S S SSSand the nature of its elements.
The degree of accuracy of R ( f ) R ( f ) R(f)R(f)R(f)is a whole m ≧ − 1 m ≧ − 1 m >= -1m \geqq-1m≧−1so that R ( f ) R ( f ) R(f)R(f)R(f)cancels out on any polynomial of degree m m mmmbut it is different from zero on the
at least a polynomial of degree m + 1 m + 1 m+1m+1m+1The degree of accuracy may not exist, but if it does exist it is well determined and is characterized by the following property:
R ( 1 ) = 0 if m = − 1 , R ( 1 ) = R ( x ) = … = R ( x m ) = 0 , R ( x m + 1 ) ≒ 0 if m ≧ 0 . R ( 1 ) = 0  if  m = − 1 , R ( 1 ) = R ( x ) = … = R x m = 0 , R x m + 1 ≒ 0  if  m ≧ 0 . {:[R(1)=0" if "m=-1","],[R(1)=R(x)=dots=R(x^(m))=0","quad R(x^(m+1))≒0" if "m >= 0.]:}\begin{gathered} R(1)=0 \text { if } m=-1, \\ R(1)=R(x)=\ldots=R\left(x^{m}\right)=0, \quad R\left(x^{m+1}\right) \fallingdotseq 0 \text { if } m \geqq 0 . \end{gathered}R(1)=0 if m=−1,R(1)=R(x)=…=R(xm)=0,R(xm+1)≒0 if m≧0.
When necessary, we will further specify the nature of the linear functional. R ( f ) R ( f ) R(f)R(f)R(f). We recall the definition of the simplicity of the linear functional R ( f ) R ( f ) R(f)R(f)R(f):
Linear functional R ( f ) R ( f ) R(f)R(f)R(f)it is said to be in simple form if there is an integer m ≧ − 1 m ≧ − 1 m >= -1m \geqq-1m≧−1, independent of function f f fff, so that for any f ∈ S f ∈ S f in Sf \in Sf∈Slet's have
(2) R ( f ) = K [ ξ 1 , ξ 2 , … , ξ m + 2 ; f ] (2) R ( f ) = K ξ 1 , ξ 2 , … , ξ m + 2 ; f {:(2)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{2} \end{equation*}(2)R(f)=K[ξ1,ξ2,…,ξm+2;f]
where K K KKKis a non-zero constant independent of the function f f fffand ξ w n == 1 , 2 , … , m + 2 ξ w n == 1 , 2 , … , m + 2 xi_(w)nu==1,2,dots,m+2\xi_{w} \nu= =1,2, \ldots, m+2ξwn==1,2,…,m+2ARE m + 2 m + 2 m+2m+2m+2distinct points of the interval and and and\mathbf{I}and, depending and n g it is n it is R A it d it is f you n c t and A ^ f and n g it is n it is R A it d it is f you n c t and A ^ f hat(generallydefunct)f\hat{in ~ general ~ of ~ function ~} fandn git isnit isRAit dit is fyounctandA ^f.
number m m mmmis completely determined and it is precisely the degree of accuracy of R ( f ) R ( f ) R(f)R(f)R(f)We have 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).
In formula (2) it is denoted by [ 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} ; right][y1,y2,…,yR;f]the divided difference, of the order R − 1 R − 1 r-1r-1R−1, of the function f f fffon the points, or nodes (distinct or 11 you 11 you 11 hours11 hours11you) y 1 y 1 y_(1)y_{1}y1, y 2 , … , y R y 2 , … , y R y_(2),dots,y_(r)y_{2}, \ldots, y_{r}y2,…,yR
3. The theory of higher-order convex functions allows us to find different criteria for the simplicity of linear functionals . R ( f ) R ( f ) R(f)R(f)R(f)Such a criterion can be stated in the following form:
theorem 1. A necessary and sufficient condition for the linear functional R ( f ) R ( f ) R(f)R(f)R(f), degree of accuracy m m mmm, to be of simple form is to have 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 the order m m mmm.
A function f f fffis called convex by the order m m mmmon and and and\mathbf{I}andif all its divided differences [ x 1 , x 2 , … , x m + 2 ; f ] x 1 , x 2 , … , x m + 2 ; f [x_(1),x_(2),dots,x_(m+2);f]\left[x_{1}, x_{2}, \ldots, x_{m+2} ; right][x1,x2,…,xm+2;f], of the order of m + 1 m + 1 m+1m+1m+1, on distinct nodes x 1 x 1 x_(1)x_{1}x1, x 2 , … , x m + 2 ∈ and x 2 , … , x m + 2 ∈ and x_(2),dots,x_(m+2)inIx_{2}, \ldots, x_{m+2} \in \mathbf{I}x2,…,xm+2∈and, are positive. If all these divided differences are nonnegative, the function is said to be nonconcave of order m m mmm(on 1). Finally, if the rents divided by the order m + 1 m + 1 m+1m+1m+1of the function f f fffare all n and all non-positive, this function is called concave, respectively the order m m mmm(on I). Moving from the function f f fffat the function − f − f -f-f−f, property of concave and non-convex functions of order m m mmmare generally deduced from the corresponding properties of convex and non-concave functions of order m m mmm. A convex (concave) function of order m m mmmis a particular case of a non-concave (non-convex) function of order m m mmmFor a function to be both non-concave and non-convex of order m m mmmit is necessary and
sufficient that all its differences divided by the order m + 1 m + 1 m+1m+1m+1, on distinct nodes, be equal to zero. Such a function is called a polynomial of order m m mmm(on I) and reduces to a polynomial of degree m m mmm, more precisely, to the restriction on and and and\mathbf{I}andof a polynomial of degree m m mmm.
Let us proceed to an outline of the proof of Theorem 1.
Let us first show that the condition in the statement is necessary. Let us assume that the linear functional R ( f ) R ( f ) R(f)R(f)R(f), degree of accuracy m m mmm, is of simple form. Let f ∈ S f ∈ S f in Sf \in Sf∈Sa convex function of order m m mmm. We then have formula (2). where K ≠ 0 K ≠ 0 K!=0K \neq 0K≠0. But, the difference divided by the second member is positive. So we have R ( f ) ≠ 0 R ( f ) ≠ 0 R(f)!=0R(f) \neq 0R(f)≠0. so R ( f ) ≠ 0 R ( f ) ≠ 0 R(f)!=0R(\mathrm{f}) \neq 0R(f)≠0.
now that the condition in the statement is also sufficient. Let us suppose that R ( f ) R ( f ) R(f)R(f)R(f)is of degree of accuracy m m mmmand is nonzero for f ∈ S f ∈ S end off \in \mathrm{~S}f∈ Sconvex of the order m m mmmFunction
(3) φ = R ( x m + 1 ) f − R ( f ) x m + 1 (3) φ = R x m + 1 f − R ( f ) x m + 1 {:(3)varphi=R(x^(m+1))fR(f)x^(m+1):}\begin{equation*} \varphi=R\left(x^{m+1}\right) fR(f) x^{m+1} \tag{3} \end{equation*}(3)φ=R(xm+1)f−R(f)xm+1
belongs to S S SSSand a simple calculation shows us that we have R ( φ ) = 0 R ( φ ) = 0 R(varphi)=0R(\varphi)=0R(φ)=0It follows that φ φ varphi\varphiφis not convex of order m m mmmIf we take into account the fact that - φ φ varphi\varphiφbelongs to S S SSSand that we have R ( − φ ) = − R ( φ ) = 0 R ( − φ ) = − R ( φ ) = 0 R(-varphi)=-R(varphi)=0R(-\varphi)=-R(\varphi)=0R(−φ)=−R(φ)=0, it follows that φ φ varphi\varphiφis not concave of the order m m mmm. There is then m + 2 m + 2 m+2m+2m+2distinct points ξ n ∈ and , n = 1 , 2 , … ξ n ∈ and , n = 1 , 2 , … xi_(nu)inI,nu=1,2,dots\xi_{\nu} \in \mathbf{I}, \nu=1,2, \ldotsξn∈and,n=1,2,…, m + 2 m + 2 m+2m+2m+2so that we have
[ ξ 1 , ξ 2 , … , ξ m + 2 ; φ ] = 0 ξ 1 , ξ 2 , … , ξ m + 2 ; φ = 0 [xi_(1),xi_(2),dots,xi_(m+2);varphi]=0\left[\xi_{1}, \xi_{2}, \ldots, \xi_{m+2} ; \varphi\right]=0[ξ1,ξ2,…,ξm+2;φ]=0
Taking into account the linearity of the divided difference, formula (2) is deduced, where K = R ( x m + 1 ) = 0 K = R x m + 1 = 0 K=R(x^(m+1))=0K=R\left(x^{m+1}\right)=0K=R(xm+1)=0.
With this Theorem 1 is proven.
If R ( f ) R ( f ) R(f)R(f)R(f)is the degree of accuracy m m mmmand is of simple form, we have
(4) R ( x m + 1 ) R ( f ) > 0 (4) R x m + 1 R ( f ) > 0 {:(4)R(x^(m+1))R(f) > 0:}\begin{equation*} R\left(x^{m+1}\right) R(f)>0 \tag{4} \end{equation*}(4)R(xm+1)R(f)>0
for any function f ∈ S f ∈ S f in Sf \in Sf∈Sconvex of the order m m mmmIndeed, x m + 1 x m + 1 x^(m+1)x^{m+1}xm+1is a convex function of order m m mmm, so if f f fffis convex of order m m mmmthe product R ( x m + 1 ) R ( f ) R x m + 1 R ( f ) R(x^(m+1))R(f)R\left(x^{m+1}\right) R(f)R(xm+1)R(f)is different from zero. Suppose that R ( x m + 1 ) R ( f ) < 0 R x m + 1 R ( f ) < 0 R(x^(m+1))R(f) < 0R\left(x^{m+1}\right) R(f)<0R(xm+1)R(f)<0Then the function R ( x m + 1 ) φ = [ R ( x m + 1 ) ] 2 f − R ( x m + 1 ) R ( f ) x m + 1 R x m + 1 φ = R x m + 1 2 f − R x m + 1 R ( f ) x m + 1 R(x^(m+1))varphi=[R(x^(m+1))]^(2)fR(x^(m+1))R(f)x^(m+1)R\left(x^{m+1}\right) \varphi=\left[R\left(x^{m+1}\right)\right]^{2} f-R\left(x^{m+1}\right) R(f) x^{m+1}R(xm+1)φ=[R(xm+1)]2f−R(xm+1)R(f)xm+1is (as a sum of two convex functions) a convex function of order m m mmmBut R ( R ( x m + 1 ) φ ) == R ( x m + 1 ) R ( φ ) = 0 R R x m + 1 φ == R x m + 1 R ( φ ) = 0 R(R(x^(m+1))varphi)==R(x_(m+1))R(varphi)=0R\left(R\left(x^{m+1}\right) \varphi\right)= =R\left(x_{m+1}\right) R(\varphi)=0R(R(xm+1)φ)==R(xm+1)R(φ)=0, which, based on Theorem 1 , is impossible. With this, inequality (4) is proven.
Under the same conditions if f f fffis a non-concave function of order m m mmmHAVE
(5) R ( x m + 1 ) R ( f ) ≧ 0 (5) R x m + 1 R ( f ) ≧ 0 {:(5)R(x^(m+1))quad R(f) >= 0:}\begin{equation*} R\left(x^{m+1}\right) \quad R(f) \geqq 0 \tag{5} \end{equation*}(5)R(xm+1)R(f)≧0
Indeed, for anything ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, function f + ε x m + 1 f + ε x m + 1 f+epsix^(m+1)f+\varepsilon x^{m+1}f+εxm+1is convex of order m m mmmand so we have R ( x m + 1 ) R ( f + ε x m + 1 ) = R ( x m + 1 ) R ( f ) + ε [ R ( x m + 1 ) ] 2 > 0 R x m + 1 R f + ε x m + 1 = R x m + 1 R ( f ) + ε R x m + 1 2 > 0 R(x^(m+1))R(f+epsix^(m+1))=R(x^(m+1))R(f)+epsi[R(x^(m+1))]^(2) > 0R\left(x^{m+1}\right) R\left(f+\varepsilon x^{m+1}\right)=R\left(x^{m+1}\right) R(f)+\varepsilon\left[R\left(x^{m+1}\right)\right]^{2}>0R(xm+1)R(f+εxm+1)=R(xm+1)R(f)+ε[R(xm+1)]2>0, whence, acting as ε ε epsi\varepsilonεto tend to 0, inequality (5) is deduced.
For the properties of higher-order convex functions, for the notion of simplicity of a linear functional and for various other properties used in this paper, one can consult my previous works. For example, my paper in "Studii și Cercetări", Cluj [4].
If m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0it can even be said that the points ξ , v = 1 , 2 , … , m + 2 ξ , v = 1 , 2 , … , m + 2 xi,v=1,2,dots,m+2\xi, v=1,2, \ldots, m+2ξ,V=1,2,…,m+2from formula (2) are within the interval I I I\mathbf{I}and,
If m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0, if R ( f ) R ( f ) R(f)R(f)R(f)is the degree of accuracy m m mmmof simple form and if f f fffhas a derivative f ( m + 1 ) f ( m + 1 ) f^((m+1))f^{(m+1)}f(m+1)of the order m + 1 m + 1 m+1m+1m+1on the inside of it I I I\mathbf{I}and, we have
(6)
R ( f ) = R ( x m + 1 ) f ( m + 1 ) ( ξ ) ( m + 1 ) ! R ( f ) = R x m + 1 f ( m + 1 ) ( ξ ) ( m + 1 ) ! R(f)=R(x^(m+1))(f^((m+1))(xi))/((m+1)!)R(f)=R\left(x^{m+1}\right) \frac{f^{(m+1)}(\xi)}{(m+1)!}R(f)=R(xm+1)f(m+1)(ξ)(m+1)!
where ξ ξ xi\xiξit is inside him I I I\mathbf{I}and
Formulas (2) and (6) allow, in the case of simplicity, to delimit the functional R ( f ) R ( f ) R(f)R(f)R(f)if the boundaries of the difference divided by the order are known m + 1 m + 1 m+1m+1m+1of the function f f fff, or its derivative of the order m + 1 m + 1 m+1m+1m+1, assumed to exist.
4. Suppose that the linear functional R ( f ) R ( f ) R(f)R(f)R(f)is defined on the set S S SSSof continuous functions on I and having a derivative f ( m + 1 ) f ( m + 1 ) f^((m+1))f^{(m+1)}f(m+1)of the order m + 1 m + 1 m+1m+1m+1on the inside of the door. We assume that m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0and that R ( f ) R ( f ) R(f)R(f)R(f)is of the degree of accuracy given function inside it I I I\mathbf{I}and, functional
(7) R ( f ) − R ( x m + 1 ) f ( m + 1 ) ( ξ ) ( m + 1 ) ! (7) R ( f ) − R x m + 1 f ( m + 1 ) ( ξ ) ( m + 1 ) ! {:(7)R(f)-R(x^(m+1))(f^((m+1))(xi))/((m+1)!):}\begin{equation*} R(f)-R\left(x^{m+1}\right) \frac{f^{(m+1)}(\xi)}{(m+1)!} \tag{7} \end{equation*}(7)R(f)−R(xm+1)f(m+1)(ξ)(m+1)!
is linear and vanishes on any polynomial of degree m + 1 m + 1 m+1m+1m+1. Putting f == x m + 2 f == x m + 2 f==x^(m+2)f= =x^{m+2}f==xm+2and taking into account (6), it is seen that there is a well-determined value c c ^(c){ }^{c}c(from the internal force it cancels out on any polynomial of grain m + 2 m + 2 m+2m+2m+2. Number c c cccis given by equation
(8)
R ( x m + 2 ) − ( m + 2 ) R ( x m + 1 ) c = 0 R x m + 2 − ( m + 2 ) R x m + 1 c = 0 R(x^(m+2))-(m+2)R(x^(m+1))c=0R\left(x^{m+2}\right)-(m+2) R\left(x^{m+1}\right) c=0R(xm+2)−(m+2)R(xm+1)c=0
We have the following
I ema 1. In addition to the previous assumptions, the linear functional
(9)
R 1 ( f ) = R ( f ) − R ( x m + 1 ) f ( m + 1 ) ( c ) ( m + 1 ) ! R 1 ( f ) = R ( f ) − R x m + 1 f ( m + 1 ) ( c ) ( m + 1 ) ! R_(1)(f)=R(f)-R(x^(m+1))(f^((m+1))(c))/((m+1)!)R_{1}(f)=R(f)-R\left(x^{m+1}\right) \frac{f^{(m+1)}(c)}{(m+1)!}R1(f)=R(f)−R(xm+1)f(m+1)(c)(m+1)!
is defined on S S SSSand it is of accuracy m + 2 m + 2 m+2m+2m+2
It is enough to show that R 1 ( x m + 3 ) R 1 x m + 3 R_(1)(x^(m+3))R_{1}\left(x^{m+3}\right)R1(xm+3)is not equal to 0 .
Taking into account (8), we have
(10) R ( x m + 1 ) R 1 ( x m + 3 ) = 1 2 ( m + 2 ) [ 2 ( m + 2 ) R ( x m + 1 ) R ( x m + 3 ) − ( m + 3 ) R 2 ( x m + 2 ) ] R x m + 1 R 1 x m + 3 = 1 2 ( m + 2 ) 2 ( m + 2 ) R x m + 1 R x m + 3 − ( m + 3 ) R 2 x m + 2 R(x^(m+1))R_(1)(x^(m+3))=(1)/(2(m+2))[2(m+2)R(x^(m+1))R(x^(m+3))-(m+3)R^(2)(x^(m+2))]R\left(x^{m+1}\right) R_{1}\left(x^{m+3}\right)=\frac{1}{2(m+2)}\left[2(m+2) R\left(x^{m+1}\right) R\left(x^{m+3}\right)-(m+3) R^{2}\left(x^{m+2}\right)\right]R(xm+1)R1(xm+3)=12(m+2)[2(m+2)R(xm+1)R(xm+3)−(m+3)R2(xm+2)].

If we put

(11) P ( x ) = x m + 3 + ( m + 3 ) z x m + 2 + ( m + 2 ) ( m + 3 ) 2 z 2 x m + 1 (11) P ( x ) = x m + 3 + ( m + 3 ) z x m + 2 + ( m + 2 ) ( m + 3 ) 2 z 2 x m + 1 {:(11)P(x)=x^(m+3)+(m+3)zx^(m+2)+((m+2)(m+3))/(2)z^(2)x^(m+1):}\begin{equation*} P(x)=x^{m+3}+(m+3) z x^{m+2}+\frac{(m+2)(m+3)}{2} z^{2} x^{m+1} \tag{11} \end{equation*}(11)P(x)=xm+3+(m+3)zxm+2+(m+2)(m+3)2z2xm+1
where z z zzzis an independent parameter of x x xxx, we have
P ( m + 1 ) ( x ) = ( m + 3 ) ! 2 ( x + z ) 2 . P ( m + 1 ) ( x ) = ( m + 3 ) ! 2 ( x + z ) 2 . P^((m+1))(x)=((m+3)!)/(2)(x+z)^(2).P^{(m+1)}(x)=\frac{(m+3)!}{2}(x+z)^{2} .P(m+1)(x)=(m+3)!2(x+z)2.
So we have P ( m + 1 ) ( x ) > 0 P ( m + 1 ) ( x ) > 0 P^((m+1))(x) > 0P^{(m+1)}(x)>0P(m+1)(x)>0for x ≠ − z x ≠ − z x!=-zx \neq-zx≠−zIt follows that the polynomial (11) is convex of order m m mmm(everywhere). Based on inequality (4), we have
R ( x m + 1 ) R ( P ) = = R ( x m + 1 ) [ R ( x m + 3 ) + ( m + 3 ) R ( x m + 2 ) z + ( m + 2 ) ( m + 3 ) 2 R ( x m + 1 ) z 2 ] > 0 R x m + 1 R ( P ) = = R x m + 1 R x m + 3 + ( m + 3 ) R x m + 2 z + ( m + 2 ) ( m + 3 ) 2 R x m + 1 z 2 > 0 {:[R(x^(m+1))R(P)=],[=R(x^(m+1))[R(x^(m+3))+(m+3)R(x^(m+2))z+((m+2)(m+3))/(2)R(x^(m+1))z^(2)] > 0]:}\begin{gathered} R\left(x^{m+1}\right) R(P)= \\ =R\left(x^{m+1}\right)\left[R\left(x^{m+3}\right)+(m+3) R\left(x^{m+2}\right) z+\frac{(m+2)(m+3)}{2} R\left(x^{m+1}\right) z^{2}\right]>0 \end{gathered}R(xm+1)R(P)==R(xm+1)[R(xm+3)+(m+3)R(xm+2)z+(m+2)(m+3)2R(xm+1)z2]>0
whatever z z zzzIt follows that the discriminant of this second degree trinomial in z z zzzis negative, so we have
( m + 3 ) [ ( m + 3 ) R 2 ( x m + 2 ) − 2 ( m + 2 ) R ( x m + 1 ) R ( x m + 3 ) ] < 0 ( m + 3 ) ( m + 3 ) R 2 x m + 2 − 2 ( m + 2 ) R x m + 1 R x m + 3 < 0 (m+3)[(m+3)R^(2)(x^(m+2))-2(m+2)R(x^(m+1))R(x^(m+3))] < 0(m+3)\left[(m+3) R^{2}\left(x^{m+2}\right)-2(m+2) R\left(x^{m+1}\right) R\left(x^{m+3}\right)\right]<0(m+3)[(m+3)R2(xm+2)−2(m+2)R(xm+1)R(xm+3)]<0
and equality (10) shows us that
(12) R ( x m + 1 ) R 1 ( x m + 3 ) > 0 (12) R x m + 1 R 1 x m + 3 > 0 {:(12)R(x^(m+1))R_(1)(x^(m+3)) > 0:}\begin{equation*} R\left(x^{m+1}\right) R_{1}\left(x^{m+3}\right)>0 \tag{12} \end{equation*}(12)R(xm+1)R1(xm+3)>0
Lemma 1 follows from this.
We will see below that the linear functional (9) is of the simple form.
5. We will now assume that the interval I reduces to the bounded and closed interval [ a , b ] ( a < b ) [ a , b ] ( a < b ) [a,b](a < b)[a, b](a<b)[A,b](A<b)and that the elements f f fffhis/hers S S SSShave a continuous derivative of order m + 1 m + 1 m+1m+1m+1on [ a , b ] [ a , b ] [a,b][a, b][A,b].
We continue to assume that m ≧ 0 m ≧ 0 m >= 0m \geqq 0m≧0.
Be it then R ( f ) R ( f ) R(f)R(f)R(f)a linear functional defined on S S SSS, degree of accuracy m m mmmand of simple form. Let us consider the linear functional (9), the number c c cccbeing determined by equation (8). We then have a < c < b a < c < b a < c < ba<c<bA<c<b.
We have the following
Le ma 2. In addition to the previous hypotheses, if there is an integer k , 0 ≤ k ≤≦ m + 1 k , 0 ≤ k ≤≦ m + 1 k,0 <= k≤≦m+1k, 0 \leq k \leq \leqq m+1k,0≤k≤≦m+1so that the linear functional R ( f ) R ( f ) R(f)R(f)R(f)to be limited compared to the norm
(13) ∑ y = 0 k max x ∈ [ a , b ] | f ( y ) ( x ) | , (13) ∑ y = 0 k   max x ∈ [ a , b ]   f ( y ) ( x ) , {:(13)sum_(y=0)^(k)max_(x in[a,b])|f^((y))(x)|",":}\begin{equation*} \sum_{y=0}^{k} \max _{x \in[a, b]}\left|f^{(y)}(x)\right|, \tag{13} \end{equation*}(13)∑y=0kMAXx∈[A,b]|f(y)(x)|,
then we have
(14) R ( x m + 1 ) R 1 ( f ) ≧ 0 (14) R x m + 1 R 1 ( f ) ≧ 0 {:(14)R(x^(m+1))R_(1)(f) >= 0:}\begin{equation*} R\left(x^{m+1}\right) R_{1}(f) \geqq 0 \tag{14} \end{equation*}(14)R(xm+1)R1(f)≧0
for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave of the order m + 2 m + 2 m+2m+2m+2
Let the functions
φ m + 3 , λ = ( x − λ + | x − λ | 2 ) m + 2 , φ m + 3 , λ = x − λ + | x − λ | 2 m + 2 , varphi_(m+3,lambda)=((x-lambda+|x-lambda|)/(2))^(m+2),\varphi_{m+3, \lambda}=\left(\frac{x-\lambda+|x-\lambda|}{2}\right)^{m+2},φm+3,λ=(x−λ+|x−λ|2)m+2,
where λ λ lambda\lambdaλis an independent parameter of x x xxxand between a a aaA, and b b bbb.
  • function φ m + 3 , λ φ m + 3 , λ varphi_(m+3,lambda)\varphi_{m+3, \lambda}φm+3,λbelongs to S S SSSand is non-concave of the order m + 2 m + 2 m+2m+2m+2for anything λ λ lambda\lambdaλWe have
φ m + 3 , λ ( m + 1 ) = ( m + 2 ) ! ( x − λ + | x − λ | 2 ) = ( m + 2 ) ! φ 2 , λ φ m + 3 , λ ( m + 1 ) = ( m + 2 ) ! x − λ + | x − λ | 2 = ( m + 2 ) ! φ 2 , λ varphi_(m+3,lambda)^((m+1))=(m+2)!((x-lambda+|x-lambda|)/(2))=(m+2)!varphi_(2,lambda)\varphi_{m+3, \lambda}^{(m+1)}=(m+2)!\left(\frac{x-\lambda+|x-\lambda|}{2}\right)=(m+2)!\varphi_{2, \lambda}φm+3,λ(m+1)=(m+2)!(x−λ+|x−λ|2)=(m+2)!φ2,λ
We will prove that inequality (14) is verified for this function, so if we put f = φ m + 3 , λ f = φ m + 3 , λ f=varphi_(m+3,lambda)f=\varphi_{m+3, \lambda}f=φm+3,λ. Indeed
R 1 ( φ m + 3 , λ ) = R ( φ m + 3 , λ ) − ( m + 2 ) R ( x m + 1 ) φ 2 , λ ( c ) R 1 φ m + 3 , λ = R φ m + 3 , λ − ( m + 2 ) R x m + 1 φ 2 , λ ( c ) R_(1)(varphi_(m+3,lambda))=R(varphi_(m+3,lambda))-(m+2)R(x^(m+1))varphi_(2,lambda)(c)R_{1}\left(\varphi_{m+3, \lambda}\right)=R\left(\varphi_{m+3, \lambda}\right)-(m+2) R\left(x^{m+1}\right) \varphi_{2, \lambda}(c)R1(φm+3,λ)=R(φm+3,λ)−(m+2)R(xm+1)φ2,λ(c)
and if we take into account (8), we have
R 1 ( φ m + 3 , λ ) = { R ( φ m + 3 , λ − x m + 2 + ( m + 2 ) λ x m + 1 ) dacă λ ≦ c , R ( φ m + 3 , λ ) dacă λ ≧ c . R 1 φ m + 3 , λ = R φ m + 3 , λ − x m + 2 + ( m + 2 ) λ x m + 1  dacă  λ ≦ c , R φ m + 3 , λ  dacă  λ ≧ c . R_(1)(varphi_(m+3),lambda)={[R(varphi_(m+3,lambda)-x^(m+2)+(m+2)lambdax^(m+1))" dacă "lambda <= c","],[R(varphi_(m+3),lambda)" dacă "lambda >= c.]:}R_{1}\left(\varphi_{m+3}, \lambda\right)=\left\{\begin{array}{l} R\left(\varphi_{m+3, \lambda}-x^{m+2}+(m+2) \lambda x^{m+1}\right) \text { dacă } \lambda \leqq c, \\ R\left(\varphi_{m+3}, \lambda\right) \text { dacă } \lambda \geqq c . \end{array}\right.R1(φm+3,λ)={R(φm+3,λ−xm+2+(m+2)λxm+1) if λ≦c,R(φm+3,λ) if λ≧c.
But the functions
φ m − 1 − 3 , λ , φ m + 3 , λ − x m + 2 + ( m + 2 ) λ x m + 1 φ m − 1 − 3 , λ , φ m + 3 , λ − x m + 2 + ( m + 2 ) λ x m + 1 varphi_(m-1-3,lambda),quadvarphi_(m+3,lambda)-x^(m+2)+(m+2)lambdax^(m+1)\varphi_{m-1-3, \lambda}, \quad \varphi_{m+3, \lambda}-x^{m+2}+(m+2) \lambda x^{m+1}φm−1−3,λ,φm+3,λ−xm+2+(m+2)λxm+1
are non-concave of the order m m mmmbecause their derivatives of the order m + 1 m + 1 m+1m+1m+1are respectively
( m + 2 ) ! ( x − λ + | x − λ | 2 ) , ( m + 2 ) ! ( | x − λ | − x + λ 2 ) ( m + 2 ) ! x − λ + | x − λ | 2 , ( m + 2 ) ! | x − λ | − x + λ 2 (m+2)!((x-lambda+|x-lambda|)/(2)),quad(m+2)!((|x-lambda|-x+lambda)/(2))(m+2)!\left(\frac{x-\lambda+|x-\lambda|}{2}\right), \quad(m+2)!\left(\frac{|x-\lambda|-x+\lambda}{2}\right)(m+2)!(x−λ+|x−λ|2),(m+2)!(|x−λ|−x+λ2)
and they are both ≧ 0 ≧ 0 >= 0\geqq 0≧0.
So we have R ( x m + 1 ) R 1 ( φ m + 3 , λ ) ≧ 0 R x m + 1 R 1 φ m + 3 , λ ≧ 0 R(x^(m+1))R_(1)(varphi_(m+3),lambda) >= 0R\left(x^{m+1}\right) R_{1}\left(\varphi_{m+3}, \lambda\right) \geqq 0R(xm+1)R1(φm+3,λ)≧0, and, taking into account ( 12 ) , R 1 ( x m + 3 ) R 1 ( φ m + 3 , λ ) ≧ 0 ( 12 ) , R 1 x m + 3 R 1 φ m + 3 , λ ≧ 0 (12),R_(1)(x^(m+3))R_(1)(varphi_(m+3),lambda) >= 0(12), R_{1}\left(x^{m+3}\right) R_{1}\left(\varphi_{m+3}, \lambda\right) \geqq 0(12),R1(xm+3)R1(φm+3,λ)≧0for anything λ λ lambda\lambdaλbetween a a aaAand b b bbb.
From theorem 15 of our cited work [4] it follows that the linear functional R 1 ( f ) R 1 ( f ) R_(1)(f)R_{1}(f)R1(f)is of simple form, so inequality (14) is true for any function f ∈ S f ∈ S f in Sf \in Sf∈Snon-concave of the order m + 2 m + 2 m+2m+2m+2(and even without equality possible if f f fffis convex of order m + 2 m + 2 m+2m+2m+2).
Lemma 2 is proven.
6. We can now prove the following
theorem 2. If the following assumptions are verified:
  1. m m mmmis a nonnegative integer.
  2. S S SSSis the set of functions f f fffhaving a continuous derivative of order m + 1 m + 1 m+1m+1m+1on the bounded and closed interval [ a , b ] , ( a < b ) [ a , b ] , ( a < b ) [a,b],(a < b)[a, b],(a<b)[A,b],(A<b).
  3. R ( f ) R ( f ) R(f)R(f)R(f)is a linear functional defined on S S SSS, of accuracy degree m, of simple form and bounded with respect to the norm (13) for a certain integer k k kkkso that 0 ≦ k ≦ m + 1 0 ≦ k ≦ m + 1 0 <= k <= m+10 \leqq k \leqq m+10≦k≦m+1.
  4. c c cccis the point determined by equation (8) (We then have a < c < b a < c < b a < c < ba<c<bA<c<b).
  5. Function f f fffverify one of the following 4 properties:
    A. is non-concave of order m + 1 m + 1 m+1m+1m+1and non-concave of the order m + 2 m + 2 m+2m+2m+2,
    B. is nonconvex of order m + 1 m + 1 m+1m+1m+1and non-concave of the order m + 2 m + 2 m+2m+2m+2,
    C. is non-concave of the order m + 1 m + 1 m+1m+1m+1and nonconvex of the order m + 2 m + 2 m+2m+2m+2.
    D. is nonconvex of order m + 1 m + 1 m+1m+1m+1and nonconvex of the order m + 2 m + 2 m+2m+2m+2, then the average formula (6) is verified, in cases A and D, by at least one point ξ ξ xi\xiξof the interval [ c , b ] [ c , b ] [c,b][c, b][c,b]and i n ^ i n ^ hat(in)\hat{i n}andn^cases B and C, by at least one point ξ ξ xi\xiξof the interval [ a , c ] [ a , c ] [a,c][a, c][A,c].
It is sufficient to do the proof in case A. In this case the function
(15) g ( x ) = R ( x ˙ ± 4 m + 1 ) [ R ( f ) − R ( x m + 1 ) f m + 1 ( x ) ( m + 1 ) ! ] (15) g ( x ) = R x ˙ ± 4 m + 1 R ( f ) − R x m + 1 f m + 1 ( x ) ( m + 1 ) ! {:(15)g(x)=R(x^(˙)_(+-4)^(m+1))[R(f)-R(x^(m+1))(f^(m+1)(x))/((m+1)!)]:}\begin{equation*} g(x)=R\left(\dot{x}_{ \pm 4}^{m+1}\right)\left[R(f)-R\left(x^{m+1}\right) \frac{f^{m+1}(x)}{(m+1)!}\right] \tag{15} \end{equation*}(15)g(x)=R(x˙±4m+1)[R(f)−R(xm+1)fm+1(x)(m+1)!]
is non-increasing on [ a , b ] [ a , b ] [a,b][a, b][A,b]and it cancels out at least one point inside the interval [ a , b ] [ a , b ] [a,b][a, b][A,b]So we have g ( a ) ≧ 0 , g ( b ) ≤ 0 g ( a ) ≧ 0 , g ( b ) ≤ 0 g(a) >= 0,g(b) <= 0g(a) \geqq 0, g(b) \leq 0g(A)≧0,g(b)≤0, and from Lemma 2 it follows that we also have g ( c ) ≥ 0 g ( c ) ≥ 0 g(c) >= 0g(c) \geq 0g(c)≥0. The property from the statement of the theorem follows. We can observe that the points ξ ξ xi\xiξwhich verifies (6) forms an interval and the obtained property means that this interval has at least one point in common with [cb]. If, in particular, the function f f fffis convex of order m + 1 m + 1 m+1m+1m+1, the point ξ ξ xi\xiξ, from formula (6) is unique and belongs to the interval [ c , b ] [ c , b ] [c,b][c, b][c,b].
otherwise cases D, C are deduced respectively from cases A, B by passing from the function f f fffto the function -f.
7. As a first application we have Gauss,
Consequence 1. If R ( f ) R ( f ) R(f)R(f)R(f)is the remainder of the quadrature formula of type
(16) ∫ a b f ( x ) d V ( x ) = ∑ v = 1 n λ v f ( x v ) + R ( f ) (16) ∫ a b   f ( x ) d V ( x ) = ∑ v = 1 n   λ v f x v + R ( f ) {:(16)int_(a)^(b)f(x)dV(x)=sum_(v=1)^(n)lambda_(v)f(x_(v))+R(f):}\begin{equation*} \int_{a}^{b} f(x) d V(x)=\sum_{v=1}^{n} \lambda_{v} f\left(x_{v}\right)+R(f) \tag{16} \end{equation*}(16)∫Abf(x)dV(x)=∑V=1nλVf(xV)+R(f)
where n is a natural number, V V VVVa non-decreasing function, having at least n + 1 n + 1 n+1n+1n+1growing points and f f fffa function that admits a continuous derivative of order 2 n 2 n 2n2 n2non the bounded and closed interval [ a , b a , b a,ba, bA,b], average formula
R ( f ) = R ( x 2 n ) f ( 2 n ) ( ξ ) ( 2 n ) ! R ( f ) = R x 2 n f ( 2 n ) ( ξ ) ( 2 n ) ! R(f)=R(x^(2n))(f^((2n))(xi))/((2n)!)R(f)=R\left(x^{2 n}\right) \frac{f^{(2 n)}(\xi)}{(2 n)!}R(f)=R(x2n)f(2n)(ξ)(2n)!
is verified, in cases A, D of Theorem 2, for at least one point in the interval [ c , b ] [ c , b ] [c,b][c, b][c,b]and in cases B , C B , C B,C\mathrm{B}, \mathrm{C}B,Cof Theorem 2, for at least one point ξ ξ xi\xiξof the interval [ a , c ] [ a , c ] [a,c][a, c][A,c].
Here it was placed m = 2 n − 1 m = 2 n − 1 m=2n-1m=2 n-1m=2n−1and c c cccis given by the corresponding equation (8).
In formula (16), x v , v = 1 , 2 , … , n x v , v = 1 , 2 , … , n x_(v),v=1,2,dots,nx_{v}, v=1,2, \ldots, nxV,V=1,2,…,nare the roots (distinct and located inside the interval [ a , b ] [ a , b ] [a,b][a, b][A,b]) of the orthogonal polynomial of degree n n nnnrelative to the distribution d V ( x ) d V ( x ) dV(x)d V(x)dV(x). The numbers λ v , v = 1 , 2 , … , n λ v , v = 1 , 2 , … , n lambda_(v),v=1,2,dots,n\lambda_{v}, v=1,2, \ldots, nλV,V=1,2,…,nare the coefficients (all > 0 ) > 0 ) > 0)>0)>0)corresponding to Cristoffel.
This property can be generalized to more general Gaussian formulas by replacing the first term of formula (16) with a suitable inverse and nonnegative functional. Among these are those studied by us in a previous paper [3].
8. As another application of Theorem 2 , we have the following
Corollary 2. If the function f f fffis continuous and has a derivative of order m + 1 m + 1 m+1m+1m+1continues over an interval containing the m + 2 m + 2 m+2m+2m+2given points x v , ν = 1 , 2 , … , m + 2 , n u x v , ν = 1 , 2 , … , m + 2 , n u x_(v),nu=1,2,dots,m+2,nux_{v}, \nu=1,2, \ldots, m+2, n uxV,n=1,2,…,m+2,nyouall confused and where m ≥ 0 m ≥ 0 m >= 0m \geq 0m≥0, then Cauchy's average formula,
[ x 1 , x 2 , … , x m + 2 ; f ] = f m + 1 ( ξ ) ( m + 1 ) ! x 1 , x 2 , … , x m + 2 ; f = f m + 1 ( ξ ) ( m + 1 ) ! [x_(1),x_(2),dots,x_(m+2);f]=(f_(m+1)(xi))/((m+1)!)\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right]=\frac{f_{m+1}(\xi)}{(m+1)!}[x1,x2,…,xm+2;f]=fm+1(ξ)(m+1)!
is checked, in cases A , D A , D A,D\mathrm{A}, \mathrm{D}A,Dof Theorem 2, for at least one point ξ ≧ 1 m + 2 ∑ v = 1 m + 2 x v ξ ≧ 1 m + 2 ∑ v = 1 m + 2   x v xi >= (1)/(m+2)sum_(v=1)^(m+2)x_(v)\xi \geqq \frac{1}{m+2} \sum_{v=1}^{m+2} x_{v}ξ≧1m+2∑V=1m+2xV, and in cases B, C of Theorem 2, for at least one point ξ ≦ 1 m + 2 ∑ v = 1 m + 2 x v ξ ≦ 1 m + 2 ∑ v = 1 m + 2   x v xi <= (1)/(m+2)sum_(v=1)^(m+2)x_(v)\xi \leqq \frac{1}{m+2} \sum_{v=1}^{m+2} x_{v}ξ≦1m+2∑V=1m+2xV.
Divided difference [ x 1 , x 2 , … , x m + 2 ; f ] x 1 , x 2 , … , x m + 2 ; f [x_(1),x_(2),dots,x_(m+2);f]\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right][x1,x2,…,xm+2;f]where the nodes x v , v = 1 , 2 , … x v , v = 1 , 2 , … x_(v),v=1,2,dotsx_{v}, v=1,2, \ldotsxV,V=1,2,…, m + 2 m + 2 m+2m+2m+2are distinct or n u n u nun unyou, is defined as usual.
It is seen that the linear functional R ( f ) = [ x 1 , x 2 , … , x m + 2 ; f ] R ( f ) = x 1 , x 2 , … , x m + 2 ; f R(f)=[x_(1),x_(2),dots,x_(m+2);f]R(f)=\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right]R(f)=[x1,x2,…,xm+2;f]verifies all the hypotheses in theorem 2 (provided that the points x y x y x_(y)x_{\mathrm{y}}xy, so that they are not all confused), [ a , b ] [ a , b ] [a,b][a, b][A,b]being an interval containing all the nodes x v , v == 1 , 2 , … , m + 2 x v , v == 1 , 2 , … , m + 2 x_(v),v==1,2,dots,m+2x_{v}, v= =1,2, \ldots, m+2xV,V==1,2,…,m+2In this case the point c c cccis precisely the arithmetic mean 1 m + 2 ∑ v = 1 m + 2 x v 1 m + 2 ∑ v = 1 m + 2   x v (1)/(m+2)sum_(v=1)^(m+2)x_(v)\frac{1}{m+2} \sum_{v=1}^{m+2} x_{v}1m+2∑V=1m+2xVof the nodes.
For m = 0 m = 0 m=0m=0m=0we obtain the corresponding properties relative to the finite growth formula (1). It is unnecessary to state these properties here.
9. The property expressed by consequence 2 can also be demonstrated directly in the following way. To fix the ideas, let us assume that we are in case A, so that the function f f fffis non-concave of the order m + 1 m + 1 m+1m+1m+1and non-concave of the order m + 2 m + 2 m+2m+2m+2. Reasoning as was done on the function (15) for the proof of Theorem 2 and using some well-known formulas on divided differences, we have first, assuming x 1 ≦ x 2 ≦ … ≦ x m + 2 x 1 ≦ x 2 ≦ … ≦ x m + 2 x_(1) <= x_(2) <= dots <= x_(m+2)x_{1} \leqq x_{2} \leqq \ldots \leqq x_{m+2}x1≦x2≦…≦xm+2,
[ x 1 , x 2 , … , x m + 2 ; f ] − f ( m + 1 ) ( x 1 ) ( m + 1 ) ! = = ∑ v = 2 m + 2 [ [ x 1 , x 1 , … , x 1 , x 2 , x 3 , … , x v ; f ] ( x v − x 1 ) ≧ 0 , ⏟ m + 4 − v [ x 1 , x 2 , … , x m + 2 ; f ] − f ( m + 1 ) ( x m + 2 ) ( m + 1 ) ! = = − ∑ v = 1 m + 1 [ x v , x v + 1 , … , x m + 1 , x m + 2 , x m + 2 , … , x m + 2 ⏟ v + 1 ; f ] ( x m + 2 − x v ) ≦ 0 . x 1 , x 2 , … , x m + 2 ; f − f ( m + 1 ) x 1 ( m + 1 ) ! = = ∑ v = 2 m + 2   [ x 1 , x 1 , … , x 1 , x 2 , x 3 , … , x v ; f x v − x 1 ≧ 0 , ⏟ m + 4 − v x 1 , x 2 , … , x m + 2 ; f − f ( m + 1 ) x m + 2 ( m + 1 ) ! = = − ∑ v = 1 m + 1   [ x v , x v + 1 , … , x m + 1 , x m + 2 , x m + 2 , … , x m + 2 ⏟ v + 1 ; f ] x m + 2 − x v ≦ 0 . {:[[x_(1),x_(2),dots,x_(m+2);f]-(f^((m+1))(x_(1)))/((m+1)!)=],[=sum_(v=2)^(m+2)[ubrace([x_(1),x_(1),dots,x_(1),x_(2),x_(3),dots,x_(v);f](x_(v)-x_(1)) >= 0,ubrace)_(m+4-v)],[[x_(1),x_(2),dots,x_(m+2);f]-(f^((m+1))(x_(m+2)))/((m+1)!)=],[=-sum_(v=1)^(m+1)[x_(v)","x_(v+1)","dots","x_(m+1)","ubrace(x_(m+2),x_(m+2),dots,x_(m+2)ubrace)_(v+1);f](x_(m+2)-x_(v)) <= 0.]:}\begin{gathered} {\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right]-\frac{f^{(m+1)}\left(x_{1}\right)}{(m+1)!}=} \\ =\sum_{v=2}^{m+2}[\underbrace{\left[x_{1}, x_{1}, \ldots, x_{1}, x_{2}, x_{3}, \ldots, x_{v} ; f\right]\left(x_{v}-x_{1}\right) \geqq 0,}_{m+4-v} \\ {\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right]-\frac{f^{(m+1)}\left(x_{m+2}\right)}{(m+1)!}=} \\ =-\sum_{v=1}^{m+1}[x_{v}, x_{v+1}, \ldots, x_{m+1}, \underbrace{x_{m+2}, x_{m+2}, \ldots, x_{m+2}}_{v+1} ; f]\left(x_{m+2}-x_{v}\right) \leqq 0 . \end{gathered}[x1,x2,…,xm+2;f]−f(m+1)(x1)(m+1)!==∑V=2m+2[[x1,x1,…,x1,x2,x3,…,xV;f](xV−x1)≧0,⏟m+4−V[x1,x2,…,xm+2;f]−f(m+1)(xm+2)(m+1)!==−∑V=1m+1[xV,xV+1,…,xm+1,xm+2,xm+2,…,xm+2⏟V+1;f](xm+2−xV)≦0.
Here the terms in which the second member contains divided differences taken at all confused nodes are suppressed.
If now the function f f fffis non-concave of the order m + 2 m + 2 m+2m+2m+2, we have
[ x 1 , x 2 , … , x m + 2 ; f ] ≧ 1 ( m + 1 ) ! f ( m + 1 ) ( x 1 + x 2 + … + x m + 3 m + 2 ) , x 1 , x 2 , … , x m + 2 ; f ≧ 1 ( m + 1 ) ! f ( m + 1 ) x 1 + x 2 + … + x m + 3 m + 2 , [x_(1),x_(2),dots,x_(m+2);f] >= (1)/((m+1)!)f^((m+1))((x_(1)+x_(2)+dots+x_(m+3))/(m+2)),\left[x_{1}, x_{2}, \ldots, x_{m+2} ; f\right] \geqq \frac{1}{(m+1)!} f^{(m+1)}\left(\frac{x_{1}+x_{2}+\ldots+x_{m+3}}{m+2}\right),[x1,x2,…,xm+2;f]≧1(m+1)!f(m+1)(x1+x2+…+xm+3m+2),
as we have demonstrated in another paper [2].
Corollary 2 now follows immediately.
10. The property expressed by consequence 1 follows from that expressed by consequence 2. Indeed, from some formulas that we have established elsewhere [1], it follows that the rest R ( f ) R ( f ) R(f)R(f)R(f)of Gauss's formula (16) differs only by a positive constant factor from the difference divided by the order 2 n 2 n 2n2 n2nof the function f f fffwith the nodes in the roots of orthogonal polynomials of degree n n nnnand n + 1 n + 1 n+1n+1n+1.
In some cases, it is possible to proceed differently. In particular, either V = x V = x V=xV=xV=x. Then x v , v = 1 , 2 , … , n x v , v = 1 , 2 , … , n x_(v),v=1,2,dots,nx_{v}, v=1,2, \ldots, nxV,V=1,2,…,nare the roots of the polynomial
P n = ∏ ν = 1 n ( x − x ν ) P n = ∏ ν = 1 n   x − x ν P_(n)=prod_(nu=1)^(n)(x-x_(nu))P_{n}=\prod_{\nu=1}^{n}\left(x-x_{\nu}\right)Pn=∏n=1n(x−xn)
Legendre's degree n n nnn(with the highest coefficient equal to 1) relative to the interval [ a , b ] [ a , b ] [a,b][a, b][A,b]. Then if F F FFFis a primitive of the function f f fff, we have
R ( f ) = F ( b ) − F ( a ) − ∑ v = 1 n λ v F ′ ( x v ) = R ∗ ( F ) R ( f ) = F ( b ) − F ( a ) − ∑ v = 1 n   λ v F ′ x v = R ∗ ( F ) R(f)=F(b)-F(a)-sum_(v=1)^(n)lambda_(v)F^(')(x_(v))=R^(**)(F)R(f)=F(b)-F(a)-\sum_{v=1}^{n} \lambda_{v} F^{\prime}\left(x_{v}\right)=R^{*}(F)R(f)=F(b)−F(A)−∑V=1nλVF′(xV)=R∗(F)
Because R ( f ) R ( f ) R(f)R(f)R(f)is a linear functional of degree of accuracy 2 n − 1 2 n − 1 2n-12 n-12n−1, R ∗ ( F ) R ∗ ( F ) R^(**)(F)R^{*}(F)R∗(F)is a linear functional of degree of accuracy 2 n 2 n 2n2 n2n, so nut differs only by a constant (positive) factor from the divided difference of the function F F FFFon the nodes a , b , x v , v = 1 , 2 , … , n a , b , x v , v = 1 , 2 , … , n a,b,x_(v),v=1,2,dots,na, b, x_{\mathrm{v}}, v=1,2, \ldots, nA,b,xV,V=1,2,…,nLATEST n n nnneach being taken twice. It is easy to see that
R ( f ) = R ∗ ( F ) = ( b − a ) P n 2 ( b ) [ a , b , x 1 , x 1 , x 2 , x 2 , … , x n , x n ; F ] R ( f ) = R ∗ ( F ) = ( b − a ) P n 2 ( b ) a , b , x 1 , x 1 , x 2 , x 2 , … , x n , x n ; F R(f)=R^(**)(F)=(b-a)P_(n)^(2)(b)[a,b,x_(1),x_(1),x_(2),x_(2),dots,x_(n),x_(n);F]R(f)=R^{*}(F)=(b-a) P_{n}^{2}(b)\left[a, b, x_{1}, x_{1}, x_{2}, x_{2}, \ldots, x_{n}, x_{n} ; F\right]R(f)=R∗(F)=(b−A)Pn2(b)[A,b,x1,x1,x2,x2,…,xn,xn;F]
The required property results.

SUR CERTAINES FORMULES DE LA MOYENNE

RESUME

R ( f ) R ( f ) R(f)R(f)R(f)est une fonctionnelle lineare defined sur l'ensemble des fonctions f f fffayant une dérivée continuous d'ordre m + 1 ( m ≥ 0 ) m + 1 ( m ≥ 0 ) m+1(m >= 0)m+1(m \geq 0)m+1(m≥0)sur l'intervale borné et fermé [ a , b ] ( a < b ) [ a , b ] ( a < b ) [a,b](a < b)[a, b](a<b)[A,b](A<b)And R ( f ) R ( f ) R(f)R(f)R(f)is the degree of accuracy m m mmm, from simple forms et est bornée par rapport à une norme de la forms (13), alors la formula de la moyenne (6) est verificie pour au moins un point ξ ξ xi\xiξof [ c , b ] [ c , b ] [c,b][c, b][c,b]respectively of [ a , c ] [ a , c ] [a,c][a, c][A,c], where c c cccis the point of ( a , b ) ( a , b ) (a,b)(a, b)(A,b)given by (8) et suivant que 1a fonction f f fffverify en même temps, dans un ordre determined par le théorème 2, des properties de non-concavité et de non-convexité d'ordre m + 1 m + 1 m+1m+1m+1and order m + 2 m + 2 m+2m+2m+2.
BIBLIOGRAPHY
[1] Popoviciu, T., Notes sur les fonctions convexes d'ordre supérieur (IV). Disquisitiones Math. et Physicae, I, 163-171 (1940).
[2] - Notes sur les fonctions convexes d'ordre supérieur (V). Bulletin de l'Acad. Rou-
[3] maine, XX1, 351-356 (1940). of Gauss's numerical intergrave. Studies and [3] Circle. Scientific Iasi, VI, 29 − 57 29 − 57 29-5729-5729−57(1955).
[4] - On the remainder in some linear approximation formulas of analysis. Studii și Cerc. de Matematică (Cluj) X, 337-389 (1959).
Received on 2. XII. 1971.
    • This paper is a slightly modified version of a paper published in French in Spisy prirodov. fak.Univ. JE Purkyne v. Brne, 5, 147-156 (1969).
1972

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