On a generalization of the “spline” functions

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

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Tiberiu Popoviciu
Institutul de Calcul

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T. Popoviciu, Sur une généralisation des fonctions “spline”, Mathematical structures – computational mathematics – mathematical modelling, Sofia, 1975, pp. 405-410.

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1975 f -Popoviciu- Math. Structures–Comput. Math.–Math. Model. – Sur une généralisation des fonction ‘spline’_

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1975 f -Popoviciu- Math. Structures–Comput. Math.–Math. Model. - On a generalization of functions
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MATHEMATICAL STRUCTURES - COMPUTATIONAL MATHEMATICS - MATHEMATICAL MODELING Papers dedicated to Professor L. Iliev's 60th Anniversary Sofia, 1975, p. 405-410

ON A GENERALIZATION OF "SPLINE" FUNCTIONS

T. Popoviciu

Dedicated to ML Iliev on the occasion of his sixtieth birthday
Conclusion. After recalling the generalization that we previously gave to the "spline" functions [4], we give some new properties of these functions. For example:
If the function f f fffis in order ( n / k ) ( n / k ) (n//k)(n/k)(n/k)on the interval [ has , b ] [ has , b ] [a,b][a, b][has,b]and if k ≦ n k ≦ n k <= nk \leqq nk≦n, this function is bounded and if k ≦ n − r − 1 k ≦ n − r − 1 k <= nr-1k \leqq nr-1k≦n−r−1, it has a continuous derivative of order r ( r ≧ 0 ) r ( r ≧ 0 ) r(r >= 0)r(r \geqq 0)r(r≧0)on any interval [ has 1 , b 1 ] has 1 , b 1 [a_(1),b_(1)]\left[a_{1}, b_{1}\right][has1,b1], Or a < a 1 < b 1 < b a < a 1 < b 1 < b a < a_(1) < b_(1) < ba<a_{1}<b_{1}<bhas<has1<b1<b.
  1. For some time now, the attention of several authors has been focused on a special class of real functions of a real variable. These are the functions formed by a finite number of polynomial pieces suitably connected. Special cases are functions constant by segments and functions whose graphical representation is a polygonal line. These functions have been encountered for a long time in various important problems such as that of approximate integration or the approximation of arbitrary functions by simpler functions.
  2. Either f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[has,b]→Ra continuous function defined on the interval [ a , b ] [ a , b ] [a,b][a, b][has,b]bounded and closed of the real axis R R RRRand consider the points
(1) x ν = a + ν h , h = b − a m , ν = 0 , 1 , … , m , (1) x ν = a + ν h , h = b − a m , ν = 0 , 1 , … , m , {:(1)x_(nu)=a+nu h","quad h=(b-a)/(m)","quad nu=0","1","dots","m",":}\begin{equation*} x_{\nu}=a+\nu h, \quad h=\frac{b-a}{m}, \quad \nu=0,1, \ldots, m, \tag{1} \end{equation*}(1)xν=has+νh,h=b−hasm,ν=0,1,…,m,
which divide this interval into m m mmmequal parts.
Let us denote by ψ m ψ m psi_(m)\psi_{m}ψmthe polygonal line (the polygonal function) inscribed in the curve y = f ( x ) y = f ( x ) y=f(x)y=f(x)y=f(x)and whose vertices are the points ( x v , f ( x v ) x v , f x v x_(v),f(x_(v))x_{v}, f\left(x_{v}\right)xv,f(xv)), We know that, on [ a , b ] [ a , b ] [a,b][a, b][has,b], the sequel ( ψ m ψ m psi_(m)\psi_{m}ψm) tends uniformly towards the function f f fff.
The function ψ m ψ m psi_(m)\psi_{m}ψmis of the form
(2) ψ m = f ( x 0 ) + [ x 0 , x 1 ; f ] ( x − a ) + 2 h ∑ v = 0 m − 2 [ x v , x v + 1 , x v + 2 ; f ] φ 2 , x v + 1 (2) ψ m = f x 0 + x 0 , x 1 ; f ( x − a ) + 2 h ∑ v = 0 m − 2   x v , x v + 1 , x v + 2 ; f φ 2 , x v + 1 {:(2)psi_(m)=f(x_(0))+[x_(0),x_(1);f](x-a)+2hsum_(v=0)^(m-2)[x_(v),x_(v+1),x_(v+2);f]varphi_(2,x_(v+1)):}\begin{equation*} \psi_{m}=f\left(x_{0}\right)+\left[x_{0}, x_{1} ; f\right](x-a)+2 h \sum_{v=0}^{m-2}\left[x_{v}, x_{v+1}, x_{v+2} ; f\right] \varphi_{2, x_{v+1}} \tag{2} \end{equation*}(2)ψm=f(x0)+[x0,x1;f](x−has)+2h∑v=0m−2[xv,xv+1,xv+2;f]φ2,xv+1
Or λ λ lambda\lambdaλis a real number and φ 2 , λ 2 = x − λ + | x − λ | 2 = ( x − λ ) + φ 2 , λ 2 = x − λ + | x − λ | 2 = ( x − λ ) + varphi_(2,lambda^(2))=(x-lambda+|x-lambda|)/(2)=(x-lambda)+\varphi_{2, \lambda^{2}}=\frac{x-\lambda+|x-\lambda|}{2}=(x-\lambda)+φ2,λ2=x−λ+|x−λ|2=(x−λ)+is the positive part of the function x − λ x − λ x-lambdax-\lambdax−λ, :
3. We will designate by [ z 1 , z 2 , … , z k + 1 ; f ] z 1 , z 2 , … , z k + 1 ; f [z_(1),z_(2),dots,z_(k+1);f]\left[z_{1}, z_{2}, \ldots, z_{k+1} ; f\right][z1,z2,…,zk+1;f]the divided difference (of order k k kkk) of the function f f fffon the k + 1 k + 1 k+1k+1k+1points or knots z 1 , z 2 , … , z k + 1 z 1 , z 2 , … , z k + 1 z_(1),z_(2),dots,z_(k+1)z_{1}, z_{2}, \ldots, z_{k+1}z1,z2,…,zk+1.
Then the functions (2) are generalized by the functions ( m ≥ n + 1 m ≥ n + 1 m >= n+1m \geq n+1m≥n+1)
(3) Q m ( x ) + ( n + 1 ) h ∑ ν = 0 m − n − 1 [ x ν , x ν + 1 , … , x ν + n + 1 ; f ] ⋅ φ n + 1 , x ν + n (3) Q m ( x ) + ( n + 1 ) h ∑ ν = 0 m − n − 1   x ν , x ν + 1 , … , x ν + n + 1 ; f ⋅ φ n + 1 , x ν + n {:(3)Q_(m)(x)+(n+1)hsum_(nu=0)^(m-n-1)[x_(nu),x_(nu+1),dots,x_(nu+n+1);f]*varphi_(n+1,x_(nu+n)):}\begin{equation*} Q_{m}(x)+(n+1) h \sum_{\nu=0}^{m-n-1}\left[x_{\nu}, x_{\nu+1}, \ldots, x_{\nu+n+1} ; f\right] \cdot \varphi_{n+1, x_{\nu+n}} \tag{3} \end{equation*}(3)Qm(x)+(n+1)h∑ν=0m−n−1[xν,xν+1,…,xν+n+1;f]⋅φn+1,xν+n
Or n n nnnis a natural number, Q m ( x ) Q m ( x ) Q_(m)(x)Q_{m}(x)Qm(x)a polynomial of degree n n nnnand the x y x y x_(y)x_{y}xyare still the points (1). Finally we have
(4) p n + 1 , λ = ( x − λ + | x − λ | 2 ) n = ( x − λ ) + n . (4) p n + 1 , λ = x − λ + | x − λ | 2 n = ( x − λ ) + n . {:(4)p_(n+1,lambda)=((x-lambda+|x-lambda|)/(2))^(n)=(x-lambda)_(+)^(n).:}\begin{equation*} p_{n+1, \lambda}=\left(\frac{x-\lambda+|x-\lambda|}{2}\right)^{n}=(x-\lambda)_{+}^{n} . \tag{4} \end{equation*}(4)pn+1,λ=(x−λ+|x−λ|2)n=(x−λ)+n.
Divided differences of various orders, on equidistant or non-equidistant nodes, are defined (for example) by the recurrence relation
[ z 1 , z 2 , … , z k + 1 ; f ] = [ z 2 , z 3 , … , z k + 1 ; f ] − [ z 1 , z 2 , … , z k ; f ] z k + 1 − z 1 z 1 , z 2 , … , z k + 1 ; f = z 2 , z 3 , … , z k + 1 ; f − z 1 , z 2 , … , z k ; f z k + 1 − z 1 [z_(1),z_(2),dots,z_(k+1);f]=([z_(2),z_(3),dots,z_(k+1);f]-[z_(1),z_(2),dots,z_(k);f])/(z_(k+1)-z_(1))\left[z_{1}, z_{2}, \ldots, z_{k+1} ; f\right]=\frac{\left[z_{2}, z_{3}, \ldots, z_{k+1} ; f\right]-\left[z_{1}, z_{2}, \ldots, z_{k} ; f\right]}{z_{k+1}-z_{1}}[z1,z2,…,zk+1;f]=[z2,z3,…,zk+1;f]−[z1,z2,…,zk;f]zk+1−z1
And [ z 1 ; f ] = f ( z 1 ) z 1 ; f = f z 1 [z_(1);f]=f(z_(1))\left[z_{1} ; f\right]=f\left(z_{1}\right)[z1;f]=f(z1)and by which we construct the divided difference of order k k kkkusing the order one k − 1 k − 1 k-1k-1k−1.
In the following we always assume that the nodes z 1 , z 2 , … , z k + 1 z 1 , z 2 , … , z k + 1 z_(1),z_(2),dots,z_(k+1)z_{1}, z_{2}, \ldots, z_{k+1}z1,z2,…,zk+1are distinct. We know that by admitting the existence of a certain number of derivatives for the function f f fff, we can also define divided differences on nodes that are not necessarily distinct. For equidistant nodes (1) we have, moreover,
[ x y , x y + 1 , … , x v + n + 1 ; f ] = 1 ( n + 1 ) ! h n + 1 Δ h n + 1 f ( a + ν h ) = 1 ( n + 1 ) ! h n + 1 ∑ μ = 0 n + 1 ( − 1 ) n + 1 − μ ( n + 1 μ ) f ( a + ν + μ ― h ) x y , x y + 1 , … , x v + n + 1 ; f = 1 ( n + 1 ) ! h n + 1 Δ h n + 1 f ( a + ν h ) = 1 ( n + 1 ) ! h n + 1 ∑ μ = 0 n + 1   ( − 1 ) n + 1 − μ ( n + 1 μ ) f ( a + ν + μ ¯ h ) {:[[x_(y),x_(y+1),dots,x_(v+n+1);f]=(1)/((n+1)!h^(n+1))Delta_(h)^(n+1)f(a+nu h)],[quad=(1)/((n+1)!h^(n+1))sum_(mu=0)^(n+1)(-1)^(n+1-mu)((n+1)/(mu))f(a+ bar(nu+mu)h)]:}\begin{aligned} & {\left[x_{y}, x_{y+1}, \ldots, x_{v+n+1} ; f\right]=\frac{1}{(n+1)!h^{n+1}} \Delta_{h}^{n+1} f(a+\nu h)} \\ & \quad=\frac{1}{(n+1)!h^{n+1}} \sum_{\mu=0}^{n+1}(-1)^{n+1-\mu}\binom{n+1}{\mu} f(a+\overline{\nu+\mu} h) \end{aligned}[xy,xy+1,…,xv+n+1;f]=1(n+1)!hn+1Δhn+1f(has+νh)=1(n+1)!hn+1∑μ=0n+1(−1)n+1−μ(n+1μ)f(has+ν+μ―h)
The structure of function (3) depends on two important notions: A. The notion of convex functions of order n n nnn. B. The notion of spline functions.
In the following we propose to make some considerations on these two notions.

A. The notion of convex functions of order n n nnn

  1. The function f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[has,b]→Ris said to be convex, non-concave, polynomial, non-convex respectively concave of order n n nnn(on [ a , b ] [ a , b ] [a,b][a, b][has,b]) depending on whether the divided differences of order n + 1 n + 1 n+1n+1n+1on any group of n + 2 n + 2 n+2n+2n+2distinct points, or nodes, of the function f f fff, which are all > , ≥ , = , ≦ > , ≥ , = , ≦ > , >= ,=, <=>, \geq,=, \leqq>,≥,=,≦respectively < 0 < 0 < 0<0<0A function verifying one of these properties is called a doodre function. n n nnn.
The whole n n nnnEast ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1. For n = − 1 n = − 1 n=-1n=-1n=−1we have the functions of invariable sign: positive, non-negative, zero (the identically zero function), non-positive, respectively negative. For n = 0 n = 0 n=0n=0n=0we have the monotonic functions: increasing, non-decreasing, constant, non-increasing respectively decreasing. Finally
for n = 1 n = 1 n=1n=1n=1we have the usual convex, non-concave, linear, non-convex and concave functions respectively.

B. The concept of spline function

  1. Let's suppose n ≧ 1 n ≧ 1 n >= 1n \geqq 1n≧1. Then any linear combination of polynomials of degree n n nnnand of a finite number of functions of the form (4) is what we formerly called an elementary function of order n n nnn[5]. Today we call them spline functions.
A spline function is therefore of the form
(5) P ( x ) + ∑ v = 1 m c v φ n + 1 , i v ( x ) (5) P ( x ) + ∑ v = 1 m   c v φ n + 1 , i v ( x ) {:(5)P(x)+sum_(v=1)^(m)c_(v)varphi_(n+1,i_(v))(x):}\begin{equation*} P(x)+\sum_{v=1}^{m} c_{v} \varphi_{n+1, i_{v}}(x) \tag{5} \end{equation*}(5)P(x)+∑v=1mcvφn+1,iv(x)
Or P P PPPis a polynomial of degree n n nnn, THE λ ν λ ν lambda_(nu)\lambda_{\nu}λνare given points such that a < λ 1 < λ 2 < … < λ m < b a < λ 1 < λ 2 < … < λ m < b a < lambda_(1) < lambda_(2) < dots < lambda_(m) < ba<\lambda_{1}<\lambda_{2}<\ldots<\lambda_{m}<bhas<λ1<λ2<…<λm<band the c y c y c_(y)c_{y}cyare any constants.
Function (5) is continuous and has a divided difference of order n n nnnlimited. If n > 1 n > 1 n > 1n>1n>1it has a continuous derivative of order n − 1 n − 1 n-1n-1n−1on [ a , b ] [ a , b ] [a,b][a, b][has,b]. The spline function (5) is formed by a finite number of pieces of polynomials of degree n n nnnwhich have a maximum connection order without being reduced to a polynomial of degree n n nnn. There are also functions formed by pieces of polynomials of degree n n nnnwhich do not connect as completely, which, for example, are not continuous or which, for n > 1 n > 1 n > 1n>1n>1are continuous but not differentiable, etc. We can say that for a n n nnngiven the functions (5) are the simplest splines.
6. Spline functions have been encountered for a long time without being explicitly highlighted. Such are the so-called "broken kernels" or Green's functions which are involved in the resolution of certain boundary value problems for linear differential equations. It is impossible to cite here all the authors who, in one form or another, have used splines. I will simply cite G. Peano [3], J. Radon [7], G. Kowalewski [1], R. v. Mises [2]. In recent times, spline functions have been studied systematically by I.J. Schoenberg and his students. These authors have obtained, among others, numerous remarkable results concerning certain problems of approximation and optimization for quadrature formulas. One can consult the work of the Wisconsin Symposium [8].
7. Between higher-order convex functions and spline functions of the form (5) there is a very close connection. We have the following property:
The conditions c v ≧ 0 , v = 1 , 2 , … , m c v ≧ 0 , v = 1 , 2 , … , m c_(v) >= 0,v=1,2,dots,mc_{v} \geqq 0, v=1,2, \ldots, mcv≧0,v=1,2,…,m, are necessary and sufficient for function (5) to be non-concave of order n n nnn.
Taking the spline functions of the form (3), where the x 2 x 2 x_(2)x_{2}x2are the equidistant points (1) and by suitably choosing the polynomial Q m ( x ) Q m ( x ) Q_(m)(x)Q_{m}(x)Qm(x)for each value of m m mmm, we obtain the approximation theorem [5, 6]:
For n ≧ 1 n ≧ 1 n >= 1n \geqq 1n≧1, any function f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[has,b]→Rcontinuous and orderly n n nnnis indefinitely and uniformly approximable by spline functions of the form (3), of order n n nnn. Otherwise: Any function f : [ a , b ] → R f : [ a , b ] → R f:[a,b]rarr Rf:[a, b] \rightarrow Rf:[has,b]→R, continuous and non-concave of order n n nnnis the limit of a uniformly convergent sequence of functions of the form (5), with coefficients c y c y c_(y)c_{y}cynon-negative.
This theorem has many applications. It can provide, for example, a criterion for the simplicity of the remainder of certain linear approximation formulas in Analysis.

C. Generalization of spline functions

spline.
8. This being said, we have proposed a generalization of the functions
Definition 1. If E ⊆ R E ⊆ R E sube RE \subseteq RE⊆R, the function f : E → R f : E → R f:E rarr Rf: E \rightarrow Rf:E→Ris said to be of order n n nnnby segments (on E E EEE) if we can decompose the whole E E EEEinto a finite number of consecutive subsets
(6) E 1 , E 2 , … , E s (6) E 1 , E 2 , … , E s {:(6)E_(1)","E_(2)","dots","E_(s):}\begin{equation*} E_{1}, E_{2}, \ldots, E_{s} \tag{6} \end{equation*}(6)E1,E2,…,Es
so that on each one the function f f fffeither of order n n nnn.
We say that (6) is a decomposition of E E EEEin consecutive subsets if:
8.1. A ⊂ E ν ⊆ E , ν = 1 , 2 , … , s A ⊂ E ν ⊆ E , ν = 1 , 2 , … , s A subE_(nu)sube E,nu=1,2,dots,sA \subset E_{\nu} \subseteq E, \nu=1,2, \ldots, sHAS⊂Eν⊆E,ν=1,2,…,s( A = A = A=A=HAS=the empty set)
8.2. ∪ ν = 1 s E ν = E ∪ ν = 1 s E ν = E uu_(nu=1)^(s)E_(nu)=E\cup_{\nu=1}^{s} E_{\nu}=E∪ν=1sEν=E
8.3. ∀ ∀ x ′ , x ′ ′ x ′ ∈ E ν , x ′ ′ ∈ E ν + 1 ⇒ x ′ < x ′ ′ ∀ ∀ x ′ , x ′ ′ x ′ ∈ E ν , x ′ ′ ∈ E ν + 1 ⇒ x ′ < x ′ ′ AAAA_(x^('),x^(''))x^(')inE_(nu),x^('')inE_(nu+1)=>x^(') < x^('')\forall \underset{x^{\prime}, x^{\prime \prime}}{\forall} x^{\prime} \in E_{\nu}, x^{\prime \prime} \in E_{\nu+1} \Rightarrow x^{\prime}<x^{\prime \prime}∀∀x′,x′′x′∈Eν,x′′∈Eν+1⇒x′<x′′.
The whole n n nnnEast ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1.
For such a function there can exist several decompositions (6) verifying the property of the previous definition.
The number s s sssof these decompositions has a minimum h h hhhwhich is called the characteristic of the function f f fff, of order n n nnnby segments. When h = 1 h = 1 h=1h=1h=1the function reduces to an order function n n nnn. In general if the characteristic is h h hhhwe can say that the function f f fffchange ( h − 1 h − 1 h-1h-1h−1)-times of convexity appearance of order n n nnn(on E E EEE). In particular, if n = − 1 n = − 1 n=-1n=-1n=−1this means that the function changes ( h − 1 ) ( h − 1 ) (h-1)(h-1)(h−1)-times of sign on E E EEE.
9. Now be n n nnnan integer ≧ − 1 ≧ − 1 >= -1\geqq-1≧−1And
(7) ( x ν ) ν = 1 m ( m ≧ n + 2 ) (7) x ν ν = 1 m ( m ≧ n + 2 ) {:(7)(x_(nu))_(nu=1)^(m)quad(m >= n+2):}\begin{equation*} \left(x_{\nu}\right)_{\nu=1}^{m} \quad(m \geqq n+2) \tag{7} \end{equation*}(7)(xν)ν=1m(m≧n+2)
a growing series of E E EEE(THE x x xxx, are not necessarily equidistant). Consider the corresponding sequence
(8) ( [ x ν , x ν + 1 , … , x ν + n + 1 ; f ] ) ν = 1 m − n − 1 (8) x ν , x ν + 1 , … , x ν + n + 1 ; f ν = 1 m − n − 1 {:(8)([x_(nu),x_(nu+1),dots,x_(nu+n+1);f])_(nu=1)^(m-n-1):}\begin{equation*} \left(\left[x_{\nu}, x_{\nu+1}, \ldots, x_{\nu+n+1} ; f\right]\right)_{\nu=1}^{m-n-1} \tag{8} \end{equation*}(8)([xν,xν+1,…,xν+n+1;f])ν=1m−n−1
divided differences of order n + 1 n + 1 n+1n+1n+1of f f fffon consecutive points of the sequence (7).
The order functions ( n k n k nkn knk) are then defined as follows:
Definition 2. The function f : E → R f : E → R f:E rarr Rf: E \rightarrow Rf:E→Ris said to be of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) if for all (finite) increasing sequences (7) of E E EEE, the maximum number of sign variations of the corresponding sequences (8) is equal to k k kkk.
n n nnnAnd k k kkkare integers, k ≧ 0 , n ≧ − 1 k ≧ 0 , n ≧ − 1 k >= 0,n >= -1k \geqq 0, n \geqq-1k≧0,n≧−1.
We have k = 0 k = 0 k=0k=0k=0if and only if the function f f fffis in order n n nnn.
For more details see my previous work [4].
10. The set of order functions n n nnnby segments and the set of order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k) for the various possible values ​​of k k kkk, coincident, Indeed [4]:
An order function ( n ∣ k n ∣ k n∣kn \mid kn∣k) is an order function n n nnnby segments and whe order function n n nnnby segments is an order function ( n ∣ k n ∣ k n∣kn \mid kn∣k) for a
If an order function n n nnnby segments has its characteristic equal to h h hhh, k k kkksuitable. then it is a function of order ( n ∣ k ) n ∣ k ) n∣k)n \mid k)n∣k)Or h − 1 ≦ k ≦ ( h − 1 ) ( n + 2 ) h − 1 ≦ k ≦ ( h − 1 ) ( n + 2 ) h-1 <= k <= (h-1)(n+2)h-1 \leqq k \leqq(h-1)(n+2)h−1≦k≦(h−1)(n+2).
The very simple property that any monotonic function changes sign at most once and that any usual convex function changes sign at most twice is generalized by the following property:
Any order function ( n ∣ k n ∣ k n∣kn \mid kn∣k) is at most of order ( n − i ∣ k + i n − i ∣ k + i n-i∣k+in-i \mid k+in−i∣k+i) for
We say that the function is at most of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) if it is of order i = 1 , 2 , … , n + 1 i = 1 , 2 , … , n + 1 i=1,2,dots,n+1i=1,2, \ldots, n+1i=1,2,…,n+1.
11. All properties of order functions n n nnnby segments and ( n k ′ n k ′ nk^(')n k^{\prime}nk′) with k ′ ≦ k k ′ ≦ k k^(') <= kk^{\prime} \leqq kk′≦k. order functions ( n ∣ k n ∣ k n∣kn \mid kn∣k) have not yet been studied.
To give examples here are some new properties of these
If the function f f fffis of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on the interval [ a , b ] [ a , b ] [a,b][a, b][has,b]and if k ≦ n k ≦ n k <= nk \leqq nk≦n, functions. this function is bounded on any interval [ a 1 , b 1 ] a 1 , b 1 [a_(1),b_(1)]\left[a_{1}, b_{1}\right][has1,b1]Or a < a 1 < b 1 < b a < a 1 < b 1 < b a < a_(1) < b_(1) < ba<a_{1}<b_{1}<bhas<has1<b1<b.
The demonstration is quite simple. Let x ∈ [ a 1 , b 1 ] x ∈ a 1 , b 1 x in[a_(1),b_(1)]x \in\left[a_{1}, b_{1}\right]x∈[has1,b1]and the fixed points x ν , x ν ′ , ν = 1 , 2 , … , n + 1 x ν , x ν ′ , ν = 1 , 2 , … , n + 1 x_(nu),x_(nu)^('),nu=1,2,dots,n+1x_{\nu}, x_{\nu}^{\prime}, \nu=1,2, \ldots, n+1xν,xν′,ν=1,2,…,n+1such as x 1 < x 2 < … < x n + 1 < a 1 < b 1 < x 1 ′ < x 2 ′ < … < x n + 1 ′ x 1 < x 2 < … < x n + 1 < a 1 < b 1 < x 1 ′ < x 2 ′ < … < x n + 1 ′ x_(1) < x_(2) < dots < x_(n+1) < a_(1) < b_(1) < x_(1)^(') < x_(2)^(') < dots < x_(n+1)^(')x_{1}<x_{2}<\ldots<x_{n+1}<a_{1}<b_{1}<x_{1}^{\prime}< x_{2}^{\prime}<\ldots<x_{n+1}^{\prime}x1<x2<…<xn+1<has1<b1<x1′<x2′<…<xn+1′. Then we can find two consecutive terms of the sequence
( [ x 1 ′ , x 2 ′ , … , x v − 1 ′ , x v , x v + 1 , … , x n + 1 , x ; f ] ) v = 1 n + 2 x 1 ′ , x 2 ′ , … , x v − 1 ′ , x v , x v + 1 , … , x n + 1 , x ; f v = 1 n + 2 ([x_(1)^('),x_(2)^('),dots,x_(v-1)^('),x_(v),x_(v+1),dots,x_(n+1),x;f])_(v=1)^(n+2)\left(\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{v-1}^{\prime}, x_{v}, x_{v+1}, \ldots, x_{n+1}, x ; f\right]\right)_{v=1}^{n+2}([x1′,x2′,…,xv−1′,xv,xv+1,…,xn+1,x;f])v=1n+2
which are of the same sign. If, to fix the ideas, the divided differences
[ x 1 ′ , x 2 ′ , … , x ν − 1 ′ , x ν , x ν + 1 … , x n + 1 , x ; f ] = f ( x ) ( x − x 1 ′ ) ( x − x 2 ′ ) … ( x − x ν − 1 ′ ) ( x − x ν ) ( x − x ν + 1 ) … ( x − x n + 1 ) + [ x 1 ′ , x 2 ′ , … , x ν − 1 ′ , x ν , x ν + 1 , … , x n + 1 ; f ( t ) t − x ] [ x 1 ′ , x 2 ′ , … , x ν ′ , x ν + 1 , x ν + 2 , … , x n + 1 , x ; f ] = f ( x ) ( x − x 1 ′ ) ( x − x 2 ′ ) … ( x − x ν ′ ) ( x − x ν + 1 ) ( x − x ν + 2 ) … ( x − x n + 1 ) + [ x 1 ′ , x 2 ′ , … , x ν ′ x ν + 1 , x ν + 2 , … , x n + 1 ; f ( t ) t − x ] x 1 ′ , x 2 ′ , … , x ν − 1 ′ , x ν , x ν + 1 … , x n + 1 , x ; f = f ( x ) x − x 1 ′ x − x 2 ′ … x − x ν − 1 ′ x − x ν x − x ν + 1 … x − x n + 1 + x 1 ′ , x 2 ′ , … , x ν − 1 ′ , x ν , x ν + 1 , … , x n + 1 ; f ( t ) t − x x 1 ′ , x 2 ′ , … , x ν ′ , x ν + 1 , x ν + 2 , … , x n + 1 , x ; f = f ( x ) x − x 1 ′ x − x 2 ′ … x − x ν ′ x − x ν + 1 x − x ν + 2 … x − x n + 1 + x 1 ′ , x 2 ′ , … , x ν ′ x ν + 1 , x ν + 2 , … , x n + 1 ; f ( t ) t − x {:[[x_(1)^('),x_(2)^('),dots,x_(nu-1)^('),x_(nu),x_(nu+1)dots,x_(n+1),x;f]],[=(f(x))/((x-x_(1)^('))(x-x_(2)^('))dots(x-x_(nu-1)^('))(x-x_(nu))(x-x_(nu+1))dots(x-x_(n+1)))],[+[x_(1)^('),x_(2)^('),dots,x_(nu-1)^('),x_(nu),x_(nu+1),dots,x_(n+1);(f(t))/(t-x)]],[[x_(1)^('),x_(2)^('),dots,x_(nu)^('),x_(nu+1),x_(nu+2),dots,x_(n+1),x;f]],[=(f(x))/((x-x_(1)^('))(x-x_(2)^('))dots(x-x_(nu)^('))(x-x_(nu+1))(x-x_(nu+2))dots(x-x_(n+1)))],[quad+[x_(1)^('),x_(2)^('),dots,x_(nu)^(')x_(nu+1),x_(nu+2),dots,x_(n+1);(f(t))/(t-x)]]:}\begin{aligned} & {\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{\nu-1}^{\prime}, x_{\nu}, x_{\nu+1} \ldots, x_{n+1}, x ; f\right]} \\ & =\frac{f(x)}{\left(x-x_{1}^{\prime}\right)\left(x-x_{2}^{\prime}\right) \ldots\left(x-x_{\nu-1}^{\prime}\right)\left(x-x_{\nu}\right)\left(x-x_{\nu+1}\right) \ldots\left(x-x_{n+1}\right)} \\ & +\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{\nu-1}^{\prime}, x_{\nu}, x_{\nu+1}, \ldots, x_{n+1} ; \frac{f(t)}{t-x}\right] \\ & {\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{\nu}^{\prime}, x_{\nu+1}, x_{\nu+2}, \ldots, x_{n+1}, x ; f\right]} \\ & =\frac{f(x)}{\left(x-x_{1}^{\prime}\right)\left(x-x_{2}^{\prime}\right) \ldots\left(x-x_{\nu}^{\prime}\right)\left(x-x_{\nu+1}\right)\left(x-x_{\nu+2}\right) \ldots\left(x-x_{n+1}\right)} \\ & \quad+\left[x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{\nu}^{\prime} x_{\nu+1}, x_{\nu+2}, \ldots, x_{n+1} ; \frac{f(t)}{t-x}\right] \end{aligned}[x1′,x2′,…,xν−1′,xν,xν+1…,xn+1,x;f]=f(x)(x−x1′)(x−x2′)…(x−xν−1′)(x−xν)(x−xν+1)…(x−xn+1)+[x1′,x2′,…,xν−1′,xν,xν+1,…,xn+1;f(t)t−x][x1′,x2′,…,xν′,xν+1,xν+2,…,xn+1,x;f]=f(x)(x−x1′)(x−x2′)…(x−xν′)(x−xν+1)(x−xν+2)…(x−xn+1)+[x1′,x2′,…,xν′xν+1,xν+2,…,xn+1;f(t)t−x]
are of the same sign, a suitable delimitation of | f ( x ) | | f ( x ) | |f(x)||f(x)||f(x)|on [ a 1 , b 1 ] ) a 1 , b 1 {:[a_(1),b_(1)])\left.\left[a_{1}, b_{1}\right]\right)[has1,b1])follows. In the previous equalities we have designated by t t tttthe variable of the function whose divided differences are considered. :-
We can also demonstrate in the same way that:
If the function f f fffis of order ( n ∣ k n ∣ k n∣kn \mid kn∣k) on the interval [ a , b ] [ a , b ] [a,b][a, b][has,b]and if k ≦ n − r − 1 k ≦ n − r − 1 k <= n-r-1k \leqq n -r-1k≦n−r−1, this function has a continuous derivative of order r ( ≧ 0 ) r ( ≧ 0 ) r( >= 0)r(\geqq 0)r(≧0)on any interval [ a 1 , b 1 ] a 1 , b 1 [a_(1),b_(1)]\left[a_{1}, b_{1}\right][has1,b1]where a < a 1 < b 1 < b a < a 1 < b 1 < b a < a_(1) < b_(1) < ba<a_{1}<b_{1}<bhas<has1<b1<b.
The demonstration consists of showing that the function is ( r + 1 ) ( r + 1 ) (r+1)(r+1)(r+1)-th divided difference, bounded on [ a 1 , b 1 ] a 1 , b 1 [a_(1),b_(1)]\left[a_{1}, b_{1}\right][has1,b1].
For r = 0 r = 0 r=0r=0r=0the property is equivalent to the continuity of the function f f fff.

LITERATURE

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  3. G. Peano. Resto nelle formula di quadratura, espresso with a definite integrale. Returns. Lincei, 22, 1913, 562-569.
  4. T. Popoviciu. Notes on the generalizations of higher-order convex functions. First note: Disquisitiones Math. et Phys., 1, 1940, 35-42. Second note: Bull Acad. Roumaine, 22, 1940, 473-477. Third note: ibid., 24, 1942, 409-416.
  5. T. Popoviciu. Notes on convex functions of higher order (IX). Bull Math. Soc. Romanian sci., 43, 1942, 85-141.
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  7. I. Radon. Restausdrücke bei Interpolations- und Quadraturformeln durch bestimmte Integrale. Monatshefte f. Math. u. Phys., 42, 1935, 389-396.
  8. IJ Schoenberg. (Ed.) Approximation with Special Emphasis on Spline Functions, 1969.
Institutul de Calcul
Cluj Romania
Received on 28.4.1973
1975

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