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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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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
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 functionffis in order(n//k)(n/k)on the interval[a,b][a, b]and ifk <= nk \leqq n, this function is bounded and ifk <= nr-1k \leqq nr-1, it has a continuous derivative of orderr(r >= 0)r(r \geqq 0)on any interval[a_(1),b_(1)]\left[a_{1}, b_{1}\right], Ora < a_(1) < b_(1) < ba<a_{1}<b_{1}<b.
- 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.
- Eitherf:[a,b]rarr Rf:[a, b] \rightarrow Ra continuous function defined on the interval[a,b][a, b]bounded and closed of the real axisRRand consider the points
{:(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*}
which divide this interval intommequal parts.
Let us denote bypsi_(m)\psi_{m}the polygonal line (the polygonal function) inscribed in the curvey=f(x)y=f(x)and whose vertices are the points (x_(v),f(x_(v))x_{v}, f\left(x_{v}\right)), We know that, on[a,b][a, b], the sequel (psi_(m)\psi_{m}) tends uniformly towards the functionff.
Let us denote bypsi_(m)\psi_{m}the polygonal line (the polygonal function) inscribed in the curvey=f(x)y=f(x)and whose vertices are the points (x_(v),f(x_(v))x_{v}, f\left(x_{v}\right)), We know that, on[a,b][a, b], the sequel (psi_(m)\psi_{m}) tends uniformly towards the functionff.
The functionpsi_(m)\psi_{m}is of the form
{:(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*}
Orlambda\lambdais a real number andvarphi_(2,lambda^(2))=(x-lambda+|x-lambda|)/(2)=(x-lambda)+\varphi_{2, \lambda^{2}}=\frac{x-\lambda+|x-\lambda|}{2}=(x-\lambda)+is the positive part of the functionx-lambdax-\lambda, :
3. We will designate by[z_(1),z_(2),dots,z_(k+1);f]\left[z_{1}, z_{2}, \ldots, z_{k+1} ; f\right]the divided difference (of orderkk) of the functionffon thek+1k+1points or knotsz_(1),z_(2),dots,z_(k+1)z_{1}, z_{2}, \ldots, z_{k+1}.
3. We will designate by[z_(1),z_(2),dots,z_(k+1);f]\left[z_{1}, z_{2}, \ldots, z_{k+1} ; f\right]the divided difference (of orderkk) of the functionffon thek+1k+1points or knotsz_(1),z_(2),dots,z_(k+1)z_{1}, z_{2}, \ldots, z_{k+1}.
Then the functions (2) are generalized by the functions (m >= n+1m \geq n+1)
{:(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*}
Ornnis a natural number,Q_(m)(x)Q_{m}(x)a polynomial of degreennand thex_(y)x_{y}are still the points (1). Finally we have
{:(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*}
Divided differences of various orders, on equidistant or non-equidistant nodes, are defined (for example) by the recurrence relation
[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}}
And[z_(1);f]=f(z_(1))\left[z_{1} ; f\right]=f\left(z_{1}\right)and by which we construct the divided difference of orderkkusing the order onek-1k-1.
In the following we always assume that the nodesz_(1),z_(2),dots,z_(k+1)z_{1}, z_{2}, \ldots, z_{k+1}are distinct. We know that by admitting the existence of a certain number of derivatives for the functionff, 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),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}
The structure of function (3) depends on two important notions: A. The notion of convex functions of ordernn. 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 ordernn
- The functionf:[a,b]rarr Rf:[a, b] \rightarrow Ris said to be convex, non-concave, polynomial, non-convex respectively concave of ordernn(on[a,b][a, b]) depending on whether the divided differences of ordern+1n+1on any group ofn+2n+2distinct points, or nodes, of the functionff, which are all> , >= ,=, <=>, \geq,=, \leqqrespectively< 0<0A function verifying one of these properties is called a doodre function.nn.
The wholennEast>= -1\geqq-1. Forn=-1n=-1we have the functions of invariable sign: positive, non-negative, zero (the identically zero function), non-positive, respectively negative. Forn=0n=0we have the monotonic functions: increasing, non-decreasing, constant, non-increasing respectively decreasing. Finally
forn=1n=1we have the usual convex, non-concave, linear, non-convex and concave functions respectively.
forn=1n=1we have the usual convex, non-concave, linear, non-convex and concave functions respectively.
B. The concept of spline function
- Let's supposen >= 1n \geqq 1. Then any linear combination of polynomials of degreennand of a finite number of functions of the form (4) is what we formerly called an elementary function of ordernn[5]. Today we call them spline functions.
A spline function is therefore of the form
{:(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*}
OrPPis a polynomial of degreenn, THElambda_(nu)\lambda_{\nu}are given points such thata < lambda_(1) < lambda_(2) < dots < lambda_(m) < ba<\lambda_{1}<\lambda_{2}<\ldots<\lambda_{m}<band thec_(y)c_{y}are any constants.
Function (5) is continuous and has a divided difference of ordernnlimited. Ifn > 1n>1it has a continuous derivative of ordern-1n-1on[a,b][a, b]. The spline function (5) is formed by a finite number of pieces of polynomials of degreennwhich have a maximum connection order without being reduced to a polynomial of degreenn. There are also functions formed by pieces of polynomials of degreennwhich do not connect as completely, which, for example, are not continuous or which, forn > 1n>1are continuous but not differentiable, etc. We can say that for anngiven 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:
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 conditionsc_(v) >= 0,v=1,2,dots,mc_{v} \geqq 0, v=1,2, \ldots, m, are necessary and sufficient for function (5) to be non-concave of ordernn.
Taking the spline functions of the form (3), where thex_(2)x_{2}are the equidistant points (1) and by suitably choosing the polynomialQ_(m)(x)Q_{m}(x)for each value ofmm, we obtain the approximation theorem [5, 6]:
Forn >= 1n \geqq 1, any functionf:[a,b]rarr Rf:[a, b] \rightarrow Rcontinuous and orderlynnis indefinitely and uniformly approximable by spline functions of the form (3), of ordernn. Otherwise: Any functionf:[a,b]rarr Rf:[a, b] \rightarrow R, continuous and non-concave of ordernnis the limit of a uniformly convergent sequence of functions of the form (5), with coefficientsc_(y)c_{y}non-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
8. This being said, we have proposed a generalization of the functions
Definition 1. IfE sube RE \subseteq R, the functionf:E rarr Rf: E \rightarrow Ris said to be of ordernnby segments (onEE) if we can decompose the wholeEEinto a finite number of consecutive subsets
{:(6)E_(1)","E_(2)","dots","E_(s):}\begin{equation*}
E_{1}, E_{2}, \ldots, E_{s} \tag{6}
\end{equation*}
so that on each one the functionffeither of ordernn.
We say that (6) is a decomposition ofEEin consecutive subsets if:
8.1.A subE_(nu)sube E,nu=1,2,dots,sA \subset E_{\nu} \subseteq E, \nu=1,2, \ldots, s(A=A=the empty set)
8.2.uu_(nu=1)^(s)E_(nu)=E\cup_{\nu=1}^{s} E_{\nu}=E
8.3.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}.
We say that (6) is a decomposition ofEEin consecutive subsets if:
8.1.A subE_(nu)sube E,nu=1,2,dots,sA \subset E_{\nu} \subseteq E, \nu=1,2, \ldots, s(A=A=the empty set)
8.2.uu_(nu=1)^(s)E_(nu)=E\cup_{\nu=1}^{s} E_{\nu}=E
8.3.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}.
The wholennEast>= -1\geqq-1.
For such a function there can exist several decompositions (6) verifying the property of the previous definition.
For such a function there can exist several decompositions (6) verifying the property of the previous definition.
The numberssof these decompositions has a minimumhhwhich is called the characteristic of the functionff, of ordernnby segments. Whenh=1h=1the function reduces to an order functionnn. In general if the characteristic ishhwe can say that the functionffchange (h-1h-1)-times of convexity appearance of ordernn(onEE). In particular, ifn=-1n=-1this means that the function changes(h-1)(h-1)-times of sign onEE.
9. Now bennan integer>= -1\geqq-1And
9. Now bennan integer>= -1\geqq-1And
{:(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*}
a growing series ofEE(THExx, are not necessarily equidistant). Consider the corresponding sequence
{:(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*}
divided differences of ordern+1n+1offfon consecutive points of the sequence (7).
The order functions (nkn k) are then defined as follows:
Definition 2. The functionf:E rarr Rf: E \rightarrow Ris said to be of order (n∣kn \mid k) if for all (finite) increasing sequences (7) ofEE, the maximum number of sign variations of the corresponding sequences (8) is equal tokk.
nnAndkkare integers,k >= 0,n >= -1k \geqq 0, n \geqq-1.
We havek=0k=0if and only if the functionffis in ordernn.
For more details see my previous work [4].
10. The set of order functionsnnby segments and the set of order functions (n∣kn \mid k) for the various possible values ​​ofkk, coincident, Indeed [4]:
Definition 2. The functionf:E rarr Rf: E \rightarrow Ris said to be of order (n∣kn \mid k) if for all (finite) increasing sequences (7) ofEE, the maximum number of sign variations of the corresponding sequences (8) is equal tokk.
nnAndkkare integers,k >= 0,n >= -1k \geqq 0, n \geqq-1.
We havek=0k=0if and only if the functionffis in ordernn.
For more details see my previous work [4].
10. The set of order functionsnnby segments and the set of order functions (n∣kn \mid k) for the various possible values ​​ofkk, coincident, Indeed [4]:
An order function (n∣kn \mid k) is an order functionnnby segments and whe order functionnnby segments is an order function (n∣kn \mid k) for a
If an order functionnnby segments has its characteristic equal tohh,kksuitable. then it is a function of order (n∣k)n \mid k)Orh-1 <= k <= (h-1)(n+2)h-1 \leqq k \leqq(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∣kn \mid k) is at most of order (n-i∣k+in-i \mid k+i) for
We say that the function is at most of order (n∣kn \mid k) if it is of orderi=1,2,dots,n+1i=1,2, \ldots, n+1.
11. All properties of order functionsnnby segments and (nk^(')n k^{\prime}) withk^(') <= kk^{\prime} \leqq k. order functions (n∣kn \mid k) have not yet been studied.
We say that the function is at most of order (n∣kn \mid k) if it is of orderi=1,2,dots,n+1i=1,2, \ldots, n+1.
11. All properties of order functionsnnby segments and (nk^(')n k^{\prime}) withk^(') <= kk^{\prime} \leqq k. order functions (n∣kn \mid k) have not yet been studied.
To give examples here are some new properties of these
If the functionffis of order (n∣kn \mid k) on the interval[a,b][a, b]and ifk <= nk \leqq n, functions. this function is bounded on any interval[a_(1),b_(1)]\left[a_{1}, b_{1}\right]Ora < a_(1) < b_(1) < ba<a_{1}<b_{1}<b.
If the functionffis of order (n∣kn \mid k) on the interval[a,b][a, b]and ifk <= nk \leqq n, functions. this function is bounded on any interval[a_(1),b_(1)]\left[a_{1}, b_{1}\right]Ora < a_(1) < b_(1) < ba<a_{1}<b_{1}<b.
The demonstration is quite simple. Letx in[a_(1),b_(1)]x \in\left[a_{1}, b_{1}\right]and the fixed pointsx_(nu),x_(nu)^('),nu=1,2,dots,n+1x_{\nu}, x_{\nu}^{\prime}, \nu=1,2, \ldots, n+1such asx_(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}. Then we can find two consecutive terms of the sequence
([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}
which are of the same sign. If, to fix the ideas, the divided differences
{:[[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}
are of the same sign, a suitable delimitation of|f(x)||f(x)|on{:[a_(1),b_(1)])\left.\left[a_{1}, b_{1}\right]\right)follows. In the previous equalities we have designated byttthe variable of the function whose divided differences are considered. :-
We can also demonstrate in the same way that:
If the functionffis of order (n∣kn \mid k) on the interval[a,b][a, b]and ifk <= n-r-1k \leqq n -r-1, this function has a continuous derivative of orderr( >= 0)r(\geqq 0)on any interval[a_(1),b_(1)]\left[a_{1}, b_{1}\right]wherea < a_(1) < b_(1) < ba<a_{1}<b_{1}<b.
If the functionffis of order (n∣kn \mid k) on the interval[a,b][a, b]and ifk <= n-r-1k \leqq n -r-1, this function has a continuous derivative of orderr( >= 0)r(\geqq 0)on any interval[a_(1),b_(1)]\left[a_{1}, b_{1}\right]wherea < a_(1) < b_(1) < ba<a_{1}<b_{1}<b.
The demonstration consists of showing that the function is(r+1)(r+1)-th divided difference, bounded on[a_(1),b_(1)]\left[a_{1}, b_{1}\right].
Forr=0r=0the property is equivalent to the continuity of the functionff.
LITERATURE
- G. Kowalewski. Interpolation and generated quadratur. Leipzig, 1932.
- R.v. Mises. Über allgemeine Quadraturformeln. J. queen u. angew. Math., 174, 1935, 56-67.
- G. Peano. Resto nelle formula di quadratura, espresso with a definite integrale. Returns. Lincei, 22, 1913, 562-569.
- 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.
- T. Popoviciu. Notes on convex functions of higher order (IX). Bull Math. Soc. Romanian sci., 43, 1942, 85-141.
- T. Popoviciu. On the remainder in some linear approximation formulas of analysis. Mathematica, 1 (24), 1959, 95-142.
- I. Radon. Restausdrücke bei Interpolations- und Quadraturformeln durch bestimmte Integrale. Monatshefte f. Math. u. Phys., 42, 1935, 389-396.
- IJ Schoenberg. (Ed.) Approximation with Special Emphasis on Spline Functions, 1969.
Institutul de Calcul
Cluj Romania
Received on 28.4.1973
Cluj Romania
Received on 28.4.1973