On Existence of Divided Differences in Linear Spaces

Mathematica-Revue d’analyse numerique et de theorie de l’approximation

L’Analyse Numérique et la Théorie de l’Approximation

Tome 2, 1973, pp.5-9

In a large number of problems, the notion of divided difference of the mappings in different linear spaces is very important. In the investigation of the divided differences many results are known ([6],[3], [1]) but the question of their existence is still an open problem. In our paper we state certain theorems which solve that question.

Let $X$ and $Y$ be real vector spaces and $P$ a mapping defined on $X$ with values in $Y$.

Definition 1

A linear mapping ${\tilde{P}}_{u,v}$ defined on $X$ with values in $Y$ is called algebraic divided difference of $P$ in the pair of different points $u,v$ of $X$, iff

$${\tilde{P}}_{u,v}\left(u-v\right)=P\left(u\right)-P\left(v\right).$$ (1)

Proposition 2

For every mapping $P$ defined on $X$ with values in $Y$, there exists an algebraic divided difference in every pair of different point $u,v$ of $X$.

Proof. For arbitrarily selected different $u$ and $v$ of $X$, let

$$X_1=\{x\in X|x=\alpha \left(u-v\right),\alpha \in ℝ\}.$$

We define on the linear subspace $X_1$ of $X$ the linear mapping

$${\overline{P}}_{u,v}\left[\alpha \left(u-v\right)\right]=\alpha P\left(u\right)-\alpha P\left(v\right).$$

We know that there is a linear projector $\mathrm{\Pi }$ of $X$ on $X_1$ (see, e.g.[5], p.71), i.e.

$$\mathrm{\Pi }\left(X\right)=X_1\text{ and }\mathrm{\Pi }\left(x\right)=x\text{ for every }x\text{ in }{X}_1.$$

If we put

$${\tilde{P}}_{u,v}={\overline{P}}_{u,v}\circ \mathrm{\Pi },$$

we have a linear mapping defined on $X$ with values in $Y$, and by definitions of ${\tilde{P}}_{u,v}\mathrm{\Pi }$ and ${\overline{P}}_{u,v}$

$${\tilde{P}}_{u,v}\left(u-v\right)=P\left(u\right)-P\left(v\right).$$

Example 3

Let $X=s,$ the set of the sequences of real numbers with usual linear structure, $Y$ a linear space and $P$ a mapping defined on $s$ with values in $Y$. If

$$u=\{u^1,u^2,\mathrm{\dots },u^n,\mathrm{\dots }\}\text{ and }v=\{v^1,v^2,\mathrm{\dots },v^n,\mathrm{\dots }\},u\ne v,$$

then

$$X_1=\{x_{\alpha }\in s|x_{\alpha }=\left(\alpha \left(u^1-v^1\right),\alpha \left(u^2-v^2\right),\mathrm{\dots },\alpha \left(u^n-v^n\right),\mathrm{\dots },\alpha \in R\right)\}$$

and

$${\overline{P}}_{u,v}\left[\alpha \left(u-v\right)\right]=\alpha P\left(u\right)-\alpha P\left(v\right).$$

Because $u\ne v$, we can consider that for $i_0,u^{i_0}-v^{i_0}\ne 0$. Then we have as a basis of $X$, the set of linearly independent vectors $e_1,e_2,\mathrm{\dots },e_{i_0-1},x_1,x_{i_0+1},\mathrm{\dots },$ where

$$x_1=\{u^1-v^1,u^2-v^2,\mathrm{\dots },u^n-v^n,\mathrm{\dots }\}$$

and $e_k$ is the sequence formed with $O$, excepting $k$-th element. Now we can

define the linear projector $\mathrm{\Pi }$ of $X$ on $X_1$ ina following manner:

$$\mathrm{\Pi }\left(x_1\right)=x_1,\mathrm{\Pi }\left(e_k\right)=0,\left(k=1,2,\mathrm{\dots },i_0-1,i_0+1,\mathrm{\dots }\right).$$

Then for

$$x=\{x^1,x^2,\mathrm{\dots },x^n,\mathrm{\dots }\}$$

we have

$$\mathrm{\Pi }\left(x\right)=\frac{x^{i_0}}{u^{i_0}-v^{i_0}}{x}_1=\frac{x^{i_0}}{u^{i_0}-v^{i_0}}\left(u-v\right),$$

and

$${\tilde{P}}_{u,v}x={\overline{P}}_{u,v}\circ \mathrm{\Pi }x=\frac{x^{i_0}}{u^{i_0}-v^{i_0}}\left[P\left(u\right)-P\left(v\right)\right],$$

which is an algebraic divided difference of $P$ in $u,v$.

Definition 4

An algebraic divided difference ${\tilde{P}}_{u,v}$ of a mapping $P$ defined on a topological vector space $X$, with values in a topological vector space $Y$, is called divided difference $P_{u,v}$ of $P$ in the different points $u$ and $v$ of $X,$ iff ${\tilde{P}}_{u,v}$ is a continuous mapping of $X$ in $Y$.

Theorem 5

Every mapping $P$ of a separated locally convex space $X$ in a topological vector space $Y$, has a divided difference in all different points $u$ and $v$ of $X$.

Proof. Let $X$, and ${\overline{P}}_{u,v}$ be as in the preceeding theorem. $X_1$ having the dimension 1, there exists the linear and continuous projector of $X$ on $X_1$ [5,p.96-97], and hence the mapping

$$P_{u,v}={\overline{P}}_{u,v}\circ \mathrm{\Pi }$$

is a linear and continuous mapping verifying the condition (1).  

Example 6

Let $X=l_2$ the set of the sequences of the real numbers with having the susual linear structure and norm, $P$ mapping $l_2$ in the linear normed space $Y$.

Defining $X_1$ and ${\overline{P}}_{u,v}$ in the same way as in the first example, we can consider as vector generating the subspace $X_1$, the element

$$e=\frac{1}{\sqrt{\sum _{n=1}^{\mathrm{\infty }}{\left(u^n-v^n\right)}^2}}\left(u-v\right)$$

Then for

$$x=\{x^1,x^2,\mathrm{\dots },x^n,\mathrm{\dots }\}\text{, we have}$$

$$\mathrm{\Pi }x=(x,e)e=\frac{\sum _{n=1}^{\mathrm{\infty }}{x}^n\left({}_{}{}^n-v^n\right)}{\sum _{n=1}^{\mathrm{\infty }}{\left(u^n-v^n\right)}^2}\left(u-v\right),$$

and hence

$$P_{u,v}x=\frac{\sum _{n=1}^{\mathrm{\infty }}{x}^n\left(u^n-v^n\right)}{\sum _{n=1}^{\mathrm{\infty }}}\cdot \left[P\left(u\right)-P\left(v\right)\right].$$

Theorem 7

Every mapping $P$ of a linear normed space $X$ in a linear normed space $Y$ has a divided difference $P_{u,v}$, with the norm

$$\Vert P_{u,v}\Vert =\frac{\Vert P\left(u\right)-P\left(v\right)\Vert }{\Vert u-v\Vert }.$$

Proof. Let $X_1$ and $P_{u,v}$ be as in the procedings, and let be the real valued fucntion $f_1$ defined on the $X_1$ by $f_1\left(\alpha \left(u-v\right)\right)=\alpha$. Considering the norm-preserving extension $f$ of $f_1$ to $X$ [see e.g.[7] p. 106] and

$$Ker\left(f\right)=\{x\in X|f\left(x\right)=0\},$$

every point $y$ of $X$ can be expressed by $y=x+f\left(y\right)\left(u-v\right)$, where $x\in Ker\left(f\right)$. We define

$$P_{u,v}y=f\left(y\right){\overline{P}}_{u,v}\left(u-v\right),$$

for every $y$ in $X$, and so we obtain that $P_{u,v}$ is a divided difference of $P$, and

$$\Vert P_{u,v}\Vert =\frac{\Vert P\left(u\right)-P\left(v\right)\Vert }{\Vert u-v\Vert }.$$

Remark 8

1. In this way we do not obtain the uniqueness of the divided differences, which is natural, because there are some known examples when the divided difference isn’t unique.

2. This paper gives only some principial answer to the question of the existence of divided differences, but the effective construction of divided differences in every practical case must be solved in other ways.

The authors wish to express their acknowledgement to Professor Adolf Haimovici and to A.B. Németh for some useful discussions on the subject, and for suggestions in improving of the paper.

Received, 12.XII. 1972.

Catedra de Analiză

Matematică a Facultăţii de

Matematică-Mecanică a

Universităţii Babeş-Bolyai

din Cluj