Abstract
The rest in some approximation formulae can be expressed in terms of a generalized divided di§erence on three knots. We provide an estimation of such a divided di§erence for functions deÖned on a uniformly convex space.
Authors
Mira-Cristiana Anisiu
Tiberiu Popoviciu Institut of Numerical Analysis, Romanian Academy, Romania
Valeriu Anisiu
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
uniformly convex space, FrÈchet derivative, generalized divided difference
Paper coordinates
M.-C. Anisiu, V. Anisiu, An estimation of a generalized divided difference in uniformly convex spaces, in Analysis, Functional Equations, Approximation and Convexity, Proceedings of the Conference Held in Honour of Prof. Elena Popoviciu, Cluj-Napoca, September 30, 2004, 57-62 (pdf file here)
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[1] D. Butnariu, A. N. Iusem, E. Resmerita, Total convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Analysis 7 (2000), 319-334
[2] D. Butnariu, A. N. Iusem, C. Zalinescu, On uniform convexity, total convexity and convergence of the proximinal point and outer Bregman projection algorithms in Banach spaces, J. Convex Analysis 10 (2003), 35-61
[3] M. Ivan, I. Rasa, The rest in some approximation formulae, Seminaire de la Theorie de la Meilleure Approximation, Convexite et Optimization, Cluj-Napoca, Srima 2002, 87-92
[4] E. Popoviciu, Teoreme de medie din analiza matematica ¸si legatura lor cu teoria interpolarii. Editura Dacia, Cluj, 1972
[5] I. Rasa, Sur les fonctionelles de la forme simple au sens de T. Popoviciu, Anal. Numer. Theor. Approx. 9 (1980), 261-268
[6] I. Rasa, Functionale de forma simpla in sensul lui Tiberiu Popoviciu (PhD Thesis), Cluj-Napoca, 1982
[7] C. Zalinescu, Convex Anaysis in General Vector Spaces , World Scientifc, 2002
An estimation of a generalized divided difference in uniformly convex spaces
Abstract
The rest in some approximation formulae can be expressed in terms of a generalized divided difference on three knots. We provide an estimation of such a divided difference for functions defined on a uniformly convex space.
KEY WORDS: uniformly convex space; Fréchet derivative; generalized divided difference
MSC 2000: 46B20, 46A55, 46T20
1 Introduction
The convexity properties of the functions were used by Tiberiu Popoviciu to give estimations of the rest in some approximation formulae. A synthesis of this type of results can be found in the book [4].
Several theorems of representation of linear functionals were proved by Raşa [5], [6]. To mention two of them, let EE denote a locally convex Hausdorff real space and XX a compact convex metrizable subset of EE; for f in C(X)f \in C(X), x,y in Xx, y \in X and a in[0,1]a \in[0,1], we denote
{:(1)(x","a","y;f)=(1-a)f(x)+af(y)-f((1-a)x+ay):}\begin{equation*}
(x, a, y ; f)=(1-a) f(x)+a f(y)-f((1-a) x+a y) \tag{1}
\end{equation*}
We remark that, since XX is a metrizable space, there exist strictly convex functions in C(X)C(X); we denote by varphi\varphi such a function.
Theorem 1 Let L:C(X)rarrRL: C(X) \rightarrow \mathbb{R} be a linear functional such that L(g) > 0L(g)>0 for each strictly convex function g in C(X)g \in C(X). Then for every f in C(X)f \in C(X) there exists x,y in X,x!=yx, y \in X, x \neq y and a in(0,1)a \in(0,1) such that
L(f)=L(varphi)((x,a,y;f))/((x,a,y;varphi))L(f)=L(\varphi) \frac{(x, a, y ; f)}{(x, a, y ; \varphi)}
We consider now C(X)C(X) endowed with the uniform norm.
Theorem 2 Let L:C(X)rarrRL: C(X) \rightarrow \mathbb{R} be a continuous and linear functional such that L(g) >= 0L(g) \geq 0 for each convex function g in C(X)g \in C(X). Then for every f in C(X)f \in C(X) there exists x,y in X,x!=yx, y \in X, x \neq y and a in(0,1)a \in(0,1) such that
Theorem 2 Let L:C(X)rarrRL: C(X) \rightarrow \mathbb{R} be a continuous and linear functional such that L(g) >= 0L(g) \geq 0 for each convex function g in C(X)g \in C(X). Then for every f in C(X)f \in C(X) there exists x,y in X,x!=yx, y \in X, x \neq y and a in(0,1)a \in(0,1) such that
L(f)=L(varphi)((x,a,y;f))/((x,a,y;varphi))L(f)=L(\varphi) \frac{(x, a, y ; f)}{(x, a, y ; \varphi)}
For the special case E=R,X=[0,1]E=\mathbb{R}, X=[0,1] and varphi(t)=t^(2),t in[0,1]\varphi(t)=t^{2}, t \in[0,1], Ivan and Raşa [3] showed that
(x,a,y;varphi)=(1-a)x^(2)+ay^(2)-((1-a)x+ay)^(2)=a(1-a)(x-y)^(2),(x, a, y ; \varphi)=(1-a) x^{2}+a y^{2}-((1-a) x+a y)^{2}=a(1-a)(x-y)^{2},
for all x,y,a in[0,1]x, y, a \in[0,1]. In this case it follows that
((x,a,y;f))/((x,a,y;varphi))=[x,(1-a)x+ay,y]\frac{(x, a, y ; f)}{(x, a, y ; \varphi)}=[x,(1-a) x+a y, y]
where the last expression is the classical divided difference of the real function ff on the knots x,(1-a)x+ayx,(1-a) x+a y and yy. In the general case,
{:(2)[x","a","y;f","varphi]:=((x,a,y;f))/((x,a,y;varphi)):}\begin{equation*}
[x, a, y ; f, \varphi]:=\frac{(x, a, y ; f)}{(x, a, y ; \varphi)} \tag{2}
\end{equation*}
with ( x,a,y;fx, a, y ; f ) given by (1) was then named generalized divided difference on three knots.
2 Main results
We give an estimate of the generalized divided difference (2) in the case of a real uniformly convex space.
Let (E,||*||)(E,\|\cdot\|) be a real smooth uniformly convex space and XX a compact subset of EE. Consider the (strictly convex) function varphi_(r)in C(X)\varphi_{r} \in C(X) given by
varphi_(r)(x)=||x||^(r),x in X,\varphi_{r}(x)=\|x\|^{r}, x \in X,
where 1 < r <= 21<r \leq 2.
We need upper and lower bounds for the expression ( x,a,y;fx, a, y ; f ). An upper bound for |(x,a,y;f)||(x, a, y ; f)| was found in [3], for ff twice Fréchet differentiable on an open set YY and ||f^('')(y)|| <= M\left\|f^{\prime \prime}(y)\right\| \leq M for each y in Yy \in Y, namely
We need upper and lower bounds for the expression ( x,a,y;fx, a, y ; f ). An upper bound for |(x,a,y;f)||(x, a, y ; f)| was found in [3], for ff twice Fréchet differentiable on an open set YY and ||f^('')(y)|| <= M\left\|f^{\prime \prime}(y)\right\| \leq M for each y in Yy \in Y, namely
{:(3)|(x","a","y;f)| <= (M)/(2)a(1-a)||x-y||^(2).:}\begin{equation*}
|(x, a, y ; f)| \leq \frac{M}{2} a(1-a)\|x-y\|^{2} . \tag{3}
\end{equation*}
It was proved for Hilbert case, but it can be shown that (3) holds in our setting too.
If ff is a convex function, ( x,a,y;fx, a, y ; f ) is >= 0\geq 0 and is related with the modulus of uniform strict convexity. We recall some definitions from [7], [2]. The modulus of uniform strict convexity at xx (named gage of uniform convexity in [7]) is
mu_(f)(x,t)=i n f_({:[y in dom(f)],[||x-y||=t],[lambda in(0","1)]:})((x,lambda,y;f))/(lambda(1-lambda)),t >= 0\mu_{f}(x, t)=\inf _{\substack{y \in \operatorname{dom}(f) \\\|x-y\|=t \\ \lambda \in(0,1)}} \frac{(x, \lambda, y ; f)}{\lambda(1-\lambda)}, t \geq 0
A related function is
bar(mu)_(f)(x,t)=i n f_({:[y in dom(f)],[||x-y||=t]:})(f(x)+f(y)-2f((x+y)/(2))).\bar{\mu}_{f}(x, t)=\inf _{\substack{y \in \operatorname{dom}(f) \\\|x-y\|=t}}\left(f(x)+f(y)-2 f\left(\frac{x+y}{2}\right)\right) .
One has [2]:
{:(4)(1)/(2)mu <= bar(mu) <= mu.:}\begin{equation*}
\frac{1}{2} \mu \leq \bar{\mu} \leq \mu . \tag{4}
\end{equation*}
The function ff is said to be uniformly convex at xx if mu_(f)(x,t) > 0\mu_{f}(x, t)>0 for each t > 0t>0. The modulus of total convexity of ff at xx is defined by
{:(5)nu_(f)(x","t)=i n f_({:[y in dom(f)],[||x-y||=t]:})(f(y)-f(x)-d^(+)f(x,y-x)):}\begin{equation*}
\nu_{f}(x, t)=\inf _{\substack{y \in \operatorname{dom}(f) \\\|x-y\|=t}}\left(f(y)-f(x)-d^{+} f(x, y-x)\right) \tag{5}
\end{equation*}
where d^(+)f(x,h)d^{+} f(x, h) denotes the directional derivative (it exists for ff convex).
One has
One has
{:(6)nu_(f) >= mu_(f)",":}\begin{equation*}
\nu_{f} \geq \mu_{f}, \tag{6}
\end{equation*}
but a reversed inequality holds only when ff is Fréchet differentiable, namely in this case there exists a positive constant alpha\alpha such that
{:(7) bar(mu)_(f)(x","t) >= alphanu_(f)(x,(t)/(2)):}\begin{equation*}
\bar{\mu}_{f}(x, t) \geq \alpha \nu_{f}\left(x, \frac{t}{2}\right) \tag{7}
\end{equation*}
We are interested now in the case f(x)=varphi_(r)(x)=||x||^(r)f(x)=\varphi_{r}(x)=\|x\|^{r} for r > 1r>1.
Lemma 3 If EE is a uniformly convex space for which the norm is smooth, then for each R > 0R>0 there exists a positive constant KK such that
Lemma 3 If EE is a uniformly convex space for which the norm is smooth, then for each R > 0R>0 there exists a positive constant KK such that
mu_(varphi_(r))(z,t) >= Kt^(r)\mu_{\varphi_{r}}(z, t) \geq K t^{r}
for each z in E,||z|| <= Rz \in E,\|z\| \leq R.
Proof. Denote by delta_(E)\delta_{E} the modulus of uniform convexity of the space. Using theorem 1 in [1], there exists a positive constant K_(1) > 0K_{1}>0 such that
Proof. Denote by delta_(E)\delta_{E} the modulus of uniform convexity of the space. Using theorem 1 in [1], there exists a positive constant K_(1) > 0K_{1}>0 such that
{:(8)nu_(varphi_(r))(z","t) >= rK_(1)t^(r)int_(0)^(1)tau^(r-1)delta_(E)((tau t)/(2(||z||+tau t)))d tau:}\begin{equation*}
\nu_{\varphi_{r}}(z, t) \geq r K_{1} t^{r} \int_{0}^{1} \tau^{r-1} \delta_{E}\left(\frac{\tau t}{2(\|z\|+\tau t)}\right) d \tau \tag{8}
\end{equation*}
Using the well known fact that t|->delta_(E)(t)//tt \mapsto \delta_{E}(t) / t is increasing, one obtains
delta_(E)((tau t)/(2(||z||+tau t))) >= delta_(E)((tau t)/(2(R+tau t))) > 0\delta_{E}\left(\frac{\tau t}{2(\|z\|+\tau t)}\right) \geq \delta_{E}\left(\frac{\tau t}{2(R+\tau t)}\right)>0
so the integral in (8) is >= K_(2)\geq K_{2}. It follows that nu_(varphi_(r))(z,t) >= t^(r)rK_(1)K_(2)\nu_{\varphi_{r}}(z, t) \geq t^{r} r K_{1} K_{2}. Because varphi_(r)\varphi_{r} is Fréchet differentiable, one can use (7) and get
bar(mu)_(varphi_(r))(z,t) >= alpha((t)/(2))^(r)rK_(1)K_(2).\bar{\mu}_{\varphi_{r}}(z, t) \geq \alpha\left(\frac{t}{2}\right)^{r} r K_{1} K_{2} .
Using (4) we have
mu_(varphi_(r))(z,t) >= 2alpha((t)/(2))^(r)rK_(1)K_(2)=Kt^(r)\mu_{\varphi_{r}}(z, t) \geq 2 \alpha\left(\frac{t}{2}\right)^{r} r K_{1} K_{2}=K t^{r}
Theorem 4 Let YY be an open set, X sub Y sub EX \subset Y \subset E and f:Y rarrRf: Y \rightarrow \mathbb{R} a function with continuous second order Fréchet derivative, such that ||f^('')(y)|| <= M\left\|f^{\prime \prime}(y)\right\| \leq M for all y in Yy \in Y. Then there exists a constant KK depending only on XX such that
{:(9)|[x,a,y;f,varphi_(r)]| <= KM:}\begin{equation*}
\left|\left[x, a, y ; f, \varphi_{r}\right]\right| \leq K M \tag{9}
\end{equation*}
for all x,y in X,x!=yx, y \in X, x \neq y and a in(0,1)a \in(0,1).
Proof. As we have mentioned above,
|(x,a,y;f)| <= (M)/(2)a(1-a)||x-y||^(2).|(x, a, y ; f)| \leq \frac{M}{2} a(1-a)\|x-y\|^{2} .
Using lemma 3 with RR such that X sube B(0,R)X \subseteq B(0, R), one has
{:[(x,a,y;varphi_(r)) >= a(1-a)mu_(varphi_(r))(x","||x-y||) >= ],[a(1-a)K_(1)||x-y||^(r)","]:}\begin{aligned}
& \left(x, a, y ; \varphi_{r}\right) \geq a(1-a) \mu_{\varphi_{r}}(x,\|x-y\|) \geq \\
& a(1-a) K_{1}\|x-y\|^{r},
\end{aligned}
and then
|[x,a,y;f,varphi_(r)]| <= (M)/(2K_(1))||x-y||^(2-r) <= KM\left|\left[x, a, y ; f, \varphi_{r}\right]\right| \leq \frac{M}{2 K_{1}}\|x-y\|^{2-r} \leq K M
In the special case when EE is a Hilbert space we have
(x,a,y;varphi_(2))=a(1-a)||x-y||^(2)\left(x, a, y ; \varphi_{2}\right)=a(1-a)\|x-y\|^{2}
and the following result obtained in [3] holds.
Corollary 5 Let EE be a Hilbert space and r=2r=2. In the conditions of theorem 4, the inequality (9) is satisfied with K=1//2K=1 / 2.
Corollary 5 Let EE be a Hilbert space and r=2r=2. In the conditions of theorem 4, the inequality (9) is satisfied with K=1//2K=1 / 2.
References
[1] D. Butnariu, A. N. Iusem, E. Resmerita, Total convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Analysis 7 (2000), 319-334
[2] D. Butnariu, A. N. Iusem, C. Zălinescu, On uniform convexity, total convexity and convergence of the proximinal point and outer Bregman projection algorithms in Banach spaces, J. Convex Analysis 10 (2003), 35-61
[3] M. Ivan, I. Raşa, The rest in some approximation formulae, Séminaire de la Théorie de la Meilleure Approximation, Convexité et Optimization, Cluj-Napoca, Srima 2002, 87-92
[4] E. Popoviciu, Teoreme de medie din analiza matematică şi legătura lor cu teoria interpolării. Editura Dacia, Cluj, 1972
[5] I. Raşa, Sur les fonctionelles de la forme simple au sens de T. Popoviciu, Anal. Numér. Théor. Approx. 9 (1980), 261-268
[6] I. Raşa, Funcţionale de formă simplă în sensul lui Tiberiu Popoviciu (PhD Thesis), Cluj-Napoca, 1982
[7] C. Zălinescu, Convex Anaysis in General Vector Spaces, World Scientific, 2002
[2] D. Butnariu, A. N. Iusem, C. Zălinescu, On uniform convexity, total convexity and convergence of the proximinal point and outer Bregman projection algorithms in Banach spaces, J. Convex Analysis 10 (2003), 35-61
[3] M. Ivan, I. Raşa, The rest in some approximation formulae, Séminaire de la Théorie de la Meilleure Approximation, Convexité et Optimization, Cluj-Napoca, Srima 2002, 87-92
[4] E. Popoviciu, Teoreme de medie din analiza matematică şi legătura lor cu teoria interpolării. Editura Dacia, Cluj, 1972
[5] I. Raşa, Sur les fonctionelles de la forme simple au sens de T. Popoviciu, Anal. Numér. Théor. Approx. 9 (1980), 261-268
[6] I. Raşa, Funcţionale de formă simplă în sensul lui Tiberiu Popoviciu (PhD Thesis), Cluj-Napoca, 1982
[7] C. Zălinescu, Convex Anaysis in General Vector Spaces, World Scientific, 2002
Mira-Cristiana ANISIU
T. Popoviciu Institute of Numerical Analysis
P. O. Box 68
T. Popoviciu Institute of Numerical Analysis
P. O. Box 68
400110 Cluj-Napoca, Romania
(E-mail: mira@math.ubbcluj.ro)
Valeriu ANISIU
Faculty of Mathematics and Computer Science
1, Kogălniceanu st
400084 Cluj-Napoca, Romania
(E-mail: anisiu@math.ubbcluj.ro)
(E-mail: mira@math.ubbcluj.ro)
Valeriu ANISIU
Faculty of Mathematics and Computer Science
1, Kogălniceanu st
400084 Cluj-Napoca, Romania
(E-mail: anisiu@math.ubbcluj.ro)