An estimation of a generalized divided difference in uniformly convex spaces

Mira Anisiu
(original), paper

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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https://ictp.acad.ro/anisiu/papers/2004-Anisiu-Anisiu-An%20estimation.pdf

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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

2004-Anisiu-Anisiu-An estimation

An estimation of a generalized divided difference in uniformly convex spaces

MIRA-CRISTIANA ANISIU VALERIU ANISIU(CLUJ-NAPOCA) (CLUJ-NAPOCA)

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 E E EEE denote a locally convex Hausdorff real space and X X XXX a compact convex metrizable subset of E E EEE; for f ∈ C ( X ) f ∈ C ( X ) f in C(X)f \in C(X)f∈C(X), x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X and a ∈ [ 0 , 1 ] a ∈ [ 0 , 1 ] a in[0,1]a \in[0,1]a∈[0,1], we denote
(1) ( x , a , y ; f ) = ( 1 − a ) f ( x ) + a f ( y ) − f ( ( 1 − a ) x + a y ) (1) ( x , a , y ; f ) = ( 1 − a ) f ( x ) + a f ( y ) − f ( ( 1 − a ) x + a y ) {:(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*}(1)(x,a,y;f)=(1−a)f(x)+af(y)−f((1−a)x+ay)
We remark that, since X X XXX is a metrizable space, there exist strictly convex functions in C ( X ) C ( X ) C(X)C(X)C(X); we denote by φ φ varphi\varphiφ such a function.
Theorem 1 Let L : C ( X ) → R L : C ( X ) → R L:C(X)rarrRL: C(X) \rightarrow \mathbb{R}L:C(X)→R be a linear functional such that L ( g ) > 0 L ( g ) > 0 L(g) > 0L(g)>0L(g)>0 for each strictly convex function g ∈ C ( X ) g ∈ C ( X ) g in C(X)g \in C(X)g∈C(X). Then for every f ∈ C ( X ) f ∈ C ( X ) f in C(X)f \in C(X)f∈C(X) there exists x , y ∈ X , x ≠ y x , y ∈ X , x ≠ y x,y in X,x!=yx, y \in X, x \neq yx,y∈X,x≠y and a ∈ ( 0 , 1 ) a ∈ ( 0 , 1 ) a in(0,1)a \in(0,1)a∈(0,1) such that
L ( f ) = L ( φ ) ( x , a , y ; f ) ( x , a , y ; φ ) L ( f ) = L ( φ ) ( x , a , y ; f ) ( x , a , y ; φ ) 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)}L(f)=L(φ)(x,a,y;f)(x,a,y;φ)
We consider now C ( X ) C ( X ) C(X)C(X)C(X) endowed with the uniform norm.
Theorem 2 Let L : C ( X ) → R L : C ( X ) → R L:C(X)rarrRL: C(X) \rightarrow \mathbb{R}L:C(X)→R be a continuous and linear functional such that L ( g ) ≥ 0 L ( g ) ≥ 0 L(g) >= 0L(g) \geq 0L(g)≥0 for each convex function g ∈ C ( X ) g ∈ C ( X ) g in C(X)g \in C(X)g∈C(X). Then for every f ∈ C ( X ) f ∈ C ( X ) f in C(X)f \in C(X)f∈C(X) there exists x , y ∈ X , x ≠ y x , y ∈ X , x ≠ y x,y in X,x!=yx, y \in X, x \neq yx,y∈X,x≠y and a ∈ ( 0 , 1 ) a ∈ ( 0 , 1 ) a in(0,1)a \in(0,1)a∈(0,1) such that
L ( f ) = L ( φ ) ( x , a , y ; f ) ( x , a , y ; φ ) L ( f ) = L ( φ ) ( x , a , y ; f ) ( x , a , y ; φ ) 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)}L(f)=L(φ)(x,a,y;f)(x,a,y;φ)
For the special case E = R , X = [ 0 , 1 ] E = R , X = [ 0 , 1 ] E=R,X=[0,1]E=\mathbb{R}, X=[0,1]E=R,X=[0,1] and φ ( t ) = t 2 , t ∈ [ 0 , 1 ] φ ( t ) = t 2 , t ∈ [ 0 , 1 ] varphi(t)=t^(2),t in[0,1]\varphi(t)=t^{2}, t \in[0,1]φ(t)=t2,t∈[0,1], Ivan and Raşa [3] showed that
( x , a , y ; φ ) = ( 1 − a ) x 2 + a y 2 − ( ( 1 − a ) x + a y ) 2 = a ( 1 − a ) ( x − y ) 2 , ( x , a , y ; φ ) = ( 1 − a ) x 2 + a y 2 − ( ( 1 − a ) x + a y ) 2 = a ( 1 − a ) ( x − y ) 2 , (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},(x,a,y;φ)=(1−a)x2+ay2−((1−a)x+ay)2=a(1−a)(x−y)2,
for all x , y , a ∈ [ 0 , 1 ] x , y , a ∈ [ 0 , 1 ] x,y,a in[0,1]x, y, a \in[0,1]x,y,a∈[0,1]. In this case it follows that
( x , a , y ; f ) ( x , a , y ; φ ) = [ x , ( 1 − a ) x + a y , y ] ( x , a , y ; f ) ( x , a , y ; φ ) = [ x , ( 1 − a ) x + a y , y ] ((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](x,a,y;f)(x,a,y;φ)=[x,(1−a)x+ay,y]
where the last expression is the classical divided difference of the real function f f fff on the knots x , ( 1 − a ) x + a y x , ( 1 − a ) x + a y x,(1-a)x+ayx,(1-a) x+a yx,(1−a)x+ay and y y yyy. In the general case,
(2) [ x , a , y ; f , φ ] := ( x , a , y ; f ) ( x , a , y ; φ ) (2) [ x , a , y ; f , φ ] := ( x , a , y ; f ) ( x , a , y ; φ ) {:(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*}(2)[x,a,y;f,φ]:=(x,a,y;f)(x,a,y;φ)
with ( x , a , y ; f x , a , y ; f x,a,y;fx, 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 , ‖ ⋅ ‖ ) (E,||*||)(E,\|\cdot\|)(E,‖⋅‖) be a real smooth uniformly convex space and X X XXX a compact subset of E E EEE. Consider the (strictly convex) function φ r ∈ C ( X ) φ r ∈ C ( X ) varphi_(r)in C(X)\varphi_{r} \in C(X)φr∈C(X) given by
φ r ( x ) = ‖ x ‖ r , x ∈ X , φ r ( x ) = ‖ x ‖ r , x ∈ X , varphi_(r)(x)=||x||^(r),x in X,\varphi_{r}(x)=\|x\|^{r}, x \in X,φr(x)=‖x‖r,x∈X,
where 1 < r ≤ 2 1 < r ≤ 2 1 < r <= 21<r \leq 21<r≤2.
We need upper and lower bounds for the expression ( x , a , y ; f x , a , y ; f x,a,y;fx, a, y ; fx,a,y;f ). An upper bound for | ( x , a , y ; f ) | | ( x , a , y ; f ) | |(x,a,y;f)||(x, a, y ; f)||(x,a,y;f)| was found in [3], for f f fff twice Fréchet differentiable on an open set Y Y YYY and ‖ f ′ ′ ( y ) ‖ ≤ M f ′ ′ ( y ) ≤ M ||f^('')(y)|| <= M\left\|f^{\prime \prime}(y)\right\| \leq M‖f′′(y)‖≤M for each y ∈ Y y ∈ Y y in Yy \in Yy∈Y, namely
(3) | ( x , a , y ; f ) | ≤ M 2 a ( 1 − a ) ‖ x − y ‖ 2 . (3) | ( x , a , y ; f ) | ≤ M 2 a ( 1 − a ) ‖ x − y ‖ 2 . {:(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*}(3)|(x,a,y;f)|≤M2a(1−a)‖x−y‖2.
It was proved for Hilbert case, but it can be shown that (3) holds in our setting too.
If f f fff is a convex function, ( x , a , y ; f x , a , y ; f x,a,y;fx, a, y ; fx,a,y;f ) is ≥ 0 ≥ 0 >= 0\geq 0≥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 x x xxx (named gage of uniform convexity in [7]) is
μ f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t λ ∈ ( 0 , 1 ) ( x , λ , y ; f ) λ ( 1 − λ ) , t ≥ 0 μ f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t λ ∈ ( 0 , 1 )   ( x , λ , y ; f ) λ ( 1 − λ ) , t ≥ 0 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 0μf(x,t)=infy∈dom(f)‖x−y‖=tλ∈(0,1)(x,λ,y;f)λ(1−λ),t≥0
A related function is
μ ¯ f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t ( f ( x ) + f ( y ) − 2 f ( x + y 2 ) ) . μ ¯ f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t   f ( x ) + f ( y ) − 2 f x + y 2 . 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) .μ¯f(x,t)=infy∈dom(f)‖x−y‖=t(f(x)+f(y)−2f(x+y2)).
One has [2]:
(4) 1 2 μ ≤ μ ¯ ≤ μ . (4) 1 2 μ ≤ μ ¯ ≤ μ . {:(4)(1)/(2)mu <= bar(mu) <= mu.:}\begin{equation*} \frac{1}{2} \mu \leq \bar{\mu} \leq \mu . \tag{4} \end{equation*}(4)12μ≤μ¯≤μ.
The function f f fff is said to be uniformly convex at x x xxx if μ f ( x , t ) > 0 μ f ( x , t ) > 0 mu_(f)(x,t) > 0\mu_{f}(x, t)>0μf(x,t)>0 for each t > 0 t > 0 t > 0t>0t>0. The modulus of total convexity of f f fff at x x xxx is defined by
(5) ν f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t ( f ( y ) − f ( x ) − d + f ( x , y − x ) ) (5) ν f ( x , t ) = inf y ∈ dom ( f ) ‖ x − y ‖ = t   f ( y ) − f ( x ) − d + f ( x , y − x ) {:(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*}(5)νf(x,t)=infy∈dom(f)‖x−y‖=t(f(y)−f(x)−d+f(x,y−x))
where d + f ( x , h ) d + f ( x , h ) d^(+)f(x,h)d^{+} f(x, h)d+f(x,h) denotes the directional derivative (it exists for f f fff convex).
One has
(6) ν f ≥ μ f , (6) ν f ≥ μ f , {:(6)nu_(f) >= mu_(f)",":}\begin{equation*} \nu_{f} \geq \mu_{f}, \tag{6} \end{equation*}(6)νf≥μf,
but a reversed inequality holds only when f f fff is Fréchet differentiable, namely in this case there exists a positive constant α α alpha\alphaα such that
(7) μ ¯ f ( x , t ) ≥ α ν f ( x , t 2 ) (7) μ ¯ f ( x , t ) ≥ α ν f x , t 2 {:(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*}(7)μ¯f(x,t)≥ανf(x,t2)
We are interested now in the case f ( x ) = φ r ( x ) = ‖ x ‖ r f ( x ) = φ r ( x ) = ‖ x ‖ r f(x)=varphi_(r)(x)=||x||^(r)f(x)=\varphi_{r}(x)=\|x\|^{r}f(x)=φr(x)=‖x‖r for r > 1 r > 1 r > 1r>1r>1.
Lemma 3 If E E EEE is a uniformly convex space for which the norm is smooth, then for each R > 0 R > 0 R > 0R>0R>0 there exists a positive constant K K KKK such that
μ φ r ( z , t ) ≥ K t r μ φ r ( z , t ) ≥ K t r mu_(varphi_(r))(z,t) >= Kt^(r)\mu_{\varphi_{r}}(z, t) \geq K t^{r}μφr(z,t)≥Ktr
for each z ∈ E , ‖ z ‖ ≤ R z ∈ E , ‖ z ‖ ≤ R z in E,||z|| <= Rz \in E,\|z\| \leq Rz∈E,‖z‖≤R.
Proof. Denote by δ E δ E delta_(E)\delta_{E}δE the modulus of uniform convexity of the space. Using theorem 1 in [1], there exists a positive constant K 1 > 0 K 1 > 0 K_(1) > 0K_{1}>0K1>0 such that
(8) ν φ r ( z , t ) ≥ r K 1 t r ∫ 0 1 τ r − 1 δ E ( τ t 2 ( ‖ z ‖ + τ t ) ) d τ (8) ν φ r ( z , t ) ≥ r K 1 t r ∫ 0 1   τ r − 1 δ E τ t 2 ( ‖ z ‖ + τ t ) d τ {:(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*}(8)νφr(z,t)≥rK1tr∫01τr−1δE(τt2(‖z‖+τt))dτ
Using the well known fact that t ↦ δ E ( t ) / t t ↦ δ E ( t ) / t t|->delta_(E)(t)//tt \mapsto \delta_{E}(t) / tt↦δE(t)/t is increasing, one obtains
δ E ( τ t 2 ( ‖ z ‖ + τ t ) ) ≥ δ E ( τ t 2 ( R + τ t ) ) > 0 δ E τ t 2 ( ‖ z ‖ + τ t ) ≥ δ E τ t 2 ( R + τ t ) > 0 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)>0δE(τt2(‖z‖+τt))≥δE(τt2(R+τt))>0
so the integral in (8) is ≥ K 2 ≥ K 2 >= K_(2)\geq K_{2}≥K2. It follows that ν φ r ( z , t ) ≥ t r r K 1 K 2 ν φ r ( z , t ) ≥ t r r K 1 K 2 nu_(varphi_(r))(z,t) >= t^(r)rK_(1)K_(2)\nu_{\varphi_{r}}(z, t) \geq t^{r} r K_{1} K_{2}νφr(z,t)≥trrK1K2. Because φ r φ r varphi_(r)\varphi_{r}φr is Fréchet differentiable, one can use (7) and get
μ ¯ φ r ( z , t ) ≥ α ( t 2 ) r r K 1 K 2 . μ ¯ φ r ( z , t ) ≥ α t 2 r r K 1 K 2 . 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} .μ¯φr(z,t)≥α(t2)rrK1K2.
Using (4) we have
μ φ r ( z , t ) ≥ 2 α ( t 2 ) r r K 1 K 2 = K t r μ φ r ( z , t ) ≥ 2 α t 2 r r K 1 K 2 = K t r 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}μφr(z,t)≥2α(t2)rrK1K2=Ktr
Theorem 4 Let Y Y YYY be an open set, X ⊂ Y ⊂ E X ⊂ Y ⊂ E X sub Y sub EX \subset Y \subset EX⊂Y⊂E and f : Y → R f : Y → R f:Y rarrRf: Y \rightarrow \mathbb{R}f:Y→R a function with continuous second order Fréchet derivative, such that ‖ f ′ ′ ( y ) ‖ ≤ M f ′ ′ ( y ) ≤ M ||f^('')(y)|| <= M\left\|f^{\prime \prime}(y)\right\| \leq M‖f′′(y)‖≤M for all y ∈ Y y ∈ Y y in Yy \in Yy∈Y. Then there exists a constant K K KKK depending only on X X XXX such that
(9) | [ x , a , y ; f , φ r ] | ≤ K M (9) x , a , y ; f , φ r ≤ K M {:(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*}(9)|[x,a,y;f,φr]|≤KM
for all x , y ∈ X , x ≠ y x , y ∈ X , x ≠ y x,y in X,x!=yx, y \in X, x \neq yx,y∈X,x≠y and a ∈ ( 0 , 1 ) a ∈ ( 0 , 1 ) a in(0,1)a \in(0,1)a∈(0,1).
Proof. As we have mentioned above,
| ( x , a , y ; f ) | ≤ M 2 a ( 1 − a ) ‖ x − y ‖ 2 . | ( x , a , y ; f ) | ≤ M 2 a ( 1 − a ) ‖ x − y ‖ 2 . |(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} .|(x,a,y;f)|≤M2a(1−a)‖x−y‖2.
Using lemma 3 with R R RRR such that X ⊆ B ( 0 , R ) X ⊆ B ( 0 , R ) X sube B(0,R)X \subseteq B(0, R)X⊆B(0,R), one has
( x , a , y ; φ r ) ≥ a ( 1 − a ) μ φ r ( x , ‖ x − y ‖ ) ≥ a ( 1 − a ) K 1 ‖ x − y ‖ r , x , a , y ; φ r ≥ a ( 1 − a ) μ φ r ( x , ‖ x − y ‖ ) ≥ a ( 1 − a ) K 1 ‖ x − y ‖ r , {:[(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}(x,a,y;φr)≥a(1−a)μφr(x,‖x−y‖)≥a(1−a)K1‖x−y‖r,
and then
| [ x , a , y ; f , φ r ] | ≤ M 2 K 1 ‖ x − y ‖ 2 − r ≤ K M x , a , y ; f , φ r ≤ M 2 K 1 ‖ x − y ‖ 2 − r ≤ K M |[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|[x,a,y;f,φr]|≤M2K1‖x−y‖2−r≤KM
In the special case when E E EEE is a Hilbert space we have
( x , a , y ; φ 2 ) = a ( 1 − a ) ‖ x − y ‖ 2 x , a , y ; φ 2 = a ( 1 − a ) ‖ x − y ‖ 2 (x,a,y;varphi_(2))=a(1-a)||x-y||^(2)\left(x, a, y ; \varphi_{2}\right)=a(1-a)\|x-y\|^{2}(x,a,y;φ2)=a(1−a)‖x−y‖2
and the following result obtained in [3] holds.
Corollary 5 Let E E EEE be a Hilbert space and r = 2 r = 2 r=2r=2r=2. In the conditions of theorem 4, the inequality (9) is satisfied with K = 1 / 2 K = 1 / 2 K=1//2K=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
Mira-Cristiana ANISIU
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)
2004

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