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Rev. Anal. Num´er. Th´eor. Approx., vol. 33 (2004) no. 2, pp. 203–208 ictp.acad.ro/jnaat

CHARACTERIZATION OF ε-NEAREST POINTS IN SPACES WITH ASYMMETRIC SEMINORM

COSTIC ˘A MUST ˘AT¸ A

Dedicated to professor Elena Popoviciu on the occasion of her 80th anniversary.

Abstract. In this note we are concerned with the characterization of the ele- ments of ε-best approximation (ε-nearest points) in a subspaceY of space X with asymmetric seminorm. For this we use functionals in the asymmetric dual Xb defined and studied in some recent papers [1], [3], [5].

MSC 2000. 41A65.

Keywords. Asymmetric seminormed spaces,ε-nearest points, characterization.

1. INTRODUCTION

LetX be a real linear space. A functional p:X→[0,∞) with the proper- ties:

(1) p(x)≥0,for allxX,

(2) p(tx) =tp(x),for allxX and t≥0, (3) p(x+y)p(x) +p(y),for all x, yX,

is called asymmetric seminorm onX,and the pair (X, p) is called a space with asymmetric seminorm.

The functionalp:X →[0,∞),defined byp(x) =p(−x), xX is another asymmetric seminorm onX, called the conjugate of p.

The functionalps:X→[0,∞), defined by

ps(x) = max{p(x), p(−x)}, xX,

is a seminorm on X. If ps satisfies the axioms of a norm, then p is called an asymmetric norm on X. It follows thatp satisfies the properties (1), (2), (3), and

(4) p(x) = 0 and p(−x) = 0 imply x= 0.

The asymmetric seminormponX generates a topologyτp onX, having as a basis of neighborhoods of a point xX the open p-balls

Bp0 (x, r) =x0X:p x0x< r , r >0.

This work has been supported by the Romanian Academy under Grant GAR 13/2004.

“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: [email protected].

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The family of closed p-balls

Bp(x, r) =x0X:p x0xr , r >0,

generates the same topology. This topology τp could not be Hausdorff (see [5]), and could not be linear (the multiplication by scalars is not continuous in general, see [1]).

Let R be the set of real numbers and u : R→[0,∞), u(a) = max{a,0}, a∈R.Then the function u is an asymmetric seminorm on Rand, for a∈R, the intervals (−∞, a+ε), ε > 0, form a basis of neighborhoods of a∈ R in the topology τu. The conjugate asymmetric seminorm of u is u :R→[0,∞), u(a) =u(−a), a∈R,and us(a) = max{u(a), u(−a)}=|a|is a norm onR. Consequently, u is an asymmetric norm onR.

Let ϕ:X →R be a linear functional. The continuity ofϕwith respect to the topologies τp and τu is called (p, u)-continuity, and it is equivalent to the upper semicontinuity of ϕas a functional from (X, τp) to (R,|.|).

The linear functionalϕ: (X, τp)→(R, u) is (p, u)-continuous if and only if it is p-bounded, i.e. there existsL≥0 such that

ϕ(x)≤Lp(x), for allxX.

The set of all (p, u)-continuous functionals is denoted by Xpb.With respect to pointwise addition and multiplication by real scalars, the set Xpb is a cone, i.e. λ≥0 andϕ, ψXpb imply ϕ+ψXb and λϕXpb.

The functionalk.|:Xpb →[0,∞) defined by

kϕ|p = sup{ϕ(x) :xX, p(x)≤1}, ϕXpb

satisfies the properties of an asymmetric seminorm, and the pair (Xpb,k.|p) is called the asymmetric dual of the asymmetric seminormed space (X, p) (see [5]). Some properties of this dual are presented in [1], [3], [5]. If there is no danger of confusion we shall use the notation Xb and kϕ| instead of Xpb and kϕ|p, respectively.

Let (X, p) be an asymmetric seminormed space andY a subspace ofX.Let Yb be the asymmetric dual of (Y, p).

The following result is the analog of a well known extension result for linear functionals in normed spaces.

Theorem 1 (Hahn-Banach). Let (Y, p) be a subspace of asymmetric semi- normed space(X, p). Then for every ϕ0Yb there existsϕXb such that

ϕ|Y =ϕ0, kϕ|=kϕ0|.

Proof. We consider the functional q : X → [0,∞), q(x) = kϕ0| · p(x), xX. Obviously q is subadditive and positive homogeneous and for every yY we have

ϕ0(y)≤ kϕ0| ·p(y) =q(y)

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i.e. ϕ0 is majorized byq onY.

By Hahn-Banach extension theorem it results that there exists the linear functional ϕ:X →Rwith properties:

ϕ|Y =ϕ0 and

ϕ(x)≤ kϕ0| ·p(x) , for everyxX.

It follows kϕ| ≤ kϕ0|,and, because

kϕ|= sup{ϕ(x) :xX, p(x)≤1}

≥sup{ϕ(y) :yY, p(y)≤1}

= sup{ϕ0(y) :yY, p(y)≤1}

=kϕ0|

we have kϕ|=kϕ0|.

2. THEε-BEST APPROXIMATION IN(X, p)

LetY be a nonvoid subset of the asymmetric seminormed space (X, p). The problem of best approximation of the elementxX by elements inY is: find an element y0Y such that

(1) dp(x, Y) := inf{p(y−x) :yY}=p(y0x).

Letε >0.The problem ofε-best approximation ofxXby elements in Y is: find y0Y such that

(2) p(y0x)dp(x, Y) +ε.

Obviously, the problem of ε-best approximation always admits a solution, because for every number n ∈ N there exists ynY such that p(ynx)dp(x, Y) +1n,so that p(ynx)dp(x, Y) +ε, forn >h1εi+ 1.

In the following we denote by

(3) PY,ε(x) ={y∈Y :p(y−x)dp(x, Y) +ε}, xX the nonvoid set of the elements of ε-best approximation for xX inY.

The paper [3] contains characterizations, in terms of functionals inXb van- ishing onY, of the elements of best approximation ofxX by elements in a subspaceY ofX.Let us observe firstly that, one can consider also the problem of ε-best approximation by using the conjugate p ofp.In this case, for (4) dp(x, Y) = inf{p(x−y) :yY}

one looks for y0Y such that

(5) p(x−y0)≤dp(x, Y) +ε.

Let us denote by

(6) PY,ε(x) ={y∈Y :p(y−x) =p(x−y)dp(x, Y) +ε}

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the set of ε-best approximation of xX with respect to the conjugate asym- metric seminorm p.

In the following we obtain characterizations of elements of ε-best approxi- mation ofxX by elements of a subspaceY, with respect to the asymmetric seminormsp and p.

Results of this type, for elements of best approximations in a normed space X, using the elements of dual X are obtained in [17] (see also [3], [9], [11], [13], [16], [18]).

Concerning the characterizations of elements of ε-best approximation in normed space, see papers [14], [15].

Theorem2. Let(X, p) be an asymmetric seminormed space,Y a subspace of X and x0X \ Y, such that d=dp(x0, Y) > 0 and d= dp(x0, Y) >0.

Then

(a) An element y0Y is in PY,ε(x0) if and only if there exists ϕXpb with the properties:

(i) ϕ(y) = 0, for all yY, (iii) kϕ|p = 1,

(iii) ϕ(−x0)≥p(y0x0)−ε.

(b) An element y0 is in PY,ε(x0) if and only if there exists ψXpb with the properties:

(j) ψ(y) = 0,for all yY, (jj) kψ|p = 1,

(jjj) ψ(x0)≥p(x0y0)−ε.

Proof. Letx0X \Y and Z =Y +hx0i be the direct sum of Y with the space generated byx0.Consider the functionalϕ0:Z →Rdefined by

ϕ0(z) =ϕ(y+λx0) =−λ,

wherezZ,and z is uniquely represented in the form z=y+λx0. The functionalϕ0 is linear onZ.

Observe thatϕ0 |Y = 0,and for everyλ >0 we have p(y−λx0) =λp1λyx0

λd=d·ϕ0(y−λx0).

It follows that

ϕ0(y−λx0)≤ 1d·p(y−λx0), for everyλ >0.

Because the last inequality is also valid if ϕ0(y−tx0) =t≤0,it follows kϕ0|p1d, and consequently ϕ0Zpb.

Now, let (yn)n≥1 be a sequence in Y such thatp(ynx0) → d, forn → ∞, and such thatp(ynx0)>0 for every n∈N.Then

0|pϕ0p(yyn−x0

n−x0)

= p(y 1

n−x0)1d,

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and, consequently, kϕ0|p = 1d.

By Theorem 1, there existsϕ1Xb such that ϕ1 |Z =ϕ0,1|=kϕ0|p= d1.

Then, the functional ϕ = d·ϕ1 satisfies the properties: ϕXpb, ϕ | Y = d·ϕ1|Y = 0,

ϕ(−x0) =ϕ(y0) +ϕ(−x0)

=ϕ(y0x0)

p(y0x0)

p(y0x0)−ε.

Conversely, if y0Y and there exists ϕXpb with the properties (a) (i)-(iii), then for every yY we have

p(y0x0)≤ϕ(−x0) +ε

=ϕ(y−x0) +ε

≤ kϕ|p·p(y−x0) +ε

p(y−x0) +ε.

Taking the infimum with respect toyY,one finds p(y0x0)≤dp(x0, Y) +ε;

so that y0PY,ε(x0).

Similarly, defining ψ0 : Z = Y +hx0i → R by ψ(z) = ψ0(y+λx0) = λ, yY andλ∈R,and proceeding in the same way, one obtains the claim

(b) of the theorem.

Theorem 2 has the following consequence:

Corollary 3. In the hypothesis of Theorem 2 we have:

(a0) MPY,ε(x0), if and only if there existsϕXb verifying (a) (i)-(ii) and the condition:

ϕ(−x0)≥p(u−x0)−ε, for all uM;

(b0) MPY,ε(x0) if and only if there exists ψXb with properties (b) (j)-(jj), and verifying the condition:

ψ(x0)≥p(x0u)ε, for all uM.

REFERENCES

[1] Borodin, P. A.,The Banach-Mazur theorem for spaces with an asymmetric norm and its applications in convex analysis, Mat. Zametki,69, no. 3, pp. 193–217, 2001.

[2] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J.

Approx. Theory,94, no. 1, pp. 54–72, 1998.

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[3] Cobzas, S. and Mustata, C.,Extension of bounded linear functionals and best ap- proximation in spaces with asymmetric norm, Rev. Anal. Num´er. Th´eor. Approx.,33, no. 1, pp. 39–50, 2004.

[4] Cobzas, S.,Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math.,27, pp. 1–22, 2004.

[5] Garcia-Raffi, L. M., Romaguera S.andanchez-P´erez, E. A.,The dual space of an asymmetric normed linear space, Quaest. Math.,26, no. 1, pp. 83–96, 2003.

[6] Garcia-Raffi, L.M., Romaguera S. and anchez-P´erez, E. A., On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29, no. 3, pp. 717–728, 2003 (electronic).

[7] Krein, M. G.and Nudel’man, A. A.,The Markov Moment Problem and Extremum Problems, Nauka, Moscow 1973 (in Russian). English translation: American Mathe- matical Society, Providence, R.I., 1997.

[8] Li, ChongandNi, Renxing,Derivatives of generalized distance functions and existence of generalized nearest points, J. Approx. Theory,115, no. 1, pp. 44–55, 2002.

[9] Mabizela, S.,Characterization of best approximation in metric linear spaces, Scientiae Mathematicae Japonica,57, 2, pp. 233–240, 2003.

[10] Must˘at¸a, C., On the best approximation in metric spaces, Mathematica – Revue d’Analyse Num´erique et de Th´eorie de l’Approximation, L’Analyse Num´erique et la Th´eorie de l’Approximation,4, pp. 45–50, 1975.

[11] Must˘at¸a, C.,On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Num´er. Th´eor. Approx.,32, no. 2, pp. 187–192, 2003.

[12] Ni, Renxing,Existence of generalized nearest points,Taiwanese J. Math.,7, no. 1, pp.

115–128, 2003.

[13] Pantelidis, G.,Approximations theorie f¨ur metrich linear R¨aume, Math. Ann.,184, pp. 30–48, 1969.

[14] Rezapour, Sh.,ε-pseudo Chebyshev and ε-quasi Chebyshev subspaces of Banach spaces, Technical Report, Azarbaidjan University of Tarbiot Moallem, 2003.

[15] Rezapour, Sh.,ε-weakly Chebyshev subspaces of Banach spaces, Analysis in Theory and Applications, 19, no. 2, pp. 130–135, 2003.

[16] Schnatz, K., Nonlinear duality and best approximation in metric linear spaces, J.

Approx. Theory,49, no. 3, pp. 201–21, 1987.

[17] Singer, I., Best Approximation in Normed Linear spaces by Elements of Linear subspaces, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New-York-Berlin, 1970.

[18] Singer, I.,Caracterisations des ´el´ements de la meilleure approximation dans un espace de Banach quelconque, Acta Sci. Math.,17, pp. 181–189, 1956.

Received by the editors: June 11, 2003.

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