Extensions of semi-Hölder real valued functions on a quasi-metric space

2 years ago

(original), Approximation Theory, asymmetric norm, paper

Abstract

In this note, the semi-Holder real valued functions on a quasi-metric (asymmetric metric) space are defined. An extension theorem for such functions and some consequences are presented.

Authors

Costica Mustata
Tiberiu Popoviciu Institute of Numerical analysis, Romania

Keywords

Semi-Holder functions; extensions.

Paper coordinates

C. Mustăţa, Extensions of semi-Hölder real valued functions on a quasi-metric space, Rev. Anal. Numer. Theor. Approx., 38 (2009), no. 2, pp. 164-169.

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https://ictp.acad.ro/jnaat/journal/article/view/2009-vol38-no2-art6/Mustata-pdf

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Journal

Revue d’Analyse Numerique et de theorie de l’Approximation

Publisher Name

Publisher House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2009-vol38-no2-art6

Print ISSN

2501-059X

Online ISSN

2457-6794

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[1] Collins, J. and Zimmer, J., An asymmetric Arzela-Ascoli theorem, Topology and its Applications, 154, pp. 2312–2322, 2007.

[2] Matouskova, E., Extensions of continuous and Lipschitz functions, Canad. Math. Bull,43, no. 2, pp. 208–217, 2000.

[3] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837–842, 1934.

[4] Menucci, A., On asymmetric distances. Technical Report, Scuola Normale Superiore, Pisa, 2007, http://cvgmt.sns.it/people/menucci.

[5] Miculescu, R., Some observations on generalized Lipschitz functions, Rocky Mountain J. Math., 37, no.3, pp. 893–903, 2007.

[6] Mustata, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001.

[7] Mustata, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, no. 3, pp. 222–230, 1977.

[8] Mustata, C., A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ. Baia Mare, Ser. B. Matematica-Informatica 18, pp. 275–280, 2002.

[9] Romaguera, S.andSanchis, M., Semi-Lipschitz functions and best approximation inquasi–metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.

[10] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitzfunctions, Acta Math Hungar,108no. 1–2, pp. 55–70, 2005.

[11] Sanchez-Alvarez, J.M., On semi-Lipschitz functions with values in a quasi-normedlinear space, Applied General Topology,6, no. 2, pp. 216–228, 2005.

[12] Weaver, N., Lattices of Lipschitz functions, Pacific Journal of Math., 164, pp. 179–193, 1994.

[13]Wells, J.H. and Williams, L.R., Embeddings and Extensions in Analysis, Springer-Verlag, Berlin, 1975.

[14]* * *, The Otto Dunkel Memorial Problem Book,New York, 1957.

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EXTENSIONS OF SEMI-H ¨ OLDER REAL VALUED FUNCTIONS ON A QUASI–METRIC SPACE COSTIC ˘ A MUST ˘ AT¸A * Abstract. In this note the semi-H¨ older real valued functions on a quasi–metric (asymmetric metric) space are deﬁned. An extension theorem for such functions and some consequences are presented. MSC 2000. 26A16, 46A22. Keywords. Semi-H¨ older functions, extensions. 1. PRELIMINARIES Let X be a non-empty set. A function d : X × X → [0, ∞) is called a quasi–metric on X [9] (see also [1]) if the following conditions hold: AM1) d(x, y)= d(y,x)=0 ⇔ x = y AM2) d(x, z ) ≤ d(x, y)+ d(y,z ), for all x, y, z ∈ X. When X is non-empty set and d a quasi–metric on X, the pair (X, d) is called a quasi–metric space. The function d : X ×X → [0, ∞) deﬁned by d(x, y)= d(y,x) for all x, y ∈ X is also a quasi–metric on X, called the conjugate quasi–metric of d. Obviously, the function d s (x, y) = max{d(x, y), d(x, y)} is a metric on X. Each quasi–metric d on X induces a topology τ (d), which has as a base the family of balls (forward open balls [4]). (1.1) B + (x, ε): {y ∈ X : d(x, y) <ε},x ∈ X, ε > 0. This topology is called the forward topology of X ([4]), [1]) and is denoted also by τ + . The topology induced by the quasi–metric d is called the backward topology and is denoted by τ - . The topology τ + is a T 0 topology. If the condition AM1) is replaced by the condition AM0) d(x, y) ≥ 0 and d(x, y)=0, for all x, y ∈ X then the topology τ + is a T 1 topology. * Institutul de Calcul “T. Popoviciu”, Cluj-Napoca, Romania, e-mail: cmustata2001@ yahoo.com, cmustata@ictp.acad.ro.

Let (X, d) be quasi–metric space. A sequence (x k ) k≥1 d -converge to x 0 ∈ X, respectively d -converge to x 0 ∈ X, iﬀ lim k→∞ d(x 0 ,x k )=0, respectively (1.2) lim k→∞ d(x 0 ,x k ) = lim k→∞ d(x k ,x 0 )=0. A set K ⊂ X is called d- compact if every open cover of K with respect to the topology τ + has a ﬁnite subcover. We say that K is d-sequentially compact if every sequence in K has a d-converget subsequence with limit in K (Deﬁnition 4.1 in [1]). Finally, the set Y in (X, d) is called (d, d)-sequentially compact if every sequence (y n ) n≥1 in Y has a subsequence (y n k ) k≥1 d-convergent to u ∈ Y and d-convergent to v ∈ Y. By Lemma 3.1 in [1] if the topology of X is T 1 , i.e. d veriﬁes the axioms AM0) and AM2), it follows that u = v. The following deﬁnition of a d-semi-H¨older function (of exponent α ∈ (0, 1)) is inspired by the deﬁnition of semi-Lipschitz function in [9]. Definition 1. Let Y be a non-empty subset of a quasi–metric space (X, d), and α ∈ (0, 1) arbitrarily chosen, but ﬁxed. A function f : Y → R is called d-semi-H¨older (of exponent α) if there exists L = L(f,Y ) ≥ 0 (named a d- semi-H¨older constant for f ) such that (1.3) f (x) - f (y) ≤ Ld α (x, y), for all x, y ∈ Y. The smallest d-semi-H¨older constant for f, verifying (1.3), is denoted by kf | α,Y and (1.4) kf | α,Y = sup n (f (x)-f (y))∨0 d α (x,y) : d(x, y) > 0, x, y ∈ Y o . This means that kf | α,Y = inf {L ≥ 0: L verifying (1.3)}. The set of all d-semi-H¨older function f : Y → R is denoted by Λ α (Y ), i.e. (1.5) Λ α (Y ) := {f : Y → R,f is d-semi-H¨older of exponent α}. This set is a cone in the linear space R Y of all functions f : Y → R, i.e. Λ α (Y ) is closed with respect to pointwise operations of multiplication with nonnega- tive real numbers of a function in Λ α (Y ), and of addition of two functions in Λ α (Y ). The functional k·| α,Y :Λ α (Y ) → [0, ∞) is nonnegative and sublinear, and the pair (Λ α (Y ), k·| α,Y ) is called the asymmetric normed cone of d-semi H¨older functions on Y (compare with [10]). The cone (Λ α (Y ), k·| α,Y ) is diﬀerent from the cone of d-semi-Lipschitz func- tion (α = 1) considered in [9]. For example, if one considers Y = [0, 1] d(x, y)= |x - y| and f : [0, 1] → R,f (x)= x sin 1 x ,x ∈ (0, 1]; f (0) = 0, then it is known that f ∈ Λ α (X, d) if and only if α ∈ (0, 1/2] (see[14], Problem 153) and in this case kf | α,Y ≤{1 + 2 ln(1 + 2π)+2π} 1/2 .

2. EXTENSIONS OF d-SEMI-H ¨ OLDER FUNCTIONS Let (X, d) be a quasi–metric space, Y ⊂ X and f ∈ Λ α (Y ). A function F ∈ Λ α (X ) is called an extension of f (preserving the semi-H¨older constant L(f,Y ) if (2.1) F | Y = f and L(F,X )= L(f,Y ) The existence of extension in Λ α (X ) for each f ∈ Λ α (X ) is assured by the following theorem. Theorem 2. Let (X, d) be a quasi–metric space, Y ⊂ X and f ∈ Λ α (Y ) with d-semi-H¨older constant L(f,Y ). Then there exist F ∈ Λ α (X ) such that F | Y = f and L(F,X )= L(f,Y ). Proof. Let G : X → R deﬁned by (2.2) G(x) = sup y∈Y {f (y) - L(f,Y ) · d α (y,x)},x ∈ X. Let y 0 ∈ Y be a ﬁxed element, and x ∈ X. For every y ∈ Y, f (y) - L(f,Y ) · d α (y,x)= f (y) - f (y 0 ) - L(f,Y ) d α (y,x)+ f (y 0 ) ≤ L(f,Y )d α (y,y 0 ) -L(f,Y )d α (y,x)+ f (y 0 ) = f (y 0 )+ L(f,Y )[d α (y,y 0 ) - d α (y,x)] ≤ f (y 0 )+ L(f,Y ) d α (x, y 0 ). Then it follows that the set {f (y) - L(f,Y )d α (y,x): y ∈ Y } is bounded from above, and G(x) exists for every x ∈ X. By the deﬁnition of G(x), for every y ∈ Y G(x) ≥ f (y) - L(f,Y )d α (y,x),x ∈ X, and for x = y one obtains G(y) ≥ f (y). On the other hand, for y ∈ Y and every y 0 ∈ Y, f (y 0 ) - f (y) ≤ L(f,Y ) · d α (y 0 ,y). It follows f (y 0 ) - L(f,Y ) · d α (y 0 ,y) ≤ f (y), and taking the supremum with respect to y 0 ∈ Y one obtains G(y) ≤ f (y),y ∈ Y. Consequently G| Y = f. Now, let u, v ∈ X and ε> 0. Choosing y ∈ Y such that G(u) ≤ f (y) - L(f,Y ) d α (y,u)+ ε,

it follows G(u) - G(v) ≤ f (y) - L(f,Y )d α (y,u)+ ε - f (y)+ L(f,Y )d α (y,v) = L(f,Y )[d α (y,v) - d α (y,u)] + ε ≤ L(f,Y )d α (u, v)+ ε. Because ε> 0 is arbitrarily chosen, one obtains: G(u) - G(v) ≤ L(f,Y ) d α (u, v), for u, v ∈ X, i.e. G ∈ Λ α (X ). Moreover L(G, X ) ≤ L(f,Y ). Because G| Y = f one obtains also L(f,Y )= L( G| Y, Y ) ≤ L(G, X ) and consequently, L(f,Y )= L(G, X ).  Remark 3. Observe that the function F : X → R, (2.3) F (x) = inf y∈Y {f (y)+ L(f,Y ) d α (x, y)},x ∈ X. is another extension of f ∈ Λ α (f,Y ). Moreover, if H is any extension of f, i.e. H | Y = f and L(H, X )= L(f,Y ) then (2.4) G(x) ≤ H (x) ≤ F (x),x ∈ X where G is deﬁned by (2.2) and F is deﬁned by (2.3). From (2.4) it follows that G is the minimal extension of f, and F is the maximal extension of f.  Indeed let H an extension of f ∈ Λ α (Y ). Then, for arbitrary x ∈ X and y ∈ Y we have H (x) - H (y) ≤ L(f,Y )d α (x, y) implying H (x) ≤ H (y)+ L(f,Y )d α (x, y) = f (y)+ L(f,y)d α (x, y) Taking the inﬁmum with respect to y ∈ Y one obtain H (x) ≤ F (x),x ∈ X Analogously, H (y) - H (x) ≤ L(f,Y ) · d α (y,x) implies f (y) - L(f,Y )d α (y,x) ≥ H (x), and taking the supremum with respect to y ∈ Y one obtain G(x) ≥ H (x),x ∈ X. It follows (2.4).

Remark 4. For f ∈ Λ α (Y ) denote by E (f ) the (non-empty) set of all extensions of f i.e. (2.5) E (f ) := {H ∈ Λ α (X ): H | Y = f and L(H, X )= L(f,Y )} Obviously, E (f ) is a convex subset of the cone Λ α (X ).  Remark 5. Let f ∈ Λ α (Y ) and let kf | α·Y be the smallest d-semi-H¨older constant for f on Y (see (1.4)). Then the functions G and F deﬁned by (2.2) and (2.3), where L(f,Y )= kf | α,Y are extensions for f, preserving the constant kf | α,Y .  Consider a ﬁxed element y 0 ∈ Y, and let (2.6) Λ α,0 (Y ) := {f ∈ Λ α (Y ): f (y 0 )=0}. Then the functional k| α,Y :Λ α,0 (Y ) → [0, ∞) deﬁned by (2.7) kf | α,Y = sup{ (f (x)-f (y))∨0 d α (x,y) : d(x, y) > 0; x, y ∈ Y } is a quasi-norm on Λ α,0 (Y ), i.e. the following properties hold: a) kf | α,Y ≥ 0; f = 0 iﬀ -f ∈ Λ α,0 (Y ) and kf | α,Y = k-f | α,Y =0,f ∈ Λ α,0 (Y ); b) kaf | α,Y = a kf | α,Y for every f ∈ Λ α,0 (Y ) and a ≥ 0; c) kf + g| α,Y ≤kf | α,Y + kg| α,Y for all f,g ∈ Λ α,0 (Y ). Remark 6. Let f ∈ Λ α (X ) and Y 1 ⊂ Y 2 ⊂ X. Suppose that q ≥kf | α,X and let G 1 (x) = sup y∈Y 1 {f (y) - qd α (y,x)},x ∈ X, G 2 (x) = sup y∈Y 2 {f (y) - qd α (y,x)},x ∈ X Then G 1 (x) ≤ G 2 (x) ≤ f (x), for all x ∈ X and G 1 | Y 1 = G 2 | Y 1 = f | Y 1 . Also, if F 1 (x) = inf y∈Y 1 {f (y)+ qd α (x, y)},x ∈ X and F 2 (x) = inf y∈Y 2 {f (y)+ qd α (x, y)},x ∈ X then F 1 (x) ≥ F 2 (x) ≥ f (x), for all x ∈ X and F 1 | Y 1 = F 2 | Y 1 = f | Y 1 . Consequently, if (Y n ) n≥1 is a sequence in 2 X such that Y 1 ⊂ Y 2 ⊂ ... ⊂ Y n ⊂ ..., f ∈ Λ α,0 (X ) and q ≥kf | α,X then the sequences (G n ) n≥1 and (F n ) n≥1 , where G n (x) = sup y∈Yn {f (y) - qd α (y,x)},x ∈ X

and F n (x) = inf y∈Yn {f (y)+ qd α (x, y)},x ∈ X are monotonically increasing, respectively decreasing, G n ,F n ∈ Λ α (X ), n =1, 2, ..., and G n (x) ≤ f (x) ≤ F n (x), for all x ∈ X.  Because, for every y ∈ Y n f (y) - qd α (y,x) ≤ G n (x) ≤ F n (x) ≤ f (y)+ qd α (x, y) it follows that, F n (x) - G n (x) ≤ q[d α (x, y)+ d α (y,x)]. Taking the inﬁmum with respect to y ∈ Y n one obtains: (2.8) F n (x) - G n (x) ≤ q inf y∈Yn [d α (x, y)+ d α (y,x)]. for every x ∈ X. If Y n is d s -dense in X, where d s (x, y)= d(x, y) ∨ d(y,x) for every x, y ∈ X then, by (2.8) it follows that F n (x)= G n (x),x ∈ X. Consequently, a function f ∈ Λ α (Y ) where Y is d s -dense in X has an unique extension F ∈ Λ α (X ). REFERENCES [1] Collins, J. and Zimmer, J., An asymmetric Arzel` a-Ascoli theorem, Topology and its Applications, 154 , pp. 2312–2322, 2007. [2] Matouskova, E., Extensions of continuous and Lipschitz functions, Canad. Math. Bull, 43, no. 2, pp. 208–217, 2000. [3] McShane, E.T., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837– 842, 1934. [4] Menucci, A., On asymmetric distances. Technical Report, Scuola Normale Superiore, Pisa, 2007, http://cvgmt.sns.it/people/menucci. [5] Miculescu, R., Some observations on generalized Lipschitz functions, Rocky Mountain J. Math., 37, no.3, pp. 893–903, 2007. [6] Must˘ at¸a, C., Extension of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Number. Th´eor. Approx., 30, no. 1, pp. 61–67, 2001. [7] Must˘ at¸a, C., Best approximation and unique extension of Lipschitz functions, J. Ap- prox. Theory, 19, no. 3, pp. 222–230, 1977. [8] Must˘ at¸a,C., A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint ¸. Univ. Baia Mare, Ser. B. Matematic˘ a-Informatic˘ a 18, pp. 275–280, 2002. [9] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi–metric spaces, J. Approx. Theory, 103, pp. 292–301, 2000. [10] Romaguera, S. and Sanchis, M., Properties of the normed cone of semi-Lipschitz functions, Acta Math Hungar, 108 no. 1–2, pp. 55–70, 2005. [11] S´ anchez- ´ Alvarez, J.M., On semi-Lipschitz functions with values in a quasi-normed linear space, Applied General Topology, 6, no. 2, pp. 216–228, 2005. [12] Weaver, N., Lattices of Lipschitz functions, Paciﬁc Journal of Math., 164, pp. 179–193, 1994. [13] Wells, J.H. and Williams, L.R. Embeddings and Extensions in Analysis, Springer- Verlag, Berlin, 1975. [14] * * *, The Otto Dunkel Memorial Problem Book, New York, 1957.