Best Approximation in Real Linear 2-Normed Spaces

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: ,p-ISSN: X, Volume 6, Issue 3 (May. - Jun. 13), PP 16-4 Best Approximation in Real Linear -Normed Spaces R.Vijayaragavan Applied Analysis Division, School of Advanced Sciences, VIT University, Vellore 63 14, Tamilnadu, India. Abstract: This pape r d e l i n e a t e s existence, characteriations and st rong unicity of best uniform approximations in real linear -normed spaces. AMS Su ject Classification: 41A5, 41A5, 41A99, 41A8. Key Words and Phrases: Best approximation, existence, -normed linear spaces. I. Introduction The concepts of linear -normed spaces were initially introduced by Gahler [5] in Since then many researchers (see also [,4]) have studied the geometric structure of -normed spaces and obtained various results. This paper mainly deals with existence, characteriations and unicity of best uniform approximation with respect to -norm. Section provides some important definitions and results that are used in the sequel. Some main results of the set of best uniform approximation in the context of linear -normed spaces are established in Section 3. II. Preliminaries Definition.1. Let X be a linear space over real numbers with dimension greater than one and let, be a real-valued function on X X satisfying the following properties for all x, y, in X. (i) x, y = if and only if x and y are linearly dependent, (ii) x, y = y, x, (iii) αx, y = α x, y, where α is a real number, (iv) x, y + x, y + x,. Then, is called a -norm and the linear space X equipped with -norm is called a linear - normed space. It is clear that -norm is non-negative. Example.. Let X = R 3 with usual component wise vector additions and scalar multiplications. For x = (a1, b1, c1 ) and y = (a, b, c ) in X, define x, y = max{ a1 b a b1, b1 c b c1, a1 c a c1 }. Then clearly, is a -norm on X. Definition.3. Let G be a subset of a real linear -normed space X and x X. Then g G is said to be a best approximation to x from the elements of G if x g, = inf x g,, X \V,G) g G where V (x, G) is the subspace generated by x and G. The set of all elements of best approximation to x X fr om G with respect to the set is denoted by PG,Z (x). Definition.4. A linear -normed space (X,, ) is said to be strictly convex if or a + b, c = a, c + b, c, a, c = b, c = 1 and c / V (a, b) a = b. 16 Page

2 Best Approximation in Real Linear -Normed Spaces A linear -normed space (X,, ) is said to be strictly convex if and only if x, = y, = 1, x y and X \V (x, y) 1 ( ), 1 x y Example.5. Let X = R 3 with -norm defined as follows: For x = (a1, b1, c1 ) and y = (a, b, c ) in X, let 1 x, y = {(a1b ab1 ) + (b1c b c1 ) + (a1 c ac1 )} Then the space (X,.,. ) is strictly convex linear -normed space. Definition.6. For all functions h C ([a, b] [c, d]) h = {sup h(t, t ) : t [a, b], t [c, d]}. The set of extreme points of a function h C ([a, b] [c, d]) is defined by E(h) = {x [a, b], y [c, d] : h(x, y) = h }. Best approximation with respect to this norm is called best uniform approximation. Definition.7. Let G be a subspace of C ([a, b] [c, d]) = {f: [a, b] [c, d]r A function g G is called a strongly unique best uniform approximation of f C ([a, b] [c, d]) if there exists a constant kf > such that for all g G, f g f g + kf g g. Example.8. Consider the space G = span{g1 } of C ([ 1, 1] [ 1, 1]), where g1 (t, t ) = t [ 1, 1] and f (C [ 1, 1] [ 1, 1]). Then (, 1) is the best approximation of [1, 1]. Definition.9. Let G be a subset of C ([a, b] [c, d]) and let f C ([a, b] [c, d]) have a unique best uniform approximation g G. Then t h e projection PG : C ([a, b] [c, d]) P OW(G) is called Lipchit-continuous at f if there exists a const kf > such that for all f C ([a, b] [c, d]) and all gf PG (f ), gf gf kf f f. III. Main Results Theorem 3.1. Let G be a finite-dimensional subspace of a real linear -normed space X. Then for every x X, there exists a best approximation from G. Proof. x X. Then by the definition of the infimum there exists a sequence {gn } G such that x g, n inf x g,. gg This implies that there exists a constant k> such that for all n, g, x, x g, inf x g, k n n gg x, k. Let Hence for all n, gn, x, + k. {gn } is bounded sequence. Then t h e r e exists a subsequence {gnk } of {g n } converging to g G. 17 Page

3 Best Approximation in Real Linear -Normed Spaces x g, lim x g inf x g,, X \ V ( x, G) k n k gg g PG,Z (x), wh i c h c om p l e t e s t h e p r o o f. Theorem 3.. Let G be a finite dimensional subspace of a strictly convex linear - normed space X. Then for every x X,there exists a unique best approximation from G. Proof. Let x X.Since G is a finite-dimensional, by Theorem 3.1 there exists an element Such that, g PG,Z (x), X \ V( x, G). N o w w e s h o w t h a t, PG,Z (x) = {g }. For that first we prove that PG, (x) is convex. Let g1, g PG,Z (x) and α 1. Then, x (αg1 + (1 α)g ), = α(x g1) + (1 α)(x g ), α x g1, + (1 α) x g, inf x g, (1 )inf x g, gg inf x g, gg x g,, for all g G. gg Since αg1 + (1 α)g G, αg1 + (1 α)g PG,Z (x). We shall suppose that g PG,Z (x). Then 1 (g + g ) PG,Z (x), which implies that 1 1 {( x g) ( x g )}, x ( g g ), = inf x g, 1 gg g G Since x g, = x g, = inf x g, and X is strictly convex, we obtain x g = x g g = g. This proves that PG,Z (x) ={g }. Theorem 3.3. Let G be a finite-dimensional subspace of a real linear -normed space X w i t h the property that every function with domain X X h a s a unique best approximation from G. Then for all f1, f X X, inf f g, inf f g, f f,, X \ G and P : X x X Gis continuous.proof. gg 1 1 gg g G Suppose that PG is not continuous. Then there exists an element f X X and a sequence {fn} X X such that {fn gn }. PG, (fn ) does not converge to PG (f ). Since G is finite dimensional, there ex- ists a subsequence {fnk} of {fn} such that PG,Z (fnk) g G, g PG (f) and we shall show that the mapping f inf f g, is continuous ( f X X ) gg Let f1, fx X. Then there exists a g G such that f g, = inf f g, G 18 Page

4 Best Approximation in Real Linear -Normed Spaces inf g G f1 g, f1 g, f1 f, + f g, f f, inf f g, = 1 gg inf f1 g, inf f g, f1 f,, X \G. g G This proves that g G inf f g, inf f g, f f,, X \ G gg 1 1 gg By continuity it follows that fnk PG (fnk ), = inf fnk g, inf lim f g, gg k n k inf f g, gg and lim f n P G ( f n ), f g, k lim( f P ( f ), f P ( f ), g k nk G nk G and P ( f) are two distinct best approximation of f and G which contradicts the uniqueness of best approximation. G Therefore PG : X X G is continuous. A functional is a real valued mapping with domain AC with A and C are linear manifolds of a - normed space X. A linear - functionals is - functional such that (i) F( a c, b d) F( a, b) F( a, d) F( c, b) F( c, d) (ii) F( a, b) F( a, b). F is called a bounded -functional if there is a real constant k such that F( a, b) k a, b the domain of F and F inf k : f ( a, b) k a, b,( a, b D( F) for all a,b in sup f ( a, b) : a, b 1,( a, b) D( F) 19 Page

5 f ( a, b) sup : a, b,( a, b) D( F) ab, Best Approximation in Real Linear -Normed Spaces Theorem 3.4. Let G be a subspace of C ([a, b] [a, b]), f C ([a, b] [a, b]) and g G. Then the following statements are equivalent: ((i) The function g is a best uniform approximation of f from G. (ii) For every function g G, min t,t E(f g )f (t, t ) g (t, t )) (g(t, t )) Proof. (ii) (i). Suppose that (ii) holds and let g G. Then by (ii) there exist the points t, t E (f g) such that (f (t, t ) g(t, t ))(g(t, t ) g (t, t )). Then we have f g f (t, t ) g(t, t ) = f (t, t ) g(t, t ) + g(t, t ) g(t, t ) = f (t, t ) g(t, t ) + g(t, t ) g (t, t ) f (t, t ) g (t, t ) which shows that (i) holds. (i) (ii). Suppose that (i) holds and assume that (ii) fails. Then there exists a function g1 G such that for all t, t E(f g), (f(t, t ) g(t, t ))g1(t, t ) >. Since E(f g) is compact, there exists a real number c > such that for all t, t E(f g ) (f(t, t ) g(t, t ))g1(t, t ) > c. (1) Further, there exists an open neighborhood U of E(f g ) such that for all t, t U,andc(f(t, t ) g (t, t ))g1(t, t ) > () f (t, t ) g (t, t ) f g (3) Since [ a, b]\u is compact, there exists a real number d > s u c h that for all t, t [a, b]\u, f (t, t ) g(t, t ) < f g d (4) Now we shall assume that g1 min {d, f g }. (5) Let g = g + g1. Then by (4) and (5) for all t, t [a, b]\u, f (t, t ) g (t, t ) = (f (t, t ) g(t, t )) g1 (t, t ) f (t, t ) g (t, t ) + g1 (t, t ) f (t, t ) g (t, t ) d + g1 (t, t ) f (t, t ) g (t, t ). Page

6 Best Approximation in Real Linear -Normed Spaces For all t U, by (), (3) and (5), f (t, t ) g (t, t ) = (f (t, t ) g (t, t )) g1 (t, t ) f (t, t ) g(t, t ) g1 (t, t ) f g. f g f g g is not the best uniform approximation of f which is a contradiction. Hence the proof. Theorem 3.5. Let G be a subset of C ([a, b] [c, d]) and f has a strongly unique best uniform approximation from G, then PG : C ([a, b] [c, b]) POW (G) is Lipschit-continuous at f. Proof. Let f X = C ([a, b] [c, b]) have a strongly unique best uniform approximation gf G. Then there exists kf > such that for all g G. f g f gf + kf g gf. Then for all f X and for all g f PG (f ). We obtain kf gf g f f g f f gf f f f f + f g f f gf + f f = f f. L f is the desired constant. k Theorem 3.6. Let G be a finite dimensional subspace of X = C ([a, b] [c, d]), f X \G and g G. Then the following statements are equivalent: (i) The function g is a strongly unique best uniform approximation of f from G. (ii) For every nontrivial function g G, min x,y E(f g )(f (x, y) g (x, y))g(x, y) < (iii) There exists a constant kf > such that for every function g G, min x,y E(f g )(f (x, y) g (x, y))g(x, y) kf f g g. Proof. (iii) (i). We shall suppose that (iii) holds and let g G. Then by (iii) there exist the points x, y E(f g ) such that (f (x, y) g(x, y))(g(x, y) g (x, y)) kf f g g g. This implies that f f g f (x, y) g(x, y) = f (x, y) g (x, y) (g(x, y) g(x, y)) = f (x, y) g (x, y) + g(x, y) g (x, y) f g + k f g g g f f ( x, y) g ( x, y) 1 Page

7 Best Approximation in Real Linear -Normed Spaces = f g + kf g g. (i) (iii). Suppose that (iii) fails, i.e. there exists a function g1 G such that for all x, y E(f g ), (f (x, y) g(x, y))g1 (x, y) > kf f g g1. Since E(f g ) is compact, there exists an open neighborhood U of E(f g) such that for all x, y U (f (x, y) g(x, y))g1 (x, y) > kf f g g1 and 1 f (x, y) g(x, y) f g. (6) Further, we can choose U sufficiently small such that for all x, y U with (f (x, y) g (x, y))g1 (x, y) < g1 (x, y) < kf g1. (7) Since [a, b] [c, d]\u is compact, there exists a real number c > such that for all x, y [a, b] [c, d]\u, f (x, y) g (x, y) f g c. (8) We may assume that without loss of generality 1 g1 1 min, c f g (9) Let g = g + g1. Then by (8) and (9) for all x, y [a, b] [c, d]\u, f (x, y) g(x, y) = f (x, y) g(x, y) g1 (x, y) f g c + g1 f g. Again, by (6) and (7), for all x, y U with (f (x, y) g(x, y))g1 (x, y) <, f (x, y) g(x, y) = (f (x, y) g(x, y)) g1 (x, y) = f (x, y) g (x, y) + g1(x, y) f g + kf g1 = f g + kf g g. By (6) and (9) for all x, y U with (f (x, y) g(x, y))g1 (x, y), f (x, y) g(x, y) = (f (x, y) g(x, y)) g1 (x, y) = f (x, y) g (x, y) g1 (x, y) f g Page

8 Best Approximation in Real Linear -Normed Spaces f g < f g + kf g g, (i) fails. (ii) (iii) suppose that (ii) holds. Let F: {g G : g =1}R RBe the mapping, defined by Fg ( ) min x, ye ( f g ) ( f ( x, y) g( x, y)) g( x, y) f g Since G is finite dimensional, the set {g G : g = 1} is compact. Therefore, since by (ii) F (g) < for all {g G : g = 1}, there exists a constant kf > Such that F g g kf for all g G, which pr oves (iii) ( iii) ( ii) is obvious Hence the proof of the theorem is complete. Theorem 3.7. Let G be a finite dimensional subspace of a real -Hilbert space X. Then for every x X, there exists a unique best approximation from G. Proof. Let (X,, ) be a -Hilbert space and l et G be a finite dimensional subspace of X. Let x, y X and x y then by parallelogram law x + y, + x y, = ( x, + y, ). (1) Let x, = y, = 1. Then by (1) x + y, = 4 x y, < 4 x y, 1 X is strictly convex.therefore, by Theorem 3. there exists a unique best approximation to x X \G from G. Theorem 3.8. Let G be a subspace of a real -Hilbert space X, x X \G and g G. Then the following statements are equivalent: (i) The element g is a best approximation of x from G. (ii) For all g G, (x g, g/) = \V (x, G). Proof. (ii) (i). Suppose that (ii) holds and let g G. Then by (ii) x g, x g, = (x g, x g/) (x g, x g /) = (x, x/) + (g, g/) (x, g/) (x x/) (g, g /) + (x, g /) = (g g, g g /) + (x g, g g/) = g g, X That is x g, x g, which proves (i) 3 Page

9 Best Approximation in Real Linear -Normed Spaces (i) (ii). Suppose that (ii) fails, i.e., there exists a function g G such that (x g, g ). x g, g/ x g g / = x g / / g, g/ x g, g/ g, g/ x g Which implies that g is not a best approximation of x. Hence the proof. Corollary 3.9. Let G = span (g1, g,, gn ) be an n -dimensional subspace of a real -Hilbert space X, x X \G and g = statements are equivalent: n i1 a g i i G. Then the following (i) The element g is a best approximation of x from G. (ii) The coefficients a1, a, an satisfy the following system of linear equations n a i (gi, gj /) = (x, gj /), j = 1,,, n. i1 Proof. The condition (f g, g/) = in Theorem 3.8 is equivalent to (f g, g/) =, j = 1,,..., n. n n Since g ai gi, ai gi, g j / f, g j / is equivalent to(f g, gj ) =, j = 1,,..., n. i1 i1 References [1] M. W. Barelt, H.W. McLaughlin, Characteriations of s t r o n g unicity in approximation theory, Journal of Approximation Theory, 9 (1973), [] C. Diminnie, S. Gahler, A. White, Strictly convex -normed space, Mathematische N a chrichten, 59 (1974), [3] C. Franchetti, M. Furi, Some c h a r a c t e r i s t i c properties of r e a l Hilbert spaces,romanian Journal of Pure and Applied Mathematics, 17 (197), [4] R. W. Freese, Y. J. Cho, G eometry of linear -normed s p a c e s, N o v a science publishers, Inc., New York, (1). [5] S.Gahler, Linear -normierte Raume, Mathematische Nachrichten, 8 (1964),1-43. [6] I. Singer, On the set of best approximation of an element in a normed linear space, Romanian Journal of Pure and Applied Mathematics, 5 (196), [7] D. E. Wulbert, Uniqueness and differential characteriation of approximation from manifolds, American J ourna l of Mathematics, 18 (1971), Page