ANNIHILATOR CONCEPT AND ITS APPLICATION TO BEST APPROXIMATION THEORY

In Constructive Theory of Functions , two sections that look different but in fact have strong relations are very important: 1) Theory of best approximation of functions and, 2) Extremal problems for linear functionals that dened in di¤erent function classes. Both of them studied independently started from the work [2] of P. L. Chebyshev and developed as systematic theories until the middle of 20th century. After that the relationship between these two theories has been realized and studied as connected theories. As a result, important ndings for both problems obtained (S.M. Nikolskiy, M.G Kreyin, S. Ya Havinson, G.Ts.Tumarkin, W. Rogosinsky) ([1], [4], [5], [11-13]). Duality term coined for the relations between two problems such as these. In such duality relations, annihilator concept took place and played an important role. But no one studied or interested with the annihilator and its structure. F. H. Nasibov rst one who studied and determined the annihilatorsstructure, and showed how to use it to solve the linear extremal problems ([6],[7],[8],[9],[10]). In this paper, we will present our current ndings about this topic. 1. Definition of Class and Determining the Structure of Annihilator Let E be a subset of R1 = ( 1;+1) and (x) 0 is a function (weighting function) dened on E. We use the notation for the function space of f(x) as L2; (E) which satises the condition: kfk2; = Z E jf(x)j (x)dx 1/2 < +1: (1.1) We could also use L2; (E) class which satises kfk2; = Z E jf(x)j d (x) 1/2 < +1; (1.2) Received by the editors June 15, 2009; Accepted: Sept. 18, 2009. Key words and phrases. Space, Functional, Annihilator, Approximation, Constructive Theory of Function. c 2009 Ankara University 31 32 FERHAD HÜSEYÍNO 1⁄4 GLU NASÍBOV AND AHMET KAÇAR where (x) satises R E d < +1. It is explicit that L2; (E) L2; (E). Now consider the system functions = f'k(t)g 1 0 that are orthonormal to (x) weighting function. In this case, each function f(x) 2 L2; (E) can be expand to the Fourier series: f(x) 1 X k=0 Ck'k(x): (1.3) This Fourier series converges to f(x) in the sense of L2; (E) norm-metric, and satises the Parvesal formula kfk22; = 1 X k=0 jCkj : (1.4) After this point, we will use notation n for polynomial set pn(t) = Pn k=0 ak'k(t) (an 6= 0) , where ak (k = 0; 1; 2; ::: ;n) are arbitrary constants: n := ( Pn(t) = n X k=o ak'k(t) (an 6= 0); ak (k = 0; 1; :::; n) are arbitrary constants ) Problem 1. For n L2; (E), dene the structures of annihilator n . According to the denition n := fl 2 (L2; ) : 8Pn 2 n; l(Pn) = 0g : Now the problem becomes the dening the structure of functionals l, which satises the conditions mentioned above. On the other hand, we can represent each l 2 (L2; ) linearly bounded (or continuous) functional as l(f) = Z E f(t) g(t) (t)dt (f; g) (1.5) and k l k = sup f2L2; (E) jl(f)j = kgk2; (1.6) formula is correct. In fact, we need to dene the structure of functions g(t) 2 L2; (E), which satises the condition l(Pn) = Z E Pn(t) g(t) (t)dt = 0 (8Pn 2 n): (1.7) Let ak = ak(Pn) is the arbitrary coe¢ cients of Pn(t) 2 n polynomial, and Ck = Ck(g) is the Fourier coe¢ cients of g(t) 2 L2; (E) according to = f'k(t)g 1 0 system. According to Parseval formula, we get l(Pn) = Z E Pn(t) g(t) (t)dt = n X k=0 akCk: (1.8) From that we see Ck= 0 (k = 0; 1; 2; :::; n) so each fakg0 (8Pn 2 n) satises the equation (1.7). This is the result we needed. (For example, if we have c1 6= 0, then from a1c1 = 0 we have a1 = 0 . Since a1 is arbitrary and if we choose a1 = 1 then c1 = 0). ANNIHILATOR CONCEPT AND ITS APPLICATION 33 Hence we proved the theorem below which dene the n annihilator. Theorem 1. For subspace n of L2; (E) annihilator n consists of functions g(t) 2 L2; (E) if and only if rst n + 1 Fourier coe¢ cients of g(t) (according to system) satises the conditions Ck = Ck(g) = Z E g(t)'k(t) (t)dt = 0; k = 0; 1; 2; :::; n: (1.9) 2. Application to Best Approximation Problem Denition 1. Best approximation to element of f 2 L2; (E) by Pn(t) 2 n polynomials is the En(f ;L2; ) = inf Pn2 n kf Pnk2; : (2.1) There is an element on the n subspace which satises the equality En(f ;L2; ) = f Pn 2; (2.2) ( n is nite dimensional). On the other hand, since L2; (E) is strictly normed space, there is only one P 0 n(t). In the approximation theory, it is one of the di¢ cult problems to nd an element that gives the best approximation to a given element. That why it is important to learn the characteristics of given element (P. L. Chebyshev, S. N. Bernstein, A. N. Kolmogorov Theorems and others, [1], [3], [10-13]). For that purpose we want to remind the theorem below. Theorem 2. (Zinger, [13]) P 0 n(t) 2 n is the best approximation polynomial to f(t) 2 L2; if and only if for every Pn(t) 2 n holds the conditions: Z E Pn(t) [f(t) P 0 n(t)] (t)dt = 0: (2.3) It is apparent that Theorem 1 is equivalent to Theorem 2: T1, T2. When we compare (1.7) and (2.3), we see g(t) = f(t) P 0 n(t) 2 n . Then, by Theorem 1, Ck(f P 0 n) = 0 (k = 0; 1; 2; :::; n) or Ck(P 0 n) = Ck(f) (k = 0; 1; 2; :::; n): (2.4) According to this, coe¢ cients of polynomial which gives best approximation to f(t) 2 L2; on L2; metric are the rst n+ 1 coe¢ cient of Fourier Series of function f on the = f'kg system. So we proved the A. Teopler Theorem which is below in the other method. Theorem 3. (A. TEOPLER). Polinom of the best approximation to f 2 L2; n n on the L2; metric (between Pn 2 n polynomials) is P 0 n(t) = Sn(f; t) = n X


Definition of Class and Determining the Structure of Annihilator
Let E be a subset of R 1 = ( 1; +1) and (x) 0 is a function (weighting function) de…ned on E. We use the notation for the function space of f(x) as L 2; (E) which satis…es the condition: We could also use L 2; (E) class which satis…es FERHAD HÜSEYÍNO ¼ GLU NASÍBOV AND AHM ET KAÇAR where (x) satis…es R E d < +1.It is explicit that L 2; (E) L 2; (E).Now consider the system functions = f' k (t)g 1 0 that are orthonormal to (x) weighting function.In this case, each function f (x) 2 L 2; (E) can be expand to the Fourier series: This Fourier series converges to f(x) in the sense of L 2; (E) norm-metric, and satis…es the Parvesal formula After this point, we will use notation n for polynomial set p n (t) = P n k=0 a k ' k (t) (a n 6 = 0) , where a k (k = 0; 1; 2; ::: ;n) are arbitrary constants: For n L 2; (E), de…ne the structures of annihilator ?
n .According to the de…nition ?n := fl 2 (L 2; ) : 8P n 2 n ; l(P n ) = 0g : Now the problem becomes the de…ning the structure of functionals l, which satis…es the conditions mentioned above.On the other hand, we can represent each l 2 (L 2; ) linearly bounded (or continuous) functional as jl(f )j = kgk 2; (1.6) formula is correct.In fact, we need to de…ne the structure of functions g(t) 2 L 2; (E), which satis…es the condition Let a k = a k (P n ) is the arbitrary coe¢ cients of P n (t) 2 n polynomial, and system.According to Parseval formula, we get From that we see C k = 0 (k = 0; 1; 2; :::; n) so each fa k g n 0 (8P n 2 n ) satis…es the equation (1.7).This is the result we needed.(For example, if we have c 1 6 = 0, then from a 1 c 1 = 0 we have a 1 = 0 .Since a 1 is arbitrary and if we choose a 1 = 1 then c 1 = 0).
Hence we proved the theorem below which de…ne the ?n annihilator.

Application to Best Approximation Problem
There is an element on the n subspace which satis…es the equality (2.2) ( n is …nite dimensional).On the other hand, since L 2; (E) is strictly normed space, there is only one P 0 n (t).In the approximation theory, it is one of the di¢ cult problems to …nd an element that gives the best approximation to a given element.That why it is important to learn the characteristics of given element (P.L. Chebyshev, S. N. Bernstein, A. N. Kolmogorov Theorems and others, [1], [3], [10][11][12][13]).For that purpose we want to remind the theorem below.
Theorem 2. (Zinger, [13]) P 0 n (t) 2 n is the best approximation polynomial to f (t) 2 L 2; if and only if for every P n (t) 2 n holds the conditions: It is apparent that Theorem 1 is equivalent to Theorem 2: T1 , T2.
When we compare (1.7) and (2.3), we see g(t) = f (t) P 0 n (t) 2 ?n .Then, by Theorem 1, C k (f P 0 n ) = 0 (k = 0; 1; 2; :::; n) or (2.4) According to this, coe¢ cients of polynomial which gives best approximation to f (t) 2 L 2; on L 2; metric are the …rst n+ 1 coe¢ cient of Fourier Series of function f on the = f' k g system.So we proved the A. Teopler Theorem which is below in the other method.
(A. TEOPLER).Polinom of the best approximation to f 2 L 2; n n on the L 2; metric (between P n 2 n polynomials) is which is the Fourier series partial sum order n of f (t) function according to = f' k g system.Other than that one, best approximation value to f (t) function with P n 2 n polynomials is de…ned by equality To validate what we said before, su¢ ciently to add investigation the following equalities:

Application to Solution to an Extremal Problem
Let g 0 (t) 2 L 2; be given.It de…nes a functional since n L 2; 1 .
Is to be found the norm The solution of this problem is based on the following duality principle: ).For every linear functional l 0 that de…ned on an subspace of F of any normed linear space X, where l 0 2 F , F 1 is a unit sphere on F.
Since X = L 2; , F = n , F 1 = n;1 and l 0 $ g 0 (t) 2 L 2; , instead of (3.5) we get jjg 0 gjj 2; : By Parseval formula, we can rearrange (3.6) as At this point, g is the extremal element for the right side of (3.6).On the other hand, n is …nite dimensional.According to this, there is a P 0 n (t) 2 n;1 polynomial that gives sup (extremal) to the left side of the (3.6).Hence the theorem below is true.
Theorem 5.There is a unique P 0 n (t) 2 n;1 polynomial that supplies condition kl 0 k = l 0 (P 0 n ) of problem (3.6) and of left hand side of (3.4).As well as max Pn2 n;1 duality relations are true.We get max for such a P 0 n (t) 2 n;1 polynomial that its coe¢ cients can be computed if and only if where C k (g 0 ) s are the Fourier coe¢ cients of g 0 (t) function according to = f' k g system.
Validity of (3.8) can be seen from the operations below.Let P 0 n (t) = where g (t) 2 n;1 function gives inf to the right side of (3.6).Because of, we have Consequently,

Application to Solution to Another Extremal Problem
Duality relations, which we will consider in this section is: If an element !(t) 2 XnF is given, then is true and F ? 1 = fl 2 F ? : klk 1g.In our case Now, suppose that and if we take g(t) P 1 k=0 C k (g)' k (t) , C k (g 0 ) = 0 (k = 0; 1; 2; :::; n), (means g 2 ? n;1 ) we get max Thus we proved the theorem below.