A NEW APPROACH IN OBTAINING A BETTER ESTIMATION IN APPROXIMATION BY POSITIVE LINEAR OPERATORS

In this study, without preserving some test functions, we present a new approach in obtaining a better error estimation in the approximation by means of positive linear operators. We also show that our method can be applied to many well-known approximation operators. 1. Introduction Obtaining better error estimations in approximation to a function by a sequence of positive linear operators is an important problem in the approximation theory. So far, some relating results have been presented for Bernstein polynomials [5], Szász-Mirakjan operators [4], Meyer-König and Zeller operators [7] and BernsteinChlodovsky operators [1] by preserving some test functions in the approximation. Recently, Agratini [2] has applied a similar idea to more general summation-type positive linear operators. However, in this note, without preserving the test functions we introduce a di¤erent approach in order to get a faster approximation. We show that our new method can easily be applied to many well-known positive linear operators. Let R := [0;1) and R+b := [0; b] with b > 0: Consider the function space E(R) dened by E(R) := f 2 C(R) : lim x!1 f(x) 1 + x2 is nite endowed with the norm kfk+ = sup x2R+ jf(x)j 1 + x2 : However, for the bounded interval R+b ; we will consider the function space C(R + b ) and the usual maximum norm k kon R+b : Received by the editors March 16, 2009, Accepted: June. 05, 2009. 1991 Mathematics Subject Classication. 41A25, 41A36. Key words and phrases. The Korovkin theorem, Bernstein polynomials, Szász-Mirakjan operators, Bernstein-Kantorovich operators, rate of convergence. c 2009 Ankara University 17 18 M. ALI ÖZARSLAN AND OKTAY DUMAN Throughout the paper we use the test functions fi(x) = x for i = 0; 1; 2: Assume that a sequence fLng of positive linear operators dened on E(R) (or, C(R+b )) satises the following conditions: Ln(f0;x) = 1; Ln(f1;x) = anx+ bn; Ln(f2;x) = cnx 2 + dnx+ en; (1.1) where (an); (bn); (cn); (dn) and (en) are sequences of non-negative real numbers satisfying the following conditions: lim n!1 an = lim n!1 cn = 1 (cn 6= 0); lim n!1 bn = lim n!1 dn = lim n!1 en = 0: (1.2) Actually, many well-known approximation operators, such as Bernstein polynomials, Szász-Mirakjan operators, Bernstein-Kantorovich operators etc., satisfy the conditions (1.1) and (1.2). Our primary interest of this paper is to construct positive linear operators providing better error estimates than the operators Ln as given above. Now consider the lattice homomorphism Tb : C(R) ! C(R+b ) dened by Tb(f) := f jR+b for every f 2 C(R ): In this case, we see from the classical Korovkin theorem (see [6, p.14]) that lim n!1 Tb (Ln(f)) = Tb(f) uniformly on R+b : (1.3) On the other hand, with the universal Korovkin-type property with respect to monotone operators (see Theorem 4:1:4 (vi) of [3, p. 199]) we have the following: Let X be a compact set and H be a conal subspace of C(X): If E is a Banach lattice, S : C(X) ! E is a lattice homomorphism and if fLng is a sequence of positive linear operators from C(X) into E such that limn!1 Ln(h) = S(h) for all h 2 H; then limn!1 Ln(f) = f provided that f belongs to the Korovkin closure of H. Hence, by using (1.3) and the above property we obtain the following result. Theorem 1.1. Let fLng be a sequence of positive linear operators dened on E(R) (resp. C(R+b )) satisfying the conditions in (1:1) and (1:2): Then, for all f 2 E(R) (resp: for all f 2 C(R+b )); we have limn!1 Ln(f) = f uniformly on the interval R+b with b > 0: 2. Better Error Estimates Let A denote R or R+b : For each x 2 A; consider the rst central moment function x dened by x(y) = y x: Assume that I be a subinterval of A. Now in order to get a better error estimation in the approximation by means of the operators Ln (cf. Theorem 1:1) we look for a functional sequence (un); un : I ! A; such that n(x) := q Ln( 2 x;un(x)) q Ln( 2 x;x) =: n(x) for x 2 I: (2.1) By (1:1); this is equivalent to cnu 2 n(x) + (dn 2anx)un(x) (cn 2an)x dnx 0: (2.2) A BETTER ESTIMATION IN APPROXIMATION OF POSITIVE LINEAR OPERATORS 19 Now let n(x) := (dn 2anx) + 4cn (cn 2an)x + dnx : Assume that there exist a subinterval I A and a number n0 2 N such that n(x) 0 (2.3) and 2anx dn 2cn 2 A (2.4) hold for every x 2 I and for every n n0: In this case, from (2.2), (2.3) and (2.4), we get sn(x) := 2anx dn p n(x) 2cn un(x) 2anx dn + p n(x) 2cn =: tn(x): Hence, we can choose, e.g., un(x) := sn(x) + tn(x) 2 = 2anx dn 2cn : In this case, we can dene a new positive linear operator as follows: L n(f ;x) := Ln(f ;un(x)); x 2 I: It is well known that if a positive linear operator U dened on CB(K); the space of all continuous bounded functions on an interval K R; preserves the test function f0; then it satises jU(f ;x) f(x)j 2! f; q U( x;x) ; where !(f; ); > 0; denotes the modulus of continuity of a continuous (and bounded) function f on K. Now let A = R (or R+b ) as stated before. Then, the last inequality implies that jL n(f ;x) f(x)j 2! (f; n(x)) , x 2 I A; and jLn(f ;x) f(x)j 2! (f; n(x)) , x 2 A: Therefore, this means that the error estimation in the approximation by the modied operators L n is better than the approximation by the original operators Ln: 3. Applications 3.1. Bernstein Polynomials. Take A = [0; 1] and consider the classical Bernstein polynomials


Introduction
Obtaining better error estimations in approximation to a function by a sequence of positive linear operators is an important problem in the approximation theory.So far, some relating results have been presented for Bernstein polynomials [5], Szász-Mirakjan operators [4], Meyer-König and Zeller operators [7] and Bernstein-Chlodovsky operators [1] by preserving some test functions in the approximation.Recently, Agratini [2] has applied a similar idea to more general summation-type positive linear operators.However, in this note, without preserving the test functions we introduce a di¤erent approach in order to get a faster approximation.We show that our new method can easily be applied to many well-known positive linear operators.
Let R + := [0; 1) and R + b := [0; b] with b > 0: Consider the function space E(R + ) de…ned by However, for the bounded interval R + b ; we will consider the function space C(R + b ) and the usual maximum norm k kon R + b : Throughout the paper we use the test functions f i (x) = x i for i = 0; 1; 2: Assume that a sequence fL n g of positive linear operators de…ned on E(R + ) (or, C(R + b )) satis…es the following conditions: where and (e n ) are sequences of non-negative real numbers satisfying the following conditions: Actually, many well-known approximation operators, such as Bernstein polynomials, Szász-Mirakjan operators, Bernstein-Kantorovich operators etc., satisfy the conditions (1.1) and (1.2).Our primary interest of this paper is to construct positive linear operators providing better error estimates than the operators L n as given above.
Now consider the lattice homomorphism In this case, we see from the classical Korovkin theorem (see [6, p.14]) that On the other hand, with the universal Korovkin-type property with respect to monotone operators (see Theorem 4:1:4 (vi) of [3, p. 199]) we have the following: "Let X be a compact set and H be a co…nal subspace of C(X): Hence, by using (1.3) and the above property we obtain the following result.

Better Error Estimates
Let A denote R + or R + b : For each x 2 A; consider the …rst central moment function x de…ned by x (y) = y x: Assume that I be a subinterval of A. Now in order to get a better error estimation in the approximation by means of the operators L n (cf.Theorem 1:1) we look for a functional sequence (u n ); u n : By (1:1); this is equivalent to Assume that there exist a subinterval I A and a number n 0 2 N such that hold for every x 2 I and for every n n 0 : In this case, from (2.2), (2.3) and (2.4), we get Hence, we can choose, e.g., In this case, we can de…ne a new positive linear operator as follows: It is well known that if a positive linear operator U de…ned on C B (K); the space of all continuous bounded functions on an interval K R; preserves the test function f 0 ; then it satis…es where !(f; ); > 0; denotes the modulus of continuity of a continuous (and bounded) function f on K.
Now let A = R + (or R + b ) as stated before.Then, the last inequality implies that jL n (f ; x) f (x)j 2! (f; n (x)) , x 2 I A; Therefore, this means that the error estimation in the approximation by the mod-i…ed operators L n is better than the approximation by the original operators L n : 3. Applications 3.1.Bernstein Polynomials.Take A = [0; 1] and consider the classical Bernstein polynomials we take for all n 2 N. Now observe that if and only if for n 2: So, choosing we can easily show that if x 2 I and n 2, then u n (x) 2 [0; 1]: Furthermore, the choices in (3.2) are the best possible with respect to the functions u n (x) in (3.1).With these choices, our new operators are de…ned as follows: where f 2 C[0; 1], x 2 [ 1 4 ; 3  4 ] and n 2: According to this application, observe that and ] and n 2: Then one can say that the convergence to zero, as n !1; of the sequence fB n (f ; x) f (x)g is faster than that of fB n (f ; x) f (x)g on the best possible interval [ 1 4 ; 3  4 ] with respect to the functions u n (x) given by (3.1).We know that the values n (x) and n (x) actually control the rate of the approximation a function by means of B n and B n ; respectively.Therefore, one can say that the error estimation in the approximation by B n is more sensitive than by the classical Bernstein polynomials B n : 3.2.Szász-Mirakjan Operators.Take A = R + and consider the classical Szász-Mirakjan operators where f 2 E(R + ); n 2 N and x 2 R + : Then, by a simple calculation, we get and n 1: Hence, choosing if x 2 I and n 1; then we have u n (x) 2 R + : So, our modi…ed operators are de…ned as follows: where f 2 E(R + ), x 2 [ 1 2 ; 1) and n 2 N: Then, one can say that, for all f 2 E(R + ); the convergence to zero, as n !1; of the sequence fS n (f ; x) f (x)g is faster than that of fS n (f ; x) f (x)g on the best possible interval [ 1 2 ; 1) with respect to the functions u n (x) given by (3.3).

Bernstein-Kantorovich Operators. Consider the classical
Bernstein-Kantorovich operators de…ned by where f 2 C[0; 1]; x 2 A = [0; 1] and n 2 N.In this case, we have if and only if and n 2: Hence, choosing we can easily show that if x 2 I and n 2; then u n (x) 2 [0; 1]: With these choices, our new operators are de…ned as follows: where f 2 C[0; 1], x 2 [ 1 3 ; 2 3 ] and n 2: Then, we conclude that, for all f 2 C[0; 1]; the convergence to zero, as n !1; of the sequence fU n (f ; x) f (x)g is faster than that of fU n (f ; x) f (x)g on the best possible interval [ 1 3 ; 2 3 ] with respect to the functions u n (x) given by (3.4).Remark 3.1.Our new approach can also be applied to other well-known approximation operators.But, we omit the details.

Theorem 1 . 1 .
Let fL n g be a sequence of positive linear operators de…ned on E(R + ) (resp.C(R + b )) satisfying the conditions in (1:1) and (1:2): Then, for all f 2 E(R + ) (resp: for all f 2 C(R + b )); we have lim n!1 L n (f ) = f uniformly on the interval R + b with b > 0: