D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 3 3 IS S N 1 3 0 3 –5 9 9 1
SOME PROPERTIES OF THE GENERALIZED BLEIMANN, BUTZER AND HAHN OPERATORS
DILEK SÖYLEMEZ-ÖZDEN
Abstract. In the present paper, we introduce sequences of Bleimann, Butzer and Hahn operators which are based on a function . This function is a con-tinuously di¤erentiable function on R+;such that (0) = 0; inf 0(x) 1:We
give a Korovkin-type theorem and prove uniform approximation of the general-ized Bleimann, Butzer and Hahn operator. We also investigate the monotonic convergence property of the sequence of the operators under convexity.
1. Introduction
Let, as usual, C[0; 1) denote the space of all continuous and real valued functions de…ned on [0; 1) and CB[0; 1) denote the space of all bounded functions from
C[0; 1): Obviously
kfkCB = sup
x 0jf (x)j
de…nes a norm on CB[0; 1): In [6]; Bleimann, Butzer and Hahn proposed a sequence
of positive linear operators Ln de…ned by
Ln(f ; x) = 1 (1 + x)n n X k=0 f k n k + 1 n k x k; x 0; n 2 N (1.1) for f 2 C[0; 1): Here, the authors proved that Ln(f ; x) ! f (x) as n ! 1
pointwise on [0; 1) when f 2 CB[0; 1): Moreover, the convergence being
uni-form on each compact subset of [0; 1) : In [9], using test functions 1+tt v
for v = 0; 1; 2; Gadjiev and Çakar stated a Korovkin-type theorem for the uniform convergence of functions belonging to some suitable function space by some linear positive operators. As an application of this result, they proved uniform approxi-mation of Bleimann, Butzer and Hahn operators.
Received by the editors: June 06, 2015; Accepted: August 08, 2015.
2010 Mathematics Subject Classi…cation. Primary ; 41A36; Secondary 41A20.
Key words and phrases. Generalized Bleimann, Butzer and Hahn operator; Korovkin-type theorem; convexity; monotonicity property.
c 2 0 1 5 A n ka ra U n ive rsity
Monotonicity properties of the Bleimann, Butzer and Hahn operators was inves-tigated by Della Vecchia in [8]: The operator Ln and its generalizations have been
studied by several authors some are in [1], [4], [11], [2].
In [7], Cárdenas-Moreles, Garrancho and Ra¸sa introduce a new type generaliza-tion of Bernstein polynomials denoted by Bn and de…ned as
Bn(f ; x) : = Bn f 1; (x) (1.2) = n X k=0 n k k(x) (1 (x))n k (f 1)(k n);
where Bn is the n th Bernstein polynomial, f 2 C [0; 1], x 2 [0; 1] and being
any function that is continuously di¤erentiable of in…nite order on [0; 1] such that (0) = 0; (1) = 1 and 0(x) > 0 for x 2 [0; 1] : In this work, the authors stud-ied some shape preserving and convergence properties concerning the generalized Bernstein operators Bn(f ; x) : A similar generalization for Szász-Mirakyan operator was stated in [5] by Aral, Inoan and Ra¸sa by taking as continuously di¤erentiable function on [0; 1); (0) = 0; inf
x2R+
0(x) 1: Here, weighted approximation as well
as the degree of the approximation were obtained. Among other results, they also showed that the sequence of the generalized Szász-Mirakyan operators is monoton-ically nonincreasing under the notion of convexity of the original function.
Now, accordingly, we consider the following generalized Bleimann, Butzer and Hahn operators for f 2 C[0; 1) :
Ln(f ; x) = 1 (1 + (x))n n X k=0 f 1 k n k + 1 n k (x) k ; (1.3)
where is a continuously di¤erentiable function de…ned on [0; 1) such that (0) = 0; inf
x2[0;1)
0(x) 1: (1.4)
An example of such a function is given in [5]. Note that, in the setting of the operator (1.3) we have
Lnf := Ln f 1 ;
where Ln is the n th Bleimann, Butzer and Hahn operator given by (1.1). If
(x) = x then Ln= Ln: Obviously, we have
Ln(1; x) = 1; Ln 1 + ; x = n n + 1 (x) 1 + (x); (1.5) Ln 1 + 2 ; x ! = n (n 1) (n + 1)2 (x) 1 + (x) 2 + n (n + 1)2 (x) 1 + (x):
The …rst purpose of this paper is to extend the results of Gadjiev and Çakar in [9] to the generalized Bleimann, Butzer and Hahn operators (1:3) : In this direction, we …rst give a generalized Korovkin-type theorem to obtain uniform convergence by fLn(f ; x)gn2N to f (x) on [0; 1) for f belonging to some suitable subspace
of continuous and bounded functions that we will denote it by H!. Next, we study the monotonic convergence under the convexity of the function which is approximated.
For this purpose, we de…ne the following class of functions
Let ! be a general functions of modulus of continuity, satisfying the following properties:
(a) ! is a continuous nonnegative increasing function on [0; 1) ; (b) ! ( 1+ 2) ! ( 1) + ! ( 2)
(c) lim !0! ( ) = 0:
Suppose that H!denote the space of all real valued functions f de…ned on [0; 1)
satisfying
jf (x) f (y)j ! (x)
1 + (x)
(y)
1 + (y) (1.6)
for all x; y 2 [0; 1) : It readily follows from (c) that if f 2 H!; then it is continuous
on [0; 1) : Moreover, if f 2 H!; then we have
jf (x)j jf (0)j + ! 1 +(x)(x) jf (0)j + ! (1) ; (x 0) ;
by the assumption on ; which clearly gives that f is bounded on [0; 1) : Therefore we have the following inclusion:
H! CB[0; 1) :
When ! (t) = M t ; 0 < 1; the space of H!will be denoted by H : From (1:6) we get that jf (x) f (y)j M j (x) (y)j (1 + (x)) (1 + (y)) : Hence we reach to H LipM( (x) ; ) ;
where LipM( (x) ; ) ; 0 < 1; M > 0; is the set of all functions f 2 C[0; 1)
satisfying the inequality
jf (t) f (x)j M j (t) (x)j ; x; t 0 (see [10]) :
De…nition 1. A continuous, real valued function f is said to be convex in D [0; 1); if f m X i=1 ixi ! m X i=1 if (xi)
for every x1; x2; :::; xm2 D and for every nonnegative numbers 1; 2; :::; msuch
that 1+ 2+ ::: + m= 1:
In [7] Cárdenas-Morales, Garrancho and Ra¸sa introduced the following de…nition of convexity of a continuous function.
De…nition 2. A continuous, real valued function f is said to be convex in D; if f 1 is convex in the sense of De…nition 1.
Here, we give a Korovkin-type theorem in the sense of Gadjiev and Çakar ([9]) : 2. Main Results
Here, we give a Korovkin-type theorem in the sense of Gadjiev and Çakar ([9]) : Theorem 1. Let fTn(f )gn2Nbe a sequence of linear positive operators from H!to CB[0; 1): If lim n!1 Tn (t) 1 + (t) v ; x (x) 1 + (x) v CB = 0; (2.1)
is satis…ed for v = 0; 1; 2; then for f 2 H! we have
lim
n!1kTn (f ) f kCB = 0:
Proof. Supposing that f 2 H!; we deduce from (1:6) that for any > 0 there exist
a positive > 0 such that
jf (t) f (x)j <
whenever 1+ (t)(t) 1+ (x)(x) < : On the other hand, from the boundedness of f , we get
jf (t) f (x)j <2M2
(t) (x)
(1 + (t)) (1 + (x))
2
when 1+ (t)(t) 1+ (x)(x) : In this case, for all t; x 2 [0; 1) we can write
jf (t) f (x)j < +2M2
(t) (x)
(1 + (t)) (1 + (x))
2
: (2.2)
Since Tn is linear and positive, then applying the operator on f (t) f (x) ; we obtain
kTn(f ; x) f (x)kCB kTn(jf (t) f (x)j ; x)kCB+ kf (x)kCBkTn(1; x) 1kCB = : In1+ In2:
From (2:1) and the boundedness of f; kfkCB M; we get lim
n!1I
2
n = 0:
On the other hand, (2:1) gives that
Tn (t) (x) (1 + (t)) (1 + (x)) 2 ; x ! CB < C n (2.3)
where n ! 0 as n ! 1 and C is a positive constant independent of n.
Moreover, from (2:1) ; (2:2) and (2:3) it follows that
In1 kTn(1; x)kCB+2M2 Tn (t) (x) (1 + (t)) (1 + (x)) 2 ; x ! CB (1 + n) + 2M 2 C n: Hence we deduce lim n!1I 1 n = 0;
which completes the proof.
Theorem 2. Let Ln be the operator de…ned by (1:3) : Then for any f 2 H! we have
lim
n!1kLnf f kCB = 0:
Proof. From Theorem 1 it su¢ ces to show that (2:1) hold for Ln: Indeed, from (1:5) we easily obtain that
Ln(1; x) = 1; (2.4) Ln (t) 1 + (t); x (x) 1 + (x) CB (2.5) = n n + 1 1 (x) 1 + (x) CB 1 n + 1; and Ln (t) 1 + (t) 2 ; x ! (x) 1 + (x) 2 CB n (n 1) (n + 1)2 + n (n + 1)2 + 1 CB 4n + 1 (n + 1)2: (2.6)
Therefore, using (2:4) (2:6) we get that (2:1) holds. By Theorem 1, the proof is completed.
3. Monotonicity Result for Ln
Theorem 3. If f is convex and non-increasing on [0; 1) ; then we have Ln(f ; x) Ln+1(f ; x)
for n 2 N:
Proof. From (1:3),we can write Ln(f ; x) Ln+1(f ; x) = 1 (1 + (x))n+1 n X k=0 f 1 k n k + 1 n k (x) k + 1 (1 + (x))n+1 n X k=0 f 1 k n k + 1 n k (x) k+1 (3.1) 1 (1 + (x))n+1 n+1 X k=0 f 1 k n k + 2 n + 1 k (x) k : (3.2)
removing, n. term and n + 1:term from (3:1) ; (3:2) respectively and taking into account of the fact n+1 kn+1 nk = n+1k ; kn1 = nk n+1 kk , we have
Ln(f ; x) Ln+1(f ; x) = (x) 1 + (x) n+1 f 1 (n) f 1 (n + 1) + 1 (1 + (x))n+1 f 1 (0) f 1 (0) + 1 (1 + (x))n+1 n X k=1 f 1 k n k + 1 n k (x) k + 1 (1 + (x))n+1 n 1 X k=0 f 1 k n k + 1 n k (x) k+1 1 (1 + (x))n+1 n X k=1 f 1 k n k + 2 n + 1 n + 1 k n k (x) k
= (x) 1 + (x) n+1 f 1 (n) f 1 (n + 1) + 1 (1 + (x))n+1 n X k=1 f 1 k n k + 1 n k (x) k + 1 (1 + (x))n+1 n X k=1 f 1 k 1 n k + 2 n k 1 (x) k 1 (1 + (x))n+1 n X k=1 f 1 k n k + 2 n + 1 n + 1 k n k (x) k = (x) 1 + (x) n+1 f 1 (n) f 1 (n + 1) + 1 (1 + (x))n+1 n X k=1 n k (x) k f 1 k n k + 1 + k n + 1 k f 1 k 1 n k + 2 n + 1 n + 1 k f 1 k n k + 2 : By taking 1 = n k+1n+1 0; 2 = n+1k 0; 1+ 2 = 1, and x1 = n k+1k ; x2= n k+2k 1 one has 1x1+ 2x2 = k n + 1+ k n + 1 k 1 n k + 2 = k (n k + 2) + k 2 k (n + 1) (n k + 2) = k n k + 2: Therefore, we obtain that
Ln(f ; x) Ln+1(f ; x) 0 by convexity and non-increasingness of f for x 2 [0; 1).
References
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Address : Ankara University, Elmadag Vocational School, 06780, Elmadag, Ankara, Turkey
E-mail : dsoylemez@ankara.edu.tr
0Ba¸sl¬k: Genelle¸stirilmi¸s Bleimann, Butzer ve Hahn Operatörlerinin Baz¬ Özellikleri
Anahtar Kelimeler: Genelle¸stirilmi¸s Bleimann, Butzer ve Hahn Operatörleri, Korovkin-tipi teorem; konveksli¼gi; monotonluk özelli¼gi
