ˇ GLASNIK MATEMATICKI Vol. 36(56)(2001), 311 – 318
APPROXIMATION BY KANTOROVICH TYPE ¨ GENERALIZATION OF MEYER- KONIG AND ZELLER OPERATORS ¨ ¨n Dog ˘ ru and Nuri Ozalp Ogu Ankara University, Turkey
Abstract. In this study, we define a Kantorovich type generalization of W. Meyer- K¨ onig and K. Zeller operators and we will give the approximation properties of these operators with the help of Korovkin theorems. Then we compute the approximation order by modulus of continuity.
1. Introduction A constructive approach to the characterization of functions is to define and approximate them in terms of some defined positive operators. A large reference list in the study of the subject is cited for the intersted readers ([1]-[20]). Specifically, in [8], we generalized the Meyer - K¨ onig and Zeller operators. In this paper, we make a Kantorovich type generalization of the operators Ln (f ; x) defined in [8] . Then, we give some approximation properties of these operators.
¨ nig and Zeller 2. Kantorovich Type Generalization of Meyer-Ko Operators Let A be a real number in the interval (0, 1). Assume that a sequence of functions {ϕn } satisfies the following conditions: 2000 Mathematics Subject Classification. 41A36. Key words and phrases. Linear positive operators, Korovkin theorem, Modulus of continuity. Supported by Ankara University Resarch Fund Grant Number 99.05.02.01. 311
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i. Every element of the sequence {ϕn } is analytic on a domain D containing the disk B = {z ∈ C : |z| ≤ A} (0)
ii. ϕn (x) = ϕn (x) > 0 (k)
(k−1)
iii. ϕn (x) = γn (k + n) (1 + `n,k ) ϕn (x) , k = 1, 2, ... (k) dk where, ϕn (x) = dx ϕ (x), and ` and γ n n,k n are sequences of numbers satisk fying the conditions; 1 1 , `n,k ≥ 0, γn = 1 + O , γn ≥ 1. `n,k = O n n For 0 < αn,k ≤ 1 (n, k = 1, 2, ...),
consider the sequence of linear positive operators
(2.1)
Mn∗ (f ; x) =
Z k+αn,k ∞ 1 X 1 xk ξ dξ ϕ(k) f n (0) ϕn (x) αn,k k k+n k! k=0
where f is an integrable function on (0, 1). Mn∗ (f ; x) is a Kantorovich type generalization of Meyer - K¨ onig and Zeller operators.
3. Approximation Properties In this section, we give approximation properties of the operator Mn∗ (f ; x) with the help of Korovkin theorem. Theorem 3.1. The sequence of linear positive operators defined by (2.1) with conditions i − iii converges uniformly to the function f ∈ C [0, A] in [0, A] . Proof. It is enough to prove the conditions of Korovkin theorem [12] which are Mn∗ (tk ; x) −→ xk , k = 0, 1, 2 ,
uniformly in [0, A] . First, since
∞ 1 X (k) xk ϕn (0) = 1, ϕn (x) k!
(3.1)
k=0
we get
Mn∗ (1; x)
= 1.
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Second, we can write Mn∗ (t; x) − x
= =
1 ϕn (x) 1 ϕn (x)
∞ P
k=0 ∞ P k=0
k (k+αn,k )2 −k2 (k) ϕn (0) xk! 2(k+n)
(k) xk k k+n ϕn (0) k!
= Ln (t; x) − x +
1 1 2 ϕn (x)
+
∞ P
k=0
−x
1 1 2 ϕn (x)
∞ P
αn,k (k) xk k+n ϕn (0) k!
k=0 αn,k (k) xk k+n ϕn (0) k!
−x
where Ln (t; x) is a generalization of Meyer- K¨ onig and Zeller operators defined, in general, as ∞ k 1 X xk f( Ln (f ; x) = )ϕ(k) (0) ϕn (x) k+n n k! k=0
(see [8]). Using Ln (t; x) − x ≥ 0, we obtain Mn∗ (t; x) − x ≥ 0. αn,k On the other hand, since k+n ≤ n1 we get
1 . 2n In [8] , it is shown that lim (Ln (t; x) − x) = 0. Therefore, we obtain 0 ≤ Mn∗ (t; x) − x ≤ Ln (t; x) − x + n→∞
lim (Mn∗ (t; x) − x) = 0, and thus the second condition of Korovkin theorem n→∞ is hold. Finally, using iii in Mn∗ (t2 ; x),we obtain ∞ P k·αn,k (k) xk Mn∗ (t2 ; x) = Ln (t2 ; x) + ϕn1(x) (k+n)2 ϕn (0) k! k=0
+ 3ϕn1(x) Thus we can write Mn∗ (t2 ; x) − x2 Now,
∞ P
k=0
= Ln (t2 ; x) − x2 + ≥ 0.
α2n,k (k) xk (k+n)2 ϕn (0) k! .
1 ϕn (x)
∞ h P k·αn,k
k=0
(k+n)2
+
α2n,k 3(k+n)2
i
(k)
k
ϕn (0) xk!
Ln (t2 ; x) − x2 ≥ 0
( See [8] ). In addition, since holds:
αn,k k+n
≤
1 n
and
α2n,k (k+n)2
≤
Mn∗ (t2 ; x) − x2 ≤ (Ln (t2 ; x) − x2 ) +
1 n2
, the following inequality
1 1 Ln (t; x) + 2 . n 3n
On the other hand Ln (t; x) −→ x , Ln (t2 ; x) −→ x2
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(See [8] ). Thus Mn∗ (t2 ; x) −→ x2 . Because of Korovkin theorem, the proof is complete.
4. The Order of Approximation In this section, we compute the approximation order of Ln (f ; x) defined in [8] and Mn∗ (f ; x) with the help of modulus of continuity which is (4.1)
ω(f ; δ) = sup {|f (t) − f (x)| ; |t − x| ≤ δ, t, x ∈ [0, A]} .
The following theorems are analogous to one given in [7] for GadjievIbragimov operators. Theorem 4.1. The following inequality holds: 1 kLn (f ; x) − f (x)kC[0,A] ≤ C ω(f ; √ ) n where ω(f ; √1n ) is a modulus of continuity defined by (4 .1 ), and C is a positive constant. Proof. From well-known properties of ω(f ; δ), we can write 1 2 2 1/2 (4.2) |Ln (f ; x) − f (x)| ≤ ω(f ; δ) (Ln (t ; x) − 2xLn (t; x) + x ) + 1 δ If we use (3.1) and inequalities Ln (t; x) − x ≤ (γn − 1)x +
dxγn , n
x2 γn2 d2 + xγn d 2dx2 γn2 + xγn + , n n2 (see[8]) in (4.2) and making the simplifications, we obtain, h√ i |Ln (f ; x) − f (x)| ≤ ω(f ; δ) δ5 O( √1n ) + 1 Ln (t2 ; x) − x2 ≤ x2 (γn2 − 1) +
≤ ω(f ; δ)
h
C1 1√ δ n
i +1 .
By choosing δ = √1n , we obtain the desired result. Theorem 4.2. Let f be uniform continuous on [0, 1] . Then, the sequence of linear positive operators defined by (2 .1 ) under the conditions i−iii, satisfies the inequality 1 |Mn∗ (f ; x) − f (x)| ≤ Cω(f ; √ ). n
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Proof. By using the triangle inequality, we can write (4.3) k+α Z n,k ∞ ξ xk 1 X 1 ∗ f dξ ϕ(k) − f (x) |Mn (f ; x) − f (x)| ≤ n (0) ϕn (x) αn,k k+n k! k=0
k
On the other hand, since
ξ − x k+n ξ f − f (x) ≤ + 1 ω(f ; δ) k+n δ
we get k+α Z n,k k
k+αn,k R ξ k+n − x dξ ξ k f + αn,k ω(f ; δ). − f (x) dξ ≤ k+n δ
Using the last inequality and (3.1) in (4.3), we obtain |Mn∗ (f ; x) − f (x)| ≤ ) # ( " k+α ∞ R n,k ξ P (k) 1 1 xk 1 ≤ δ ϕn (x) k+n − x dξ ϕn (0) k! + 1 ω(f ; δ) αn,k k=0
k
∞ P ≤ 1δ ϕn1(x)
k=0
≤ ≤
h
h
1 δ
1 αn,k
k
Ln (t2 ; x) + n2 Ln (t; x) +
1 √1 δ C1 O( n )
If we choose δ =
ξ k+n − x dξ
k+α R n,k
1 2n2
!2
12
k (k) ϕn (0) xk! + 1 ω(f ; δ)
+ 2xLn (t; x) +
x n
− 3x2
i + 1 ω(f ; δ). √1 , n
21
i + 1 ω(f ; δ)
then the proof is completed.
5. A Differential Equation We refer to some papers, in which equations analogous to the following theorem are given: May [14] , Volkov [20] , Alkemade [2] and [8] . Let g satisfy the equality g(
k ak )= k+n bn
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k where a and b are arbitrary nonzero constant. Hence letting n = − k, we t at obtain g(t) = . b(1 − t) Now we give the following theorem. Theorem 5.1. Let g(t) =
at b(1 − t)
(t ∈ [0, A] , a, b 6= 0).
For each x ∈ [0, A] , f ∈ C [0, A] and αn,k = 1; Mn∗ (f ; x) satisfies the following functional differential equation for n = 2, 3, ... : h i d 1 x dx Mn∗ (f ; x) = −γn (1 + n)(1 + `n,1 )x + ln( n−1 − n Mn∗ (f ; x) n ) (5.1) − a ln(bn−1 ) Mn∗ ((f, g); x). n
where Mn∗ ((f,
k+1 Z Z ∞ k+1 xk ξ 1 X ξ dξ dξ ϕ(k) (0) g . g); x) = f n ϕn (x) k+n k+n k! k=0 k
k
Proof. Since Mn∗ (f ; x) is uniform convergent on [0, A] , we can differentiate this series term by term in this interval. Hence, ∞ Z −γn (1 + n)(1 + `n,1 )x X d ∗ xk ξ x Mn (f ; x) = dξ ϕ(k) (0) f n dx ϕn (x) k+n k! k+1
k=0 k
−γn (1+n)(1+`n,1 )x ϕn (x)
d x dx Mn∗ (f ; x) =
(5.2) + ϕn1(x)
∞ k+1 R P
f
k=0 k
∞ k+1 R P
f
k=0 k
ξ k+n
k=
k+1 R k a b
g
ξ k+n
ln( n−1 n )
ξ k+n
(k)
(k)
dξ − n.
k
dξ ϕn (0) xk! k
dξ ϕn (0)k xk! .
It can be shown that a b−
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By using this equality in (5.2), after simplification, we get i h 1 1 d − n Mn∗ (f ; x) = −γn (1 + n)(1 + `n,1 )x + ln( n−1 x dx ϕn (x) ) n
× ×
∞ k+1 R P
f
f
k=0 k
∞ k+1 R P
k=0 k
ξ k+n
dξ ϕn (0) xk! −
dξ
ξ k+n
k
(k)
k+1 R k
g
ξ k+n
a b
1 1 ϕn (x) ln( n−1 n )
(k)
k
dξ ϕn (0) xk! .
Thus the theorem is proved. Proposition 5.1. For αn,k = 1, Mn∗ (t; x) gives a solution of the differential equation a+b (5.3) xy 0 − − γn (1 + n)(1 + `n,1 )x + n−1 − n y = a ln( n ) 1 γn (1 + n)(1 + `n,1 )x − + n. ln( n−1 n ) Proof. Setting, in (5.1), f = 1 − t, it follows that 1 d ∗ x Mn (1 − t; x) = −γn (1 + n)(1 + `n,1 )x + − n Mn∗ (1 − t; x) dx ln( n−1 n ) (5.4)
−
b Mn∗ (t; x). ) a ln( n−1 n
Using the linearity of Mn∗ and Mn∗ (1; x) = 1 in (5.4) we get (5.3). Acknowledgements. Authors would like to thank to the referee for the valuable suggestions and remarks. References [1] U. Abel, The Moments for the Meyer-K¨ onig and Zeller Operators, J. Approx. Theory 82 (1995), 352-361. [2] J. A. H. Alkemade, The Second Moment for the Meyer- K¨ onig and Zeller Operators, J. Approx. Theory 40 (1984), 261-273. [3] O. Arama, Properties Concerning the Monotonicity of the Sequence of Polynomials of Interpolation of S. N. Bernstein and Their Application to the Study of Approximation of Functions, Studii si Cercetari de Matematica A.R.P.R., 8 (1957), 195-210. [4] M. Becker and R. J. Nessel, A Global Approximation Theorem for Meyer-K¨ onig and Zeller Operators, Math. Z. 160 (1978), 195-206. [5] E. W. Cheney and A. Sharma, Bernstein Power Series, Canad. J. Math. 16 (1964), 241-253. [6] O. Do˘ gru, On a Certain Family of Linear Positive Operators, Tr. J. of Mathematics 21 (1997), 387-399.
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[7] O. Do˘ gru, On the Order of Approximation of Unbounded Functions by the Family of Generalized Linear Positive Operators, Commun. Fac. Sci. Univ. Ank. Series A1, 46 (1997), 173-181. [8] O. Do˘ gru, Approximation Order and Asymptotic Approximation for Generalized Meyer-K¨ onig and Zeller Operators, Math. Balkanica, New Ser. Vol.12, Fasc. 3-4 (1998), 359-368. [9] O. Do˘ gru, On Weighted Approximation of Continuous Functions by Linear Positive Operators on Infinite Intervals, Math. Cluj Vol. 41(64), No:1 (1999) 39–46. [10] A. D. Gadjiev and I. I. Ibragimov, On a Sequence of Linear Positive Operators, Soviet Math., Dokl. 11 (1970), 1092-1095. [11] A. D. Gadjiev and H. H. Hacısaliho˘ glu, Approximation of Sequences of Linear Positive Operators, Ankara, 1995. [12] P. P. Korovkin, Linear Operators and Approximation Theory, Delhi, 1960. [13] G. G. Lorentz, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966. [14] C. P. May, Saturation and Inverse Theorems for Combinations of a Class of Exponential-Type Operators, Canad. J. Math. 28 (1976), 1224-1250. [15] W. Meyer-K¨ onigand K. Zeller, Bernsteinsche Potenzreihen, Studia Math., 19 (1960), 89-94. [16] M. W. M¨ uller, Die Folge der Gammaoperatoren, Dissertation, Stuttgart, 1967. [17] O. Shisha and B. Mond, The Degree of Convergence of Sequences of Linear Positive Operators, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1196-1200. [18] P. C. Sikkema, On the Asymptotic Approximation with operators of Meyer-K¨ onigand Zeller, Indag. Math. 32 (1970), 428-440. [19] V. Totik, Approximation by Meyer-K¨ onigand Zeller type Operators, Math. Z. 182 (1983), 425-446. [20] Yu. I. Volkov, Certain Positive Linear Operators, Mat. Zametki 23 (1978), 363-368. Ankara University Faculty of Science Department of Mathematics 06100 Tando˘ gan Ankara - Turkey E-mail :
[email protected] E-mail :
[email protected] Received : 12.06.2000. Revised : 03.11.2000.
