ON THE RATES OF CONVERGENCE OF

Georgian Mathematical Journal Volume 16 (2009), Number 4, 693–704

ON THE RATES OF CONVERGENCE OF CHLODOVSKY–DURRMEYER OPERATORS AND THEIR ´ BEZIER VARIANT HARUN KARSLI AND PAULINA PYCH-TABERSKA

Abstract. We consider the Bézier variant of Chlodovsky–Durrmeyer operators Dn,α for functions f measurable and locally bounded on the interval [0, ∞). By using the Chanturia1 modulus of variation we estimate the rate of pointwise convergence of (Dn,α f ) (x) at those x > 0 at which the one-sided limits f (x+), f (x−) exist. In the special case α = 1 the recent result of [14] concerning the Chlodovsky–Durrmeyer operators Dn is essentially improved and extended to more general classes of functions. 2000 Mathematics Subject Classification: 41A25, 41A35, 41A36. Key words and phrases: Rate of convergence, Chlodovsky–Durrmeyer operator, Bézier basis, Chanturia modulus of variation, p-th power variation.

1. Introduction It is well-known that the classical Bernstein–Durrmeyer operators (see [8]) Mn applied to f are defined as (Mn f ) (x) = (n + 1)

n X

Z1 f (t)Pn,k (t) dt, 0 ≤ x ≤ 1,

Pn,k (x)

k=0

(1)

0

µ ¶ n k where Pn,k (x) = x (1−x)n−k . Several researchers have studied approximak tion properties of the operators Mn (see, e.g., [7], [10]) for functions of bounded variation defined on the interval [0, 1]. Not long ago Zeng and Chen [22] defined the Bézier variant of Durrmeyer operators as (Mn,α f ) (x) = (n + 1)

n X k=0

(α)

Z1 (α) Qn,k

(x)

f (t)Pn,k (t) dt, 0

α α where Qn,k (x) = Jn,k (x)−Jn,k+1 (x) and Jn,k (x) =

n P

Pn,j (x) for k = 0, 1, . . . , n,

j=k

Jn,n+1 (x) = 0 are the Bézier basis functions (introduced by P. Bézier [4]), and estimated the rate of convergence of Mn,α f for functions of bounded variation on the interval [0, 1]. 1

The proper form of the family name sometimes written as Chanturiya.

c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

694

H. KARSLI AND P. PYCH-TABERSKA

Let Xloc [0, ∞) be the class of all complex valued functions measurable and locally bounded on the interval [0, ∞). For f ∈ Xloc [0, ∞) the Chlodovsky– Durrmeyer operators Dn are defined as µ ¶ Zbn µ ¶ n n+1X x t (Dn f ) (x) = Pn,k f (t)Pn,k dt, 0 ≤ x ≤ bn , (2) bn k=0 bn bn 0

where (bn ) is a positive increasing sequence with the properties bn lim bn = ∞ and lim = 0. (3) n→∞ n→∞ n Ibikli and Karsli [14] estimated the rate of convergence of Chlodovsky–Durrmeyer operators for functions in BV [0, ∞). We should also mention the recent articles which devoted to the investigation of Chlodovsky operators (for a Voronovskaya type theorem see an earlier paper by Albrycht and Radecki [2]), the Bézier variant of Chlodovsky operators and Chlodovsky–Durrmeyer operators for functions in BV [0, ∞) and DBV [0, ∞) are due to Karsli [15], Karsli and Ibikli [16, 17], Butzer and Karsli [5]. For f ∈ Xloc [0, ∞) and α ≥ 1, we introduce the Bézier variant of Chlodovsky– Durrmeyer operators Dn,α as follows: n

n + 1 X (α) Q (Dn,α f ) (x) = bn k=0 n,k

µ

x bn

¶ Zbn

µ f (t)Pn,k

0

t bn

¶ dt,

0 ≤ x ≤ bn .

(4)

Obviously, Dn,α is a positive linear operator and (Dn,α 1) (x) = 1. In particular, when α = 1 the operators (4) reduce to the operators (2). Recently Agratini [1], Aniol and Pych-Taberska [3], Pych-Taberska [21] and Gupta [11, 12] have investigated the rate of pointwise convergence for Kantorovich and Durrmeyer type Baskakov–Bézier and Bézier operators using a different approach. They have proved their theorems in terms of the Chanturia modulus of variation, which is a generalization of the classical Jordan variation. It is useful to point out that a deeper analysis of the Chanturia modulus of variation can be found in [6], but actually the modulus of variation was introduced for the first time by Lagrange [18]. Although the Chanturia modulus of variation was defined as a generalization of the classical variation nearly four decades years ago, it was not used to a sufficient extent to solve the problem mentioned above. The paper is concerned with the rate of pointwise convergence of the operators (4) when f belong to Xloc [0, ∞). Using the Chanturia modulus of variation defined in [6], we examine the rate of pointwise convergence of (Dn,α f ) (x) at the points of continuity and at the first kind discontinuity points of f . For some important papers on different operators related to the present study we refer the readers to Gupta et al. [9, 20] and Zeng and Piriou [23]. It is necessary to point out that in the present paper we extend and improve the result of [14] for Chlodovsky–Durrmeyer operators. We begin by giving

ON THE RATES OF CONVERGENCE

695

Definition 1. Let f be a bounded function on a compact interval I = [a, b]. The modulus of variation νn (f ; [a, b]) of a function f is defined for nonnegative integers n as ν0 (f ; [a, b]) = 0 and for n ≥ 1 as n−1 X νn (f ; [a, b]) = sup |f (x2k+1 ) − f (x2k )| , Πn

k=0

where Πn is an arbitrary system of n disjoint intervals (x2k , x2k+1 ), k = 0, 1, . . . , n − 1, i.e., a ≤ x0 < x1 ≤ x2 < x3 ≤ · · · ≤ x2n−2 < x2n−1 ≤ b. The modulus of variation of any function is a non-decreasing function of n. Some other properties of this modulus can be found in [6]. If f ∈ BVp (I), p ≥ 1, i.e., if f is of p-th bounded power variation on I, then for every k ∈ N, νk (f ; I) ≤ k 1−1/p Vp (f, I), (5) where Vp (f, I) denotes the total p-th power variation of f on I, defined as ³P ´1/p the upper bound of the set of numbers |f (kj ) − f (lj )|p over all finite j

systems of non-overlapping intervals (kj , lj ) ⊂ I. p We also consider the class BVloc [0, ∞), p ≥ 1, consisting of all functions of bounded p-th power variation on every compact interval I ⊂ [0, ∞). In the sequel it will be always assumed that the sequence (bn ) satisfies the fundamental conditions (3). The symbol [a] will denote the greatest integer not greater than a. Theorem 1. Let f ∈ Xloc [0, ∞) and let the one-sided limits f (x+), f (x−) exist at a fixed point x ∈ (0, ∞). Then, for all integers n such that bn > 2x and 4bn ≤ n one has ¯ ¯ p ¯ ¯ f (x+) + αf (x−) ¯(Dn,α f ) (x) − ¯ ≤ 2v1 (gx ; Hx (x bn /n)) ¯ ¯ α+1 # p ¶ "m−1 µ µ ¶ X vj (gx ; Hx (jx bn /n)) vm (gx ; Hx (x)) 32α x bn + 2 x 1− + + x bn n j3 m2 j=1 µ ¶q µ µ ¶ ¶q bn x bn 2α |f (x+) − f (x−)| 2αcq q x 1− + + + 2q µ(bn ; f ) , nx x x n bn n (1 − ) bn bn p where m := [ n/bn ], Hx (u) = [x−u, x+u] for 0 ≤ u ≤ x, µ(b; f ) := sup |f (t)| ,   f (t) − f (x+) if t > x, gx (t) := 0 if t = x,  f (t) − f (x−) if 0 ≤ t < x,

0≤t≤b

(6)

q is an arbitrary positive integer and cq is a positive constant depending only on q.

696


From Theorem 1 and inequality (5) we get p Theorem 2. Let f ∈ BVloc [0, ∞), p ≥ 1, and let x ∈ (0, ∞). Then, for all integers n such that bn > 2x and 4bn ≤ n we have ¯ ¯ p ¯ ¯ f (x+) + αf (x−) ¯(Dn,α f ) (x) − ¯ ≤ 2Vp (gx ; Hx (x bn /n)) ¯ ¯ α+1 2 ¶ (m+1) µ µ ¶ X −1 Vp (gx ; Hx ( √xk )) x bn √ x 1− + bn n ( k)1−1/p k=1 µ ¶q µ µ ¶ ¶q 2αcq bn x bn 2α |f (x+) − f (x−)| q + 2q µ(bn ; f ) x 1− + + . nx x n bn n (1 − x )

27+1/p α + 2 1+1/p xm

bn

bn

Note that Theorems 1 and 2 are approximation theorems. In order to show this it is necessary to prove that the right-hand sides of the inequalities mentioned in the hypotheses hp iof the theorems tend to zero as n → ∞. In view of (3) we have m = n/bn → ∞ as n → ∞. So, in Theorem 1 it is enough to consider only the term Λm (x) =

m−1 X j=1

vj (gx ; Hx (jxdn )) , j3

where dn =

p

bn /n.

Clearly, Λm (x) ≤

m−1 X j=1

v1 (gx ; Hx (jxdn )) ≤ 4dn j2

mdn Z

v1 (gx ; Hx (xt)) dt t2

dn m+1 Z

≤ 4dn 1

m ³ x ´´ ³ ³ x ´´ 4 X ³ ds ≤ v1 gx ; Hx . v1 g x ; H x s m k=1 k

Since the function gx is continuous at x and v1 (gx ; Hx ( xk )) denotes the oscillation of gx on the interval Hx ( xk ), we have ³ ³ x ´´ =0 lim v1 gx ; Hx k→∞ k and, consequently, lim Λm (x) = 0. m→∞

As regards Theorem 2, it is easy to verify that in view of the continuity of gx at x, 2 −1 µ µ ¶¶ m X x 1 1 √ Vp gx ; Hx √ lim = 0. 1−1/p m→∞ m1+1/p ( k) k k=1 Thus we get the following approximation theorem.


697

p Corollary. Suppose that f ∈ Xloc [0, ∞) (in particular, f ∈ BVloc [0, ∞), p ≥ 1) and that there exists a positive integer q such that µ ¶q bn lim µ(bn ; f ) = 0. n→∞ n

Then at every point x ∈ (0, ∞) at which the limits f (x+), f (x−) exist we have lim (Dn,α f ) (x) =

n→∞

f (x+) + αf (x−) . α+1

Obviously, the above relations hold true for every measurable function f bounded on [0, ∞), in particular for every function f of bounded p-th power variation (p ≥ 1) on the whole interval [0, ∞). Remark. Now, let us consider the special case α = 1, p = 1, and let us suppose that a function f is of bounded variation in the Jordan sense on the whole interval [0, ∞) (f ∈ BV [0, ∞)). Then, for all integers n such that bn > 2x and 4bn ≤ n, we have the following estimation for the rate of convergence of the Chlodovsky–Durrmeyer operators (2): ¯ ¯ p ¯ ¯ f (x+) + f (x−) ¯ ≤ 2V (gx ; Hx (x bn /n)) ¯(Dn f ) (x) − ¯ ¯ 2 µ µ ¶ ¶ 2[n/bn ] µ ¶¶ µ 210 bn x bn X x + x 1− + V gx ; Hx √ nx2 bn n k k=1 µ µ ¶ ¶ 4M bn x 2 |f (x+) − f (x−)| bn + , x 1− + q + 2 nx bn n n bxn (1 − bxn ) where M = sup |f (x)| and V (gx ; H) denotes the Jordan variation of gx on 0≤x 1, µ Wn,2q (x) =

bn n

¶q X q

µ µ ¶¶q−j µ ¶j x bn βj,q x 1 − , bn n j=0

(8)

where βj,q are real numbers independent of x and bounded uniformly in n. Moreover, for n ≥ 2 µ µ ¶ ¶ bn x bn x 1− + (9) Wn,2 (x) ≤ 2 n bn n and, for q > 1, µ Wn,2q (x) ≤ cq

bn n

¶q µ µ ¶ ¶q x bn x 1− + , bn n

(10)

where cq is a positive constant depending only on q. Proof. Formulas for Wn,0 , Wn,1 , Wn,2 and inequality (9) follow by simple calculation. Suppose q > 1 and put y := x/bn . Then y ∈ [0, 1] and n

n+1X Pn,k (y) Wn,2q (x) = bn k=0 = (n +

1) b2q n

n X

Z1 Pn,k (y)

k=0

µ

Zbn 2q

(t − ybn ) Pn,k 0

t bn

¶ dt

¡ ¢ 2q (s − y)2q Pn,k (s) ds = b2q M ψ (y) , n n y

(11)

0

where Mn is the classical Bernstein–Durrmeyer operator (1). The representation formula (8) follows at once from the known identity µ ¶q−j q X ¡ ¢ y(1 − y) 2q Mn ψy (y) = n−2j , βj,q,n n j=0 where βj,q,n are real numbers independent of y and bounded uniformly £ 1in ¤ n (see £[13], Lemma 4.8 with c = −1). Now, let us observe that for y ∈ 0, or n ¤ 1 n−1 y ∈ 1 − n , 1 , n ≥ 2, one has y(1 − y) ≤ n2 and ¡

Mn ψy2q

¢

(y) ≤

q X j=0

µ |βj,q,n |

n−1 n3

¶q−j −2j

n

≤

q X j=0

ηj,q n−2q ,


699

£1

¤ where ηj,q are non-negative numbers depending only on j and q. If y ∈ n , 1 − n1 1 n then ny(1−y) ≤ n−1 ≤ 2 and µ µ ¶q q ¶q q ¡ ¢ y(1 − y) X y(1 − y) X 1 2q Mn ψy (y) ≤ |βj,q,n | ηj,q 2j . j ≤ n n (ny(1 − y)) j=0 j=0 Consequently, ¡ ¢ cq Mn ψy2q (y) ≤ q n

µ

1 y(1 − y) + n

¶q with cq =

q X

ηj,q 2j .

j=0

Taking in (11) and in the above inequality y = x/bn we easily obtain estimation (10). ¤ Lemma 2. Let x ∈ (0, ∞) and let µ ¶ µ ¶ µ ¶ n x t n + 1 X (α) x t Kn,α , := Qn,k Pn,k . bn bn bn k=0 bn bn Then

µ

Zbn Kn,α t

and

µ

Zt Kn,α 0

x u , bn bn

x u , bn bn

¶ du ≤

¶

α Wn.2 (x) if (t − x)2

α Wn.2 (x) if (x − t)2

du ≤

x 0, we write Ix (u) := [x + u, x] ∩ [0, ∞) if u < 0 Ix (u) := [x, x + u]

if u > 0.

Lemma 3. Let f ∈ Xloc [0, ∞) and let the one-sided limits f (x+), f (x−) exist at a fixed p point x ∈ (0, ∞). Consider the function gx defined by (6) and write dn := bn /n. If h = −x or h = x, then for all integers n such that dn ≤ 1/2 we have

700


¯ Z µ ¶ ¯¯ ¯ x t ¯ ¯ gx (t)Kn,α , dt¯ ≤ v1 (gx ; Ix (hdn )) ¯ ¯ ¯ bn bn Ix (h)

"m−1 # 8αWn,2 (x) X vj (gx ; Ix (jhdn )) vm (gx ; Ix (h)) , + + 3 2 h2 d2n j m j=1

where m = [1/dn ] and Wn,2 (x) is estimated in (9). Proof. Restricting the proof to h = −x we define the points tj = x + jhdn for j = 0, 1, 2, 3, . . . , m and we denote tm+1 = 0. Put Tj = [tj , x] for j = 1, 2, 3, . . . , m + 1 and write µ

Z gx (t)Kn,α Ix (h)

+

m X

gx (tj )

Kn,α

tj+1

µ

Zx dt =

gx (t)Kn,α t1

µ

Ztj

j=1

¶

x t , bn bn

x t , bn bn

¶ dt +

t m Zj X j=1 t

x t , bn bn

¶ dt µ

(gx (t) − gx (tj )) Kn,α

x t , bn bn

¶ dt

j+1

=: I1 (n, x) + I2 (n, x) + I3 (n, x), say. Clearly, µ

Zx |I1 (n, x)| ≤

|gx (t) − gx (x)| Kn,α t1

µ

Zbn ≤ v1 (gx ; T1 )

Kn,α 0

x t , bn bn

x t , bn bn

¶ dt

¶ dt = v1 (gx ; T1 ).

By the Abel lemma on summation by parts and by (13) we have |I2 (n, x)| tj+1 µ ¶ Z x t Kn,α |gx (tj+1 ) − gx (tj )| ≤ |gx (t1 )| Kn,α dt + , dt bn bn j=1 0 0 Ã ! m−1 X 1 αWn,2 (x) |gx (tj+1 ) − gx (tj )| ≤ |gx (t1 ) − gx (x)| + h2 d2n (j + 1)2 j=1 Ã µ j m−2 XX 1 αWn,2 (x) |gx (t1 ) − gx (x)| + |gx (tk+1 ) − gx (tk )| = 2 2 h dn (j + 1)2 j=1 k=1 ! ¶ m 1 1 X − + 2 |gx (tk+1 ) − gx (tk )| (j + 2)2 m k=1

Zt1

µ

x t , bn bn

¶

m−1 X


701

m−2 X

vj+1 (gx ; Tj+1 ) vm (gx ; Tm ) + 3 m2 (j + 1) j=1 Ã m−1 ! X vj (gx ; Tj ) vm (gx ; Tm+1 ) αWn,2 (x) ≤ 2 + . h2 d2n j3 m2 j=1

≤

αWn,2 (x) h2 d2n

Ã

v1 (gx ; T1 ) + 2

!

Next, in view of (13) and the Abel transformation, |I3 (n, x)| ≤

m X

v1 (gx ; [tj+1, tj ])

j=1

≤

µ

Ztj Kn,α

tj+1

x t , bn bn

¶ dt

m αWn,2 (x) X v1 (gx ; [tj+1, tj ])

h2 d2n

αWn,2 (x) = h2 d2n

j=1

j2

Ã m X v1 (gx ; [tk+1, tk ]) Ã

k=1

m2

+

j m−1 XX

µ v1 (gx ; [tk+1, tk ])

j=1 k=1

1 1 − 2 j (j + 1)2

¶!

! m−1 X vj (gx ; Tj+1 ) vm (gx ; Tm+1 ) +6 m2 (j + 1)3 j=1 ! Ã m X vj (gx ; Tj ) αWn,2 (x) vm (gx ; Tm+1 ) +6 . ≤ h2 d2n m2 j3 j=2 αWn,2 (x) ≤ h2 d2n

Combining the results and observing that Tj = Ix (jhdn ) we get the desired estimation for h = −x. In the case of h = x the proof runs analogously; we use inequality (12) instead of (13). ¤ 3. Proof of the Main Results Now we are ready to prove our main theorems. Proof of Theorem 1. We decompose f (t) into four parts as · ¸ f (x+) + αf (x−) f (x+) − f (x−) α−1 f (t) = + sgnx (t) + α+1 2 α+1 · ¸ f (x+) + f (x−) + gx (t) + δx (t) f (x) − , 2 where gx (t) is defined as (6) and sgnx (t) := sgn(t − x), ½ 1, x = t, δx (t) = 0, x 6= t. From (14) we have f (x+) + αf (x−) f (x+) − f (x−) (Dn,α f ) (x) = + (Dn,α gx ) (x) + α + 1¸ · · ¸ 2 α−1 f (x+) + f (x−) × (Dn,α sgnx ) (x) + + f (x) − (Dn,α δx ) (x). α+1 2

(14)

(15)

702


For operators Dn,α , using (15) we can observe that the last term on the right hand side vanishes. In addition it is obvious that (Dn,α 1) (x) = 1. Hence we have ¯ ¯ ¯ ¯ f (x+) + αf (x−) ¯(Dn,α f ) (x) − ¯ ≤ |(Dn,α gx ) (x)| ¯ ¯ α+1 ¯ ¯¯ ¯ ¯ f (x+) − f (x−) ¯ ¯ α − 1 ¯¯ ¯ ¯ ¯ +¯ ¯ ¯(Dn,α sgnx ) (x) + α + 1 ¯ . (16) 2 In order to prove our theorem we need the estimates for (Dn,α gx ) (x) and α−1 . (Dn,α sgnx ) (x) + α+1 To estimate (Dn,α gx ) (x), with the help of the fixed points x and 2x, we decompose it into three parts as follows: ¯ Zbn µ µ ¶ ¯¯ ¯¯ Zx ¶ ¯¯ ¯ x x t t ¯ ¯ ¯ ¯ , dt¯ ≤ ¯ gx (t)Kn,α , dt¯ ¯ gx (t)Kn,α ¯ ¯ ¯ ¯ bn bn bn bn 0

0

¯ Z2x µ ¶ ¯¯ ¯¯ Zbn ¶ ¯¯ µ ¯ x t x t ¯ ¯ ¯ ¯ + ¯ gx (t)Kn,α , dt¯ + ¯ gx (t)Kn,α , dt¯ ¯ ¯ ¯ ¯ bn bn bn bn x

2x

=: |E1,α (n, x)| + |E2,α (n, x)| + |E3,α (n, x)| , (17) where Kn,α ( bxn , btn ) is defined in Lemma 2. The estimations for |E1,α (n, x)| and |E2,α (n, x)| are given in Lemma 3 in which we put h = −x and h = x, respectively. Using the obvious inequality vj (gx ; Ix (−u)) + vj (gx ; Ix (u)) ≤ 2vj (gx ; Hx (u)), where Hx (u) = [x − u, x + u], 0 < u ≤ x, we obtain p |E1,α (n, x)| + |E2,α (n, x)| ≤ 2v1 (gx ; Hx (x bn /n)) "m−1 # p 16αWn,2 (x)n X vj (gx ; Hx (jx bn /n)) vm (gx ; Hx (x)) + + . (18) 3 2 x2 b n j m j=1 Now, we estimate |E3,α (n, x)| . Clearly, given any q ∈ N, we have µ ¶ Zbn µ ¶ n n + 1 X (α) x t |E3,α (n, x)| ≤ 2µ(bn ; f ) Qn,k Pn,k dt bn k=0 bn bn 2x

n

n + 1 X (α) Q ≤ 2µ(bn ; f ) 2q x bn k=0 n,k n

µ

2αµ(bn ; f ) n + 1 X ≤ Pn,k x2q bn k=0 =

2αµ(bn ; f ) Wn,2q (x). x2q

x bn

µ

¶ Zbn

x bn

µ 2q

(t − x) Pn,k 2x

¶ Zbn

µ 2q

(t − x) Pn,k 0

t bn

¶

t bn

dt ¶ dt (19)


703

Finally, replacing x by bxn in the result of X. M. Zeng and W. Chen [22] (Sect. 3, pp. 9–11) we immediately get ¯ ¯ ¯ ¯ ¯(Dn,α sgnx ) (x) + α − 1 ¯ ≤ q 4α . (20) ¯ α + 1¯ n bxn (1 − bxn ) Putting (17), (18), (19) and (20) into (16), we get the required result. Thus the proof of Theorem 1 is complete. ¤ p Proofp of Theorem 2.pLet f ∈ BVloc [0, ∞), p ≥ 1. In view of (5) and the notation dn = bn /n, m = [ n/bn ], we have m−1 X j=1

mdn Z m−1 vj (gx ; Hx (jxdn )) X Vp (gx ; Hx (jxdn )) Vp (gx ; Hx (xt)) 1+1/p ≤ ≤ (2dn ) dt 3 2+1/p 2+1/p j j t j=1

µ ≤

dn

2 m

¶1+1/p

2 (m+1) Z

1

and

2 √ µ ¶1+1/p (m+1) √ X −1 Vp (gx ; Hx (x/ k) 2 Vp (gx ; Hx (x/ s) ds ≤ ³√ ´1−1/p √ 1−1/p m ( s) k=1 k

vm (gx ; Hx (x)) Vp (gx ; Hx (x)) ≤ . 2 m m1+1/p

Moreover,

p p v1 (gx ; Hx (x bn /n)) ≤ Vp (gx ; Hx (x bn /n)). The estimation given in Theorem 2 now immediately follows from Theorem 1. ¤ References 1. O. Agratini, An approximation process of Kantorovich type. Math. Notes (Miskolc) 2(2001), No. 1, 3–10. 2. J. Albrycht and J. Radecki, On a generalization of the theorem of Voronovskaya. Zeszyty Nauk. Uniw. Mickiewicza No. 25 (1960), 3–7. 3. G. Aniol and P. Pych-Taberska, Some properties of the Bezier-Durrmeyer type operators. Comment. Math. Prace Mat. 41(2001), 1–11. ´zier, Numerical Control Mathematics and Applications, Wiley, London, 1972. 4. P. Be 5. P. L. Butzer and H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein–Chlodovsky polynomials and the Szász–Mirakyan operator. Comment. Math. 49(2009), No.1, 33–57. 6. Z. A. Chanturiya, The moduli of variation of a function and its applications in the theory of Fourier series. (Russian) Dokl. Akad. Nauk SSSR 214(1974), 63–66; English transl.: Sov. Math., Dokl. 15(1974), 67–71. 7. M. M. Derriennic, Sur l’approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifiés. J. Approx. Theory 31(1981), No. 4, 325–343. 8. J. L. Durrmeyer, Une formule d’inversion de la transformée de Laplace-applications à la théorie des moments. Thèse de 3e cycle, Faculté des Sciences de l’Université de Paris, 1967.

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(Received 4.05.2009) Authors’ addresses: H. Karsli Abant Izzet Baysal University, Faculty of Science and Arts Department of Mathematics, 14280 Bolu, Turkey E-mail: karsli [email protected] P. Pych-Taberska Faculty of Mathematics and Computer Science Adam Mickiewicz University, Umultowska 87, 61-614 Pozna´ n, Poland E-mail: [email protected]