On the Iterates of Some Bernstein-Type Operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

209, 529]541 Ž1997.

AY975371

On the Iterates of Some Bernstein-Type Operators* Jose ´ A. Adell† Departamento de Metodos Estadısticos, Facultad de Ciencias, ´ ´ Uni¨ ersidad de Zaragoza, 50009, Zaragoza, Spain

F. German ´ Badıa ´ Departamento de Metodos Estadısticos, Centro Politecnico Superior, ´ ´ ´ Uni¨ ersidad de Zaragoza, 50015, Zaragoza, Spain

and Jesus ´ de la Cal ‡ Departamento de Matematica e In¨ estigacion ´ Aplicada y Estadıstica ´ ´ Operati¨ a, Facultad de Ciencias, Uni¨ ersidad del Paıs ´ Vasco, Apartado 644, 48080, Bilbao, Spain Submitted by William F. Ames Received May 31, 1995

In this paper, we establish two basic functional-type identities between the iterates of the Bleimann]Butzer]Hahn operator and those of the Bernstein operator, on the one hand, and the iterates of the Žmodified. Meyer]Konig ¨ and Zeller operator and those of the Baskakov operator, on the other. These identities allow us to transfer the properties of these operators from one to another. Attention is focused on the limit behavior of the iterates and the linear combinations of iterates of Fejer]Korovkin type. Q 1997 Academic Press ´

* Research supported by Grant PB95-0809 of the Spanish DGICYT. † E-mail address: [email protected]. ‡ E-mail address: [email protected]. 529 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

ADELL, BADIA ´ , AND DE LA CAL

530

1. INTRODUCTION The linear operator L n defined by n

Ln Ž f , x . [

f

Ý ks0

ž

k nykq1

xk n , k Ž1 q x. n

x G 0, n s 1, 2, . . . ,

/ž /

where f g R w0, `. Žfor convenience, we also set L0 Ž f, x . [ f Ž0.. was introduced by Bleimann, Butzer, and Hahn w1x to approximate continuous functions on the positive semi-axis and has been studied by several authors Žsee, for instance, w3, 9, 10, 14x.. In w1x, the authors pointed out some formal similarities and differences between L n and other operators, namely, the classical Bernstein operator given by n

Bn Ž h, x . [

h

Ý ks0

k

ny k n k x Ž1 y x. , k

x g w 0, 1 x , n s 1, 2, . . . ,

ž /ž / n

where h g R w0, 1x; the Baskakov operator Hn defined by Hn Ž f , x . [

`

Ý ks0

f

k

ž /ž n

xk nqky1 , nqk k Ž1 q x.

x G 0, n s 1, 2, . . . ,

/

where f g C w0, `. satisfies f Ž x . s O Ž x a . Ž x ª `., for some a ) 0; and the Meyer-Konig and Zeller operator Mn Žin the modified version of ¨ Cheney and Sharma w2x. defined by Mn Ž g , x . [ Ž 1 y x .

nq 1

`

Ý ks0

g

ž

k nqk

/ž

nqk k x , k

/

x g 0, 1 . , n s 1, 2, . . . , where g g C w0, 1. satisfies g Ž x . s O ŽŽ1 y x .ya ., Ž x ª 1., for some a ) 0. In a further remark Žafter w1, Lemma 1x., the authors observed that ‘‘there exists essentially the same connection between the Bn and the L n as between the Hn and the Mn ; it is basically given by the rational transformation r Ž u. [ urŽ1 q u., u g w0, `., and its inverse ry1 Ž ¨ . [ ¨ rŽ1 y ¨ ., ¨ g w0, 1.. Since this transformation is rational, one can hardly expect to carry over the well-known results of the operator Bn Žor Hn . to the transformed operator L n Žor Mn ..’’ This last assertion is true with regard to convergence results Žwith rates., which is the main topic considered in w1x. However, if one is interested in various kinds of iterates of these operators and other topics, the preceding ideas become very fruitful.

531

ITERATES OF OPERATORS

In the next section, we show that the ‘‘connection’’ suggested in w1x can be formulated in a precise way by means of the following two identities ŽTheorems 1 and 2., k Lkn s T#( Bnq1 ( S#,

Ž 1.

Mnk s T ( Hnk ( S,

Ž 2.

where Ak denotes the kth iterate of the operator A Ži.e., A0 s I is the identity operator on the corresponding function space and, for k G 1, Ak s A( Aky1 ., and S, T, S#, T# are suitable positive linear operators which will be defined below. Thanks to identity Ž1., different properties of Bn can be transferred to L n with little extra effort. Thus, the limiting behavior of the iterates of L n is immediately derived from Ž1. and the well-known results by Kelisky and Rivlin w8x and Karlin and Ziegler w7x for the iterates of Bn Žsee Theorem 5 and Remark 1 below.. On the other hand, the behavior of the iterates of Fejer]Korovkin type ´ r

Ln , r [ I y Ž I y Ln . s

r

Ý Ž y1. jy1 js1

r L nj , j

ž/

is also derived from Ž1. and the corresponding result for Bn shown by Felbecker w4x Žsee Theorem 4 in Section 3.. Identity Ž2. is used to carry over analogous properties of the operator Hn to the operator Mn ŽTheorem 3, Remark 2.. Finally, formulae Ž1. and Ž2. may have other applications, some of which are outlined in the last section.

2. THE BASIC IDENTITIES We introduce the following auxiliary operators. Firstly, let T : R w0, 1. ª R and T#: R w0, 1x ª R w0, `. be the positive linear operators defined by w0, `.

T Ž g , x . [ Ž 1 q x . g Ž xr1 q x . ,

x g 0, ` . ,

T#h [ Th#, where h# is the restriction of h to w0, 1.. Secondly, let S: R w0, `. ª R w0, 1. and S#: R w0, `. ª R w0, 1x be defined by S Ž f , x . [ Ž 1 y x . f Ž xr1 y x . , S# Ž f , x . [

½

SŽ f , x . 0

x g 0, 1 . ,

if x g 0, 1 . if x s 1.

ADELL, BADIA ´ , AND DE LA CAL

532

On the other hand, we shall use the notations M [  g g C 0, 1 . : g Ž x . s O Ž Ž 1 y x .

ya

. Ž x ª 1. , for some a ) 0 4 ,

H [  f g C 0, ` . : f Ž x . s O Ž x a . Ž x ª ` . , for some a ) 0 4 , C* 0, ` . [  f g C 0, ` . : f Ž x . s o Ž x . , Ž x ª ` . 4 . The following lemma collects some properties of the preceding operators to be used throughout the paper. All the assertions can be checked by elementary calculations. LEMMA 1. We ha¨ e Ža. T ( S s T#( S# is the identity operator on R w0, `.. Žb. S(T is the identity operator on R w0, 1.. Žc. Ž S#(T#. h s h, for e¨ ery h g R w0, 1x such that hŽ1. s 0. Žd. S, T, and T# preser¨ e continuity and con¨ ergence on compact subsets. Že. If f g C*w0, `. then S# f g C w0, 1x. Žf. Let r s 0, 1, . . . . If g g C r w0, 1. satisfies g Ž r . g M , then Tg g rw C 0, `. and ŽTg .Ž r . g H . Žg. Let r s 0, 1, . . . . If f g C r w0, `. satisfies f Ž r . g H , then Sf g rw C 0, 1. and Ž Sf .Ž r . g M . The relation between the iterates of L n and Bnq1 is given in the following theorem. For n, k s 0, 1, 2 . . . , we ha¨ e

THEOREM 1.

k Lkn s T#( Bnq1 ( S#.

Proof. If k s 0, we just have Lemma 1Ža.. If k s 1, the result follows from the equality n

Ý ks0

f

ž

k

xk n k Ž1 q x. n

/ž / . Ý ž

nykq1

n

s Ž1 q x

1y

ks0

=

ž

nq1 k

/ž

k nq1

x 1qx

k

/ž

/ž f

krn q 1 1 y Ž krn q 1 .

1 1qx

nq 1yk

/

,

/

533

ITERATES OF OPERATORS

where f g R w0, `. and x G 0. Finally, if k G 2, the conclusion follows by induction on k, taking into account Lemma 1Žc.. Before stating the identities concerning the operators Mn and Hn , we give the following two lemmata. The first one is a well-known elementary result Žcf. w6x.. LEMMA 2. Let X be a random ¨ ariable ha¨ ing the negati¨ e binomial distribution with parameters n and q g Ž0, 1., i.e., PŽ X s j. s

ž

nqjy1 n Ž1 y q. q j, j

j s 0, 1, 2, . . . .

/

Then, for k s 1, 2, . . . , k

E X Ž X y 1 . ??? Ž X y k q 1 . s n Ž n q 1 . ??? Ž n q k y 1 . Ž qr1 y q . . The second lemma guarantees that Mn and Hn can be iterated on M and H , respectively. LEMMA 3. Ža. Žb.

For n s 1, 2, . . . , we ha¨ e:

If g g M then Mn g g M . If f g H then Hn f g H .

Proof. Both parts Ža. and Žb. easily follow from Lemma 2 and the probabilistic representations Hn Ž f , x . s Ef

ž

Mn Ž g , x . s Eg

ž

Vn Ž xr1 q x . n Vnq 1 Ž x . n q Vnq 1 Ž x .

/

,

/

,

where E denotes mathematical expectation and VnŽ u. is a random variable having the negative binomial distribution with parameters n, u. THEOREM 2.

For n s 1, 2, . . . and k s 0, 1, 2, . . . , we ha¨ e Mnk g s Ž S( Hnk (T . g ,

g g M,

Hnk f s Ž T ( Mnk ( S . f ,

f g H.

and

Proof. The proof follows along the lines of that in Theorem 1. Details are omitted.

ADELL, BADIA ´ , AND DE LA CAL

534

3. ITERATES OF FEJER]KOROVKIN TYPE In order to increase the rate of convergence according to the smoothness of the functions, several authors Žcf. w4, 12, 13x. have considered the operator Bn, r defined by r

Bn , r [ I y Ž I y Bn . s

r

Ý Ž y1. jy1 js1

r Bj, j n

ž/

n, r s 1, 2, . . . .

Obviously, Bn, 1 s Bn . For r ) 1, the operator Bn, r is not positive, but it satisfies the following property shown by Felbecker w4x. THEOREM A.

Let r s 1, 2, . . . and h g C 2 r w0, 1x. Then

lim n r Bn , r Ž h, x . y h Ž x . s Ž y1 .

ry1

nª`

r B# Ž h, x .

uniformly on w0, 1x, where B# is the differential operator defined by B# Ž h, x . [

xŽ1 y x. 2

h0 Ž x . ,

x g w 0, 1 x , h g C 2 w 0, 1 x .

Ž 3.

w5x to a This result has been extended by Gawronsky and Stadtmuller ¨ large class of discrete operators. The following theorem is the specialization of w5, Theorem 1x to the case of the Baskakov operator. THEOREM B. Then

Let r s 1, 2, . . . and let f g C 2 r w0, `. such that f Ž2 r . g H .

lim n r Hn , r Ž f , x . y f Ž x . s Ž y1 .

nª`

ry1

H r Ž f, x.

uniformly on compact subsets of w0, `., where Hn, r [ I y Ž I y Hn . r and H is the differential operator defined by HŽ f, x. [

xŽ1 q x. 2

f0 Ž x. ,

x G 0, f g C 2 0, ` . .

Ž 4.

The results of Gawronsky and Stadtmuller do not apply to L n nor Mn . ¨ In this section, the analogous results for L n and Mn are derived from Theorems A and B, respectively, by using the identities shown in the preceding section. To do this, we need the following auxiliary result.

535

ITERATES OF OPERATORS

LEMMA 4. Ža. Let L be the differential operator gi¨ en by LŽ f , x . [

xŽ1 q x. 2

2

f0 Ž x. ,

x G 0, f g C 2 0, ` . .

Ž 5.

Then, we ha¨ e Lr s T ( B r ( S,

r s 1, 2, . . . , 2w

where B is the differential operator on C 0, 1. defined in the same way as B# in Ž3., i.e., BŽ g , x. [

xŽ1 y x.

g0 Ž x. ,

2

x g 0, 1 . , g g C 2 0, 1 . .

Žb. Let M be the differential operator gi¨ en by MŽ g, x. [

xŽ1 y x. 2

2

g0 Ž x. ,

x g 0, 1 . , g g C 2 0, 1 . .

Ž 6.

We ha¨ e M r s S( H r (T ,

r s 1, 2, . . . ,

where H is defined in Ž4.. Proof. Parts Ža. and Žb. have similar proofs. Actually, the case r s 1 can be easily checked and, for r G 2, the conclusion follows by induction, taking into account Lemma 1Ža., Žb.. Using Theorem 2, Lemma 1, and Lemma 4Žb., the following result is obtained as an immediate consequence of Theorem B. THEOREM 3. Then

Let r s 1, 2, . . . and let g g C 2 r w0, 1. such that g Ž2 r . g M .

lim n r Mn , r Ž g , x . y g Ž x . s Ž y1 .

nª`

ry1

M r Ž g, x.

uniformly on compact subsets of w0, 1., where r

Mn , r [ I y Ž I y Mn . s S( Hn , r (T and M is the differential operator defined in Ž6.. The corresponding result for the operator L n is stated in Theorem 4. Observe that it contains, as a particular case, the Voronovskaja-type result for L n obtained by Totik w14x.

ADELL, BADIA ´ , AND DE LA CAL

536 THEOREM 4.

Let r s 1, 2, . . . and let f g H l C 2 r w0, `.. Then

lim n r L n , r Ž f , x . y f Ž x . s Ž y1 .

nª`

ry1

Lr Ž f , x .

uniformly on compact subsets of w0, `., where r

L n , r [ I y Ž I y L n . s T#( Bnq1, r ( S#

Ž 7.

and L is the differential operator gi¨ en in Ž5.. The proof of Theorem 4, based on Theorem 1, Lemma 1, and Theorem A, is not as simple as that of Theorem 3. In fact, if f g C 2 r w0, `., then S# f g C 2 r w0, 1., but it cannot be guaranteed that S# f g C 2 r w0, 1x. This is the reason why we shall need the following auxiliary result. Let x g w0, 1x and n s 1, 2, . . . . Define, inducti¨ ely on j,

LEMMA 5.

pnŽ1., k Ž x . [

ny k n k x Ž1 y x. , k

ž /

k s 0, . . . , n,

n

pnŽ j., k Ž x . [

pnŽ1., k Ž irn . pnŽ jy1. Ž x. , ,i

Ý

k s 0, . . . , n.

is0

For j s 1, 2, . . . , we ha¨ e Ža.

Let h g R w0, 1x. Then n

Bnj Ž h, x . s

Ý h Ž krn . pnŽ j., k Ž x . . ks0

Ž j. Ž . In particular Ž take h ' 1., the quantities pn, k x , k s 0, . . . , n add up to 1. Žb. For p G 1,

n

Ý ks0

k n

p

yx

pnŽ j., k Ž x . F C j, p nyp r2 ,

n s 1, 2 . . . ,

where C j, p is some positi¨ e constant only depending upon j and p. Proof of Lemma 5. Assertion Ža. can be easily checked by induction on j. To prove Žb., we also use induction. For j s 1, the result is well known Žcf., for instance, w11, p. 15x.. On the other hand, using the inequality < a q b < p F 2 py1 Ž < a < p q < b < p . ,

a, b g R,

537

ITERATES OF OPERATORS

we have, for j ) 1, n

k

Ý

n

ks0

p

yx

n

pnŽ j., k Ž x . s

n

k

Ý Ý

n

ks0 is0 n

F2

py 1

ž

p

yx n

k

Ý Ý is0

pnŽ1., k Ž irn . pnŽ jy1. Ž x. ,i

ks0

n

i

y

p

/

pnŽ1., k Ž irn . pnŽ jy1. Ž x. ,i

n n

qÝ is0

p

i

yx

n

pnŽ jy1. Ž x. . ,i

Thus, the conclusion follows from Ža., the case j s 1, and the induction hypothesis. Now, we are in a position to show Theorem 4. Proof of Theorem 4. Let a ) 0 and fix z such that a* [ ar1 q a z - 1. Choose h g C 2 r w0, 1x such that h Ž u . s S# Ž f , u . ,

for all u g w 0, z x .

Ž 8.

By Ž7. and Ž8., we obviously have, for x g w0, ax, Ln , r Ž f , x . y f Ž x . s Qn Ž x . q R n Ž x . , where Q n Ž x . [ Ž 1 q x . Bnq1, r Ž h, xr1 q x . y h Ž xr1 q x . and R n Ž x . [ Ž 1 q x . Bnq1, r Ž S# f y h, xr1 q x . . From Theorem A, Lemma 4Ža., and Ž8., we deduce that lim n r Q n Ž x . s Ž 1 q x . Ž y1 .

nª`

ry1

r B# Ž h, xr1 q x .

s Ž y1 .

ry1

ŽT ( Br(S. Ž f , x.

s Ž y1 .

ry1

Lr Ž f , x . ,

uniformly on w0, ax. Therefore, the proof will be complete as soon as we show that, for j s 1, . . . , r, U yŽ rq1. j sup Bnq , 1 Ž S# f y h , xr1 q x . F C j n

xg w0, a x

where CUj is some positive constant.

n s 1, 2, . . . , Ž 9 .

ADELL, BADIA ´ , AND DE LA CAL

538

Since f g H , it is not hard to see that S# Ž f , krn q 1 . F Cn b ,

sup

n s 1, 2, . . . ,

Ž 10 .

kr Ž nq1 .)z

for some constants C ) 0 and b G 0. Taking into account Ž8., and using successively Ž10., Chebyshev’s inequality, and Lemma 5, we obtain, for x g w0, ax. j Bnq 1 Ž S# f y h , xr1 q x .

s

Ž j. S# Ž f , krn q 1 . y h Ž krn q 1 . pnq 1, k Ž xr1 q x .

Ý kr Ž nq1 .)z

F Ž Cn b q 5 h 5 .

Ž j. pnq 1, k Ž xr1 q x .

Ý kr Ž nq1 .yxr Ž1qx . ) Ž zya* .

F C9n Ž z y a* . b

y2 Ž bqrq1 .

nq1

k

Ý

nq1

ks0

y

x

2 Ž bqrq1 .

1qx

Ž j. = pnq 1, k Ž xr1 q x .

F C0 n b C j Ž n q 1 .

y Ž bqrq1 .

F CUj nyŽ rq1. ,

where 5 ? 5 denotes the sup-norm in C w0, 1x. This shows claim Ž9. and, therefore, the proof of Theorem 4 is complete. 4. LIMIT BEHAVIOR OF ITERATES n. The main purpose of this section is to deduce the limit behavior of LkŽ n , kŽ n. as n ª ` and k Ž n. ª `, from the limit behavior of Bn . The following theorem summarizes the main results concerning the limit behavior of the iterates of the Bernstein operator. Parts Ža., Žb., and formula Ž11. were obtained by Kelisky and Rivlin w8x, while the first assertion in part Žc. was shown by Karlin and Ziegler w7x.

THEOREM C.

Let h g C w0, 1x. We ha¨ e:

Ža. If k Ž n.rn ª 0 then 5 BnkŽ n. h y h 5 ª 0. Žb. If k Ž n.rn ª ` then 5 BnkŽ n. h y B1 h 5 ª 0. Žc. If k Ž n.rn ª t g Ž0, `. then 5 BnkŽ n. h y B Ž t . h 5 ª 0, where  B Ž t .: t G 04 is the C0-semigroup of operators acting on C w0, 1x whose infinitesimal generator is the differential operator B# defined in Ž3.. In particular, if h r Ž u. [ u r Ž r s 1, 2, . . . ., then r

B Ž t . Ž hr , x . s

Ý is1

bi x i ,

x g w 0, 1 x ,

539

ITERATES OF OPERATORS

where

bi [

i r r i

ž/

2

iqj

Ž y1 .

r

Ý jsi

ž

2jy2 jyi

/ž

ryi jyi

2

ž / jqry1 ryj

eyjŽ jy1.t r2 .

Ž 11 .

/

Combining Theorem C with Theorem 1 and Lemma 1, we immediately obtain the following result. THEOREM 5. Ža. subsets. Žb. subsets. Žc. compact

Let f g C*w0, `.. We ha¨ e:

n. Ž If k Ž n.rn ª 0 then LkŽ f, x . ª f Ž x ., uniformly on compact n n. Ž If k Ž n.rn ª ` then LkŽ f, x . ª f Ž0., uniformly on compact n n. Ž If k Ž n.rn ª t g Ž0, `. then LkŽ f, x . ª LŽ t .Ž f, x ., uniformly on n subsets, where

L Ž t . [ T#( B Ž t . ( S#.

Ž 12 .

In particular, if f r Ž u. [ Ž1 q u.1y r Ž r s 1, 2, . . . ., then r

L Ž t . Ž fr , x . s

Ý bi Ž 1 q x . 1y i ,

x G 0,

is1

where bi is defined in Ž11.. Remark 1. Karlin and Ziegler w7x have shown that 1

B Ž t . Ž h, x . s B1 Ž h, x . q

H0

h Ž y . y B1 Ž h, y . p Ž t ; x, y . dy,

where pŽ t; x, y . is the transition probability density of a diffusion process on w0, 1x with absorbing barriers whose backward equation is

­p ­t

s

xŽ1 y x. ­ 2 p 2

­ x2

Žsee w7, Ž1.10.x for the explicit formula for pŽ t; x, y ... From this fact and Ž12., it is not hard to see that L Ž t . Ž f , x . s f Ž 0. q

`

H0

f Ž y . y f Ž 0 . q Ž t ; x, y . dy,

ADELL, BADIA ´ , AND DE LA CAL

540 where

q Ž t ; x, y . [

1qx

Ž1 q y.

3

ž

p t;

x

,

y

1qx 1qy

/

.

Remark 2. The results for the Bernstein operator were extended by Karlin and Ziegler w7x to a more general context of positive linear operators. Although the Baskakov operator is not specifically mentioned in w7x, it satisfies analogous properties to the Szasz ´ operator Žsee w7, Sect. 5x.. n. Therefore, as we have shown for BnkŽ n. and LkŽ n , the properties concerning kŽ n. the limit behavior of Hn can be transferred to MnkŽ n. via Theorem 2 above. We shall not enter into the details.

5. CONCLUDING REMARKS Theorems 1 and 2 may have other applications. As an example, we give a simple proof of the following known results on convexity concerning L n Žcf. w3, Theorems 2.1 and 2.2; 9; 10x.. THEOREM 6. Let f g R w0, `. be a nonincreasing con¨ ex function. Then, for n s 0, 1, 2, . . . , we ha¨ e: Ža. Žb.

L n f is con¨ ex. L n f G L nq1 f.

To prove Theorem 6, we shall need the following elementary result. LEMMA 6. The operators S, T, and T# preser¨ e con¨ exity. Also, if f g R w0, `. is nonincreasing and con¨ ex, then S# f is con¨ ex. Proof of Theorem 6. As it is well known, if h g R w0, 1x is convex then, for n s 1, 2, . . . , Bn h is convex and, moveover, Bn h G Bnq1 h. Thus, the conclusion in Ža. follows from Theorem 1 and Lemma 6, while part Žb. is a consequence of Theorem 1, Lemma 6, and the positivity of T#. The proof is complete. Also, the operators Hn and Mn preserve convexity and have the property of monotonic convergence under convexity Žcf. w2, 10x.. In view of Theorem 2 and Lemma 6, it becomes apparent that Mn satisfies these properties if and only if Hn does. On the other hand, we can obtain characterizations of convexity for L n and Mn similar to those established by Karlin and Ziegler in w7, Sect. 7x. This can be done by using the results in Section 4 or, alternatively, by using the corresponding known results for Bn and Hn , via Theorems 1 and 2 and Lemmas 1 and 6. Details are omitted.

ITERATES OF OPERATORS

541

REFERENCES 1. G. Bleimann, P. L. Butzer, and L. Hahn, A Bernstein-type operator approximating continuous functions on the semi-axis, Indag. Math. 42 Ž1980., 255]262. 2. E. W. Cheney and A. Sharma, Bernstein power series, Canad. J. Math. 16 Ž1964., 241]252. 3. B. Della Vecchia, Some properties of a rational operator of Bernstein-type, in ‘‘Progress in Approximation Theory’’ ŽP. Nevai and A. Pinkus, Eds.., pp. 177]185, Academic Press, New York, 1991. 4. F. Felbecker, Linearkombinationen von iterierten Bernsteinoperatoren, Manuscripta Math. 29 Ž1979., 229]248. 5. W. Gawronsky and U. Stadtmuller, Linear combinations of iterated generalized Bernstein ¨ functions with an application to density estimation, Acta Sci. Math. (Szeged) 47 Ž1984., 205]221. 6. N. L. Johnson and S. Kotz, ‘‘Discrete Distributions,’’ Houghton Mifflin, Boston, 1969. 7. S. Karlin and Z. Ziegler, Iteration of positive approximation operators, J. Approx. Theory 3 Ž1970., 310]339. 8. R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 Ž1967., 511]520. 9. R. A. Khan, Some properties of a Bernstein-type operator of Bleimann, Butzer and Hahn, in ‘‘Progress in Approximation Theory’’ ŽP. Nevai and A. Pinkus, Eds.., pp. 497]504, Academic Press, New York, 1991. 10. R. A. Khan, Reverse martingales and approximation operators, J. Approx. Theory 80 Ž1995., 367]377. 11. G. G. Lorentz, ‘‘Bernstein Polynomials,’’ 2nd ed., Chelsea, New York, 1986. 12. G. Mastroianni and M. R. Occorsio, Una generalizzazione dell’operatore di Bernstein, Rend. Accad. Sci. Fis. Mat. Napoli (4) 44 Ž1977., 151]169. 13. C. A. Micchelli, The saturation class of iterates of the Bernstein polynomials, J. Approx. Theory 8 Ž1973., 1]18. 14. V. Totik, Uniform approximation by Bernstein-type operators, Indag. Math. 46 Ž1984., 87]93.