Approximation of Unbounded Functions by a New Sequence of Linear Positive Operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

225, 660᎐672 Ž1998.

AY986036

NOTE Approximation of Unbounded Functions by a New Sequence of Linear Positive Operators P. N. Agrawal and Kareem J. Thamer* Department of Mathematics, Uni¨ ersity of Roorkee, Roorkee-247 667 (U.P.), India Submitted by Mark J. Balas Received November 6, 1997

In the present paper, we introduce a new sequence of linear positive operators to 䊚 1998 Academic study the simultaneous approximation of unbounded functions. Press

1. INTRODUCTION For f g C␣ w0, ⬁. s  f g C w0, ⬁.: < f Ž t .< F M Ž1 q t . ␣ for some M ) 0, ␣ ) 04 , we define a sequence of linear positive operators Mn as Mn Ž f Ž t . ; x . s Ž n y 1 .

⬁

Ý

pn , k Ž x .

ks1

qŽ 1 q x .

yn

⬁

H0

pn , ky1 Ž t . f Ž t . dt

f Ž 0. ,

Ž 1.1.

where pn , k Ž x . s

ž

nqky1 k y Ž nqk . x Ž1 q x. , k

/

x g w 0, ⬁ . .

The space C␣ w0, ⬁. is normed by 5 f 5 C␣ s sup 0 F t -⬁ < f Ž t .<Ž1 q t .y␣ . We may also write Ž1.1. as Mn Ž f Ž t . ; x . s

⬁

H0 W Ž t , x . f Ž t . dt, n

* On study leave from the Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq. 660 0022-247Xr98 $25.00 Copyright 䊚 1998 by Academic Press All rights of reproduction in any form reserved.

661

NOTE

where Wn Ž t , x . s Ž n y 1 .

⬁

pn , k Ž x . pn , ky1 Ž t . q Ž 1 q x .

Ý

yn

␦ Ž t. ,

ks1

␦ Ž t . being the Dirac delta function. The study in simultaneous approximation Žapproximation of derivatives of functions by the corresponding order derivatives of operators. began with a remarkable result for the Bernstein polynomials BnŽ f . owing to Lorentz w7x, who proved that B Ž k . Ž f ; x . ª f Ž k . Ž x ., n ª ⬁, whenever the latter exists at the particular point x g w0, 1x, k s 1, 2, 3, . . . being arbitrary. For further research in this area, we refer the reader to w1᎐5x. The purpose of this paper is to study some results in simultaneous approximation by the operators Ž1.1.. First, we establish the basic pointwise convergence theorem and then proceed to study the degree of this approximation.

2. DEFINITIONS AND AUXILIARY RESULTS LEMMA 1 w6x. For m g N 0 Ž the set of nonnegati¨ e integers., the mth order moment of the Lupas operators is defined by ⬁

␮n , m Ž x . s

Ý

pn , k Ž x .

ks0

ž

k n

m

yx

/

.

Hence, ␮ n, 0 Ž x . s 1, ␮ n, 1Ž x . s 0, and there holds the recurrence relation n ␮ n,mq1Ž x . s x Ž1 q x .w ␮Xn, mŽ x . q m ␮ n, my1Ž x .x, m g N Ž the set of natural numbers.. Consequently, Ži. ␮ n, mŽ x . is a polynomial in x of degree at most m. Žii. For e¨ ery x g w0, ⬁., ␮ n, mŽ x . s O Ž nywŽ mq1.r2x ., where w ␤ x denotes the integral part of ␤ . LEMMA 2.

Let the function Tn, mŽ x ., m g N 0 , be defined as

Tn , m Ž x . s Ž n y 1 .

⬁

Ý

pn , k Ž x .

ks1 m

q Ž yx . Ž 1 q x .

⬁

H0

yn

pn , ky1 Ž t . Ž t y x .

.

Then Tn , 0 Ž x . s 1,

Tn , 1 Ž x . s

2x ny2

m

dt

662

NOTE

and

Ž n y m y 2 . Tn , mq1 Ž x . s x Ž 1 q x . TnX , m Ž x . q Ž 2 x q 1 . m q 2 x Tn , m Ž x . q2 mx Ž 1 q x . Tn , my1 Ž x . ,

m g N.

Hence, Ži. Žii. Žiii.

Tn, mŽ x . is a polynomial in x of degree m. For e¨ ery x g w0, ⬁., Tn, mŽ x . s O Ž nywŽ mq1.r2x .. The coefficients of nŽ kq1. in Tn, 2 kq2 Ž x . and Tn, 2 kq1Ž x . are

Ž 2 k q 2. !  x Ž 1 q x . 4 Ž k q 1. ! Ž 2 k q 1. ! k!

kq 1

and

 Ž k q 1. Ž 1 q 2 x . y 14  x Ž 1 q x . 4 k .

Making use of the relation x Ž1 q x . pXn, k Ž x . s Ž k y nx . pn, k Ž x ., the proof of the lemma is straightforward and hence is omitted. LEMMA 3 w6x. There exist the polynomials qi, j, r Ž x . independent of n and k such that x r Ž1 q x.

r

dr dx r

pn , k Ž x . s

Ý

j

n i Ž k y nx . qi , j, r Ž x . pn , k Ž x . .

2 iqjFr i , jG0

3. MAIN RESULTS In this section, first we show that the derivative MnŽ r . Ž f Ž t .; x . is an approximation process for f Ž r ., r s 1, 2, 3, . . . . THEOREM 1. If r g N, f g C␣ w0, ⬁. for some ␣ ) 0, and f Ž r . exists at a point x g Ž0, ⬁., then lim MnŽ r . Ž f Ž t . ; x . s f Ž r . Ž x . .

nª⬁

Ž 3.1.

Further, if f Ž r . exists and is continuous on Ž a y ␩ , b q ␩ . ; Ž0, ⬁., ␩ ) 0, then Ž3.1. holds uniformly in x g w a, b x.

663

NOTE

Proof. By the hypothesis, we have f Ž i. Ž x .

r

f Ž t. s

Ý

i!

is0

r

i Ž t y x. q ␧ Ž t, x. Ž t y x. ,

where ␧ Ž t, x . ª 0 as t ª x. Hence MnŽ r . Ž f Ž t . ; x . s

⬁

Ž t , x . f Ž t . dt

f Ž i. Ž x .

r

s

Žr. n

H0 W Ý

i!

is0

q

⬁

Žr. n

H0 W

⬁

Žr. n

H0 W

i Ž t , x . Ž t y x . dt

r

Ž t , x . ␧ Ž t , x . Ž t y x . dt

s I1 q I2 , say. From Lemma 2, it follows that H0⬁ WnŽ t, x . t ␯ dt is a polynomial in x of degree exactly ␯ and the coefficient of x ␯ is

Ž n q ␯ y 1. ! Ž n y ␯ y 2. ! Ž n y 1. ! Ž n y 2. ! and thus f Ž i. Ž x .

r

I1 s

Ý is0

s

i

Ý

i!

␯ s0

f Žr.Ž x . dr r!

dx

r

r i Ž yx . iy ␯ d ␯ dx r

ž /

⬁

H0 W Ž t , x . t n

␯

dt

Ž n q r y 1. ! Ž n y r y 2. ! r x Ž n y 1. ! Ž n y 2. ! qterms in lower powers of x

s f Žr.Ž x .

Ž n q r y 1. ! Ž n y r y 2. ! ª f Ž r . Ž x . , as n ª ⬁. Ž n y 1. ! Ž n y 2. !

664

NOTE

Next, using Lemma 3, we obtain ⬁

I2 s

H0 W

s

Ý

Žr. n

r

Ž t , x . ␧ Ž t , x . Ž t y x . dt qi , j, r Ž x .

ni

x Ž1 q x. r

2 iqjFr i , jG0 ⬁

=

H0

r

Ž n y 1.

⬁

Ý

pn , k Ž x . Ž k y nx .

j

ks1

r

pn , ky1 Ž t . ␧ Ž t , x . Ž t y x . dt

q Ž y1 .

r

Ž n q r y 1. ! ynyr r ␧ Ž 0, x . Ž yx . . Ž1 q x. Ž n y 1. !

Hence

< I2 < F

ni

Ý 2 iqjFr i , jG0 ⬁

=

H0

q

< qi , j, r Ž x . < x r Ž1 q x.

r

Ž n y 1.

⬁

Ý

pn , k Ž x . < k y nx < j

ks1

pn , ky1 Ž t . < ␧ Ž t , x . < < Ž t y x . < ␥ dt

Ž n q r y 1. ! yn yr < ␧ Ž 0, x . < x r Ž1 q x. Ž n y 1. !

s I3 q I4 . Since ␧ Ž t, x . ª 0 as t ª x, for a given ␧ ) 0 there exists a ␦ ) 0 such that < ␧ Ž t, x .< - ␧ , whenever 0 - < t y x < - ␦ . Further, if ␥ is any integer G maxŽ ␣ , r ., then we can find a constant K ) 0 such that < ␧ Ž t, x .Ž t y x . r < F K < t y x < ␥ for < t y x < G ␦ . Hence, I3 F C 1

Ý

ni Ž n y 1.

2 iqjFr i , jG0

= ␧

½H

< tyx <- ␦

⬁

Ý

pn , k Ž x . < k y nx < j

ks1

pn , ky1 Ž t . < t y x < r dt q

s I5 q I6 , say,

H< tyx
n , ky1

Ž t . K < t y x < ␥ dt

5

665

NOTE

where < qi , j, r Ž x . <

C1 s sup

x r Ž1 q x.

2 iqjFr i , jG0

r

.

Now, on application of Schwarz inequality for integration and then for summation, we obtain I5 F ␧ C1

Ý

ni Ž n y 1.

2 iqjFr i , jG0

=

⬁

žH

0

F ␧ C1

pn , k Ž x . < k y nx < j

ks1

ž

⬁

H0

1r2

pn , ky1 Ž t . dt

/

1r2

pn , ky1 Ž t . Ž t y x .

Ý

2r

dt

/ 1r2

⬁

ni

2 iqjFr i , jG0

žÝ

pn , k Ž x . Ž k y nx .

2j

ks1

= Ž n y 1.

ž

⬁

Ý

⬁

pn , k Ž x .

Ý ks1

⬁

H0

/ 1r2

pn , ky1 Ž t . Ž t y x .

2r

dt

/

as H0⬁ pn, ky1Ž t . dt s Ž n y 1.y1 . Using Lemma 1, we get ⬁

Ý

2j

pn , k Ž x . Ž k ynx . s n2 j

ks1

⬁

½Ý

pn , k Ž x .

ks0

k

2j

yx

y Ž 1 qx .

ž / n

s n2 j  O Ž nyj . q O Ž nys . 4

yn

Ž yx .

2j

5

Ž for any s ) 0 .

s OŽ n . . j

Ž 3.2.

Similarly, using Lemma 2

Ž n y 1.

⬁

Ý

pn , k Ž x .

ks1

⬁

H0

pn , ky1 Ž t . Ž t y x .

s Tn , 2 r Ž x . y Ž 1 q x . s O Ž nyr . q O Ž nys . s O Ž nyr . .

yn

Ž yx .

2r

dt

2r

Ž for any s ) 0 . Ž 3.3.

666

NOTE

Hence I5 F ␧ C1

n i O Ž n j r2 . O Ž nyr r2 .

Ý 2 iqjFr i , jG0

s ␧ O Ž 1. . Again, using Schwarz inequality for integration and then for summation, in view of Ž3.2. and Ž3.3., we have

I6 F C2

Ý

ni Ž n y 1.

2 iqjFr i , jG0

F C2

Ý

ž

pn , k Ž x . < k y nx < j

H< tyx
ni Ž n y 1.

⬁

Ý

pn , k Ž x . < k y nx < j

ž

ks1

n , ky1 Ž t . Ž t y x .

⬁

Ý 2 iqjFr i , jG0

ni

žÝ

= Ž n y 1.

ž

Ý

2␥

Ž t . < t y x < ␥ dt

H< tyx
n , ky1 Ž t . dt

/

1r2

dt

/ 1r2

pn , k Ž x . Ž k y nx .

ks1

⬁

Ý

n , ky1

1r2

H< tyx
F C2

s

Ý ks1

2 iqjFr i , jG0

=

⬁

pn , k Ž x .

ks1

⬁

H0

2j

/ 1r2

pn , ky1 Ž t . Ž t y x .

2␥

dt

/

n i O Ž n j r2 . O Ž ny␥ r2 .

2 iqjFr i , jG0

s O Ž nŽ ry ␥ .r2 . s o Ž 1 . , and, therefore, in view of the arbitrariness of ␧ ) 0, it follows that I3 s oŽ1.. Also, I4 ª 0 as n ª ⬁ and hence I2 s oŽ1.. Combining the estimates of I1 and I2 , we obtain Ž3.1.. To prove the uniformity assertion, it is sufficient to remark that ␦ Ž ␧ . in the above proof can be chosen to be independent of x g w a, b x and also that the other estimates hold uniformly in x g w a, b x.

667

NOTE

Next, we prove a Voronovskaja type asymptotic formula. THEOREM 2. Let f g C␣ w0, ⬁. for some ␣ ) 0. If f Ž rq2. exists at a point x g Ž0, ⬁., then Lim n MnŽ r . Ž f Ž t . ; x . y f Ž r . Ž x .

nª⬁

s r Ž r q 1 . f Ž r . Ž x . q  2 Ž r q 1 . x q r 4 f Ž rq1. Ž x . q x Ž 1 q x . f Ž rq2. Ž x . .

Ž 3.4.

Further, if f Ž rq2. exists and is continuous on Ž a y ␩ , b q ␩ . ; Ž0, ⬁., ␩ ) 0, then Ž3.4. holds uniformly on w a, b x. Proof. By the Taylor expansion of f, we have rq2

MnŽ r . Ž f Ž t . ; x . s

f Ž i. Ž x .

Ý

i!

is0 ⬁

q

Žr. n

H0 W

⬁

Žr. n

H0 W

i Ž t , x . Ž t y x . dt

rq2 dt Ž t, x. ␧ Ž t, x. Ž t y x.

s I1 q I2 , say, where ␧ Ž t, x . ª 0 as t ª x. Using Lemma 2, we get rq2

I1 s

f Ž i. Ž x .

Ý

i!

is0

s

f Žr.Ž x . r! q q

i

Ý js0

dr i iyj yx . j Ž dx r

ž/

⬁

H0 W Ž t , x . t n

j

dt

MnŽ r . Ž t r ; x .

f Ž rq1. Ž x .

Ž r q 1. !

Ž r q 1 . Ž yx . MnŽ r . Ž t r ; x . q MnŽ r . Ž t rq1 ; x .

f Ž rq2. Ž x .

Ž r q 2. Ž r q 1.

Ž r q 2. !

2

x 2 MnŽ r . Ž t r ; x .

q Ž r q 2 . Ž yx . MnŽ r . Ž t rq1 ; x . q MnŽ r . Ž t rq2 ; x . .

668

NOTE

By Lemma 2, it follows that for each x g Ž0, ⬁., Mn Ž t i ; x . s

Ž n q i y 1. ! Ž n y i y 2. ! i x Ž n y 1. ! Ž n y 2. ! q i Ž i y 1.

Ž n q i y 2 . ! Ž n y i y 2 . ! iy1 x q O Ž ny2 . , Ž 3.5. Ž n y 1. ! Ž n y 2. !

and, therefore, f Ž rq1. Ž x . Ž n q r y 1. ! Ž n y r y 2. ! I1 s f Ž x . q Ž n y 1. ! Ž n y 2. ! Ž r q 1. ! Žr.

= Ž r q 1 . Ž yx . Ž r ! . q

½

Ž n q r y 1. ! Ž n y r y 3. ! Ž r !. Ž n y 1. ! Ž n y 2. !

f Ž rq2. Ž x .

Ž r q 2. Ž r q 1. x 2

Ž r q 2. !

2

q Ž r q 2 . Ž yx .

½

½

Ž r !.

5

Ž n q r y 1. ! Ž n y r y 2. ! Ž n y 1. ! Ž n y 2. !

Ž n q r . ! Ž n y r y 3. ! Ž r q 1. ! x q r Ž r q 1. Ž n y 1. ! Ž n y 2. ! =

q

5

Ž n q r . ! Ž n y r y 3. ! Ž r q 1. ! x Ž n y 1. ! Ž n y 2. ! qr Ž r q 1 .

q

½

Ž n q r y 1. ! Ž n y r y 2. ! Ž n y 1. ! Ž n y 2. !

Ž n q r y 1. ! Ž n y r y 3. ! Ž r !. Ž n y 1. ! Ž n y 2. !

5

Ž n q r q 1. ! Ž n y r y 4. ! Ž r q 2. ! 2 x 2 Ž n y 1. ! Ž n y 2. !

q Ž r q 1. Ž r q 2.

1 Ž n q r . ! Ž n y r y 4. ! Ž r q 1. ! x q O 2 n Ž n y 1. ! Ž n y 2. !

5 ž /

.

Hence, to prove Ž3.4., it is sufficient to show that for each x g Ž0, ⬁., nI2 ª 0 as n ª ⬁, which follows on proceeding along the lines of the proof of I2 ª 0 as n ª ⬁ in Theorem 1. The uniformity assertion follows as in the proof of Theorem 1. Finally, we give an estimate of the degree of approximation by MnŽ r . Ž⭈, x . for smooth functions.

669

NOTE

THEOREM 3. Let f g C␣ w0, ⬁. for some ␣ ) 0 and r F q F r q 2. If f Ž q. exists and is continuous on Ž a y ␩ , b q ␩ . ; Ž0, ⬁., ␩ ) 0, then for sufficiently large n, MnŽ r . Ž f Ž t . ; x . y f Ž r . Ž x . q

5 f Ž i. 5 q C2 ny1r2␻ f Ž rq 1. Ž ny1r2 . q O Ž ny2 . ,

F C1 ny1

žÝ / isr

where C1 and C2 are both independent of f and n, ␻ f Ž ␦ . is the modulus of continuity of f on Ž a y ␩ , b q ␩ ., and 5 ⭈ 5 denotes the sup-norm on w a, b x. Proof. By a finite Taylor expansion of f, q

f Ž t. s

f Ž i. Ž x .

Ý

i!

is0

i Ž t y x. q

f Ž q. Ž ␰ . y f Ž q. Ž x . q!

q

Ž t y x. ␹ Ž t.

q hŽ t , x . Ž 1 y ␹ Ž t . . , where ␰ lies between t and x, and ␹ Ž t . is the characteristic function of Ž a y ␩ , b q ␩ .. For t g Ž a y ␩ , b q ␩ . and x g w a, b x, we have q

f Ž t. s

f Ž i. Ž x .

Ý

i!

is0

i Ž t y x. q

f Ž q. Ž ␰ . y f Ž q. Ž x . q!

q

Žty␰ . .

For t g w0, ⬁. _ Ž a y ␩ , b q ␩ . and x g w a, b x, we define q

hŽ t , x . s f Ž t . y

f Ž i. Ž x .

Ý

i!

is0

i Ž t y x. .

Now, MnŽ r . Ž f Ž t . ; x . y f Ž r . Ž x . q

s

f Ž i. Ž x .

Ý

i!

is0

q

⬁

H0

q

⬁

WnŽ r . Ž t , x .

⬁

Žr. n

H0 W

Žr. n

H0 W

½

i Ž t , x . Ž t y x . dt y f Ž r . Ž x .

f Ž q. Ž ␰ . y f Ž q. Ž x . q!

Ž t , x . h Ž t , x . Ž 1 y ␹ Ž t . . dt

s I1 q I2 q I3 , say.

q

5

Ž t y x . ␹ Ž t . dt

670

NOTE

Using Ž3.5., we get q

I1 s

f Ž i. Ž x .

ž/ Ž . Ý ž /Ž

Ý q

Ý

i!

is0

s

dr i iyj yx . j Ž dx r

i js0

f Ž i. x

Ý

i

i!

is0

i j

js0

yx .

⬁

H0 W Ž t , x . t n

dr

iyj

dx

r

½

j

dt y f Ž r . Ž x .

Ž n q j y 1. ! Ž n y j y 2. ! j x Ž n y 1. ! Ž n y 2. !

qj Ž j y 1 .

Ž n q j y 2. ! Ž n y j y 2. ! Ž n y 1. ! Ž n y 2. !

= x jy1 q O Ž ny2 .

5

y f Žr.Ž x . .

Hence, q

5 I1 5 F C1 ny1

ž

Ý 5 f Ž i. 5 isr

/

q O Ž ny2 .

uniformly in x g w a, b x. Next, we estimate I2 as < I2 < F F F

⬁

H0 < W

Žr. n

Ž t, x. <

␻ f Žq. Ž ␦ . q!

⬁

< f Ž q. Ž ␰ . y f Ž q. Ž x . < q!

H0 < W

␻ f Žq. Ž ␦ .

Žr. n

q!


Ž t, x. < 1 q

ž

Ž n y 1.

< t y x < q␹ Ž t . dt

␦

/

< t y x < q dt

⬁

Ý < pnŽ r, .k Ž x . < ks1

=

⬁

H0

q

pn , ky1 Ž t . Ž < t y x < q q ␦y1 < t y x < qq1 . dt

Ž n q r y 1. ! yn yr Ž1 q x. Ž < x < q q ␦y1 < x < qq1 . , Ž n y 1. !

␦ ) 0.

Now, we show that for s s 0, 1, 2, . . . ,

Ž n y 1.

⬁

Ý ks1

pn , k Ž x . < k y nx < j

⬁

H0

pn , ky1 Ž t . < t y x < s dt s O Ž nŽ jys.r2 . .

671

NOTE

The left-hand side of the above equation is less than or equal to ⬁

Ž n y 1.

pn , k Ž x . < k y nx < j

Ý

ž

ks1

⬁

H0

=

1r2

pn , ky1 Ž t . dt ⬁

žH

0

ž

1r2

pn , ky1 Ž t . Ž t y x .

2s

dt

/

1r2

⬁

F

/

pn , k Ž x . Ž k y nx .

Ý

2j

ks1

/

⬁

= Ž n y 1.

ž

pn , k Ž x .

Ý ks1

⬁

H0

1r2

pn , ky1 Ž t . Ž t y x .

2s

dt

/

s O Ž n j r2 . O Ž nys r2 . s O Ž nŽ jys.r2 . uniformly in x, in view of Ž3.2. and Ž3.3.. Therefore, using Lemma 3, we have

Ž n y 1.

⬁

⬁

Ý < pnŽ r, .k Ž x .
0

ks1

F Ž n y 1.

pn , ky1 Ž t . < t y x < s dt

⬁

n i < k y nx < j

Ý Ý ks1 2 iqjFr i , jG0

=

⬁

H0

FK

x r Ž1 q x.

r

pn , k Ž x .

pn , ky1 Ž t . < t y x < s dt

Ý

ni Ž n y 1.

2 iqjFr i , jG0

sK

< qi , j, r Ž x . <

Ý

⬁

Ý

pn , k Ž x . < k y nx < j

ks1

⬁

H0

pn , ky1 Ž t . < t y x < s dt

n i O Ž nŽ jys.r2 . s O Ž nŽ rys.r2 .

Ž 3.6.

2 iqjFr i , jG0

uniformly in x, where K s sup

sup

2 iqjFr xg w a, b x i , jG0

< qi , j, r Ž x . < x r Ž1 q x.

r

.

672

NOTE

Choosing ␦ s ny1 r2 and making use of Ž3.6., we get for any m ) 0, 5 I2 5 F

␻ f Ž q . Ž ny1 r2 . q!

O Ž nŽ ryq.r2 . q n1r2 O Ž nŽ ryqy1.r2 . q O Ž nym .

F C2 nyŽ qyr .r2␻ f Ž q . Ž ny1r2 . . Since t g w0, ⬁. _ Ž a y ␩ , b q ␩ ., we can choose a ␦ ) 0 in such a way that < t y x < G ␦ for all x g w a, b x. Applying Lemma 3, we obtain < I3 < F Ž n y 1 .

⬁

Ý Ý

r i < k y nx < j

ks1 2 iqjFr i , jG0

=

H< tyx
q

n , ky1

< qi , j, r Ž x . < x r Ž1 q x.

r

pn , k Ž x .

Ž t . < h Ž t , x . < dt

Ž n q r y 1. ! yn yr < h Ž 0, x . < . Ž1 q x. Ž n y 1. !

If ␤ is any integer greater than or equal to  ␣ , q4 , then we can find a constant M1 such that < hŽ t, x .< F M1 < t y x < ␤ for < t y x < G ␦ . Now, applying Schwarz inequality, Ž3.2., and Ž3.3., it is easy to see that I3 s O Ž nys . for any s ) 0 uniformly on w a, b x. Combining the estimates of I1 , I2 , and I3 , the required result follows. REFERENCES 1. P. N. Agrawal, Inverse theorem in simultaneous approximation by Micchelli combination of Bernstein polynomials, Demonstratio Math. 31Ž1. Ž1998., 55᎐62. 2. P. N. Agrawal and V. Gupta, On convergence of derivatives of Phillips operators, Demonstratio Math. 27Ž2. Ž1994., 501᎐510. 3. M. M. Derriennic, Sur l’approximation de fonctions integrable sur w0, 1x par des polynomes ˆ de Bernstein modifies, J. Approx. Theory 31 Ž1981., 325᎐343. 4. Z. Ditzian and K. Ivanov, Bernstein-type operators and their derivatives, J. Approx. Theory 56 Ž1989., 72᎐90. 5. H. H. Gonska and X.-L. Zhou, A global inverse theorem on simultaneous approximation by Bernstein᎐Durrmeyer operators, J. Approx. Theory 67 Ž1991., 284᎐302. 6. H. S. Kasana, P. N. Agrawal, and V. Gupta, Inverse and saturation theorems for linear combination of modified Baskakov operators, Approx. Theory Appl. 7Ž2. Ž1991., 65᎐82. 7. G. G. Lorentz, ‘‘Bernstein Polynomials,’’ Univ. Toronto Press, Toronto, 1953.