Rate of convergence for generalized Baskakov operators - Core

Arab Journal of Mathematical Sciences (2012) 18, 39–50

King Saud University

Arab Journal of Mathematical Sciences www.ksu.edu.sa www.sciencedirect.com

ORIGINAL ARTICLE

Rate of convergence for generalized Baskakov operators Vijay Gupta, Rani Yadav

*

School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3, Dwarka, New Delhi 110 078, India Received 8 March 2011; revised 13 May 2011; accepted 7 August 2011

Available online 21 September 2011

KEYWORDS Bounded variation; Baskakov operators; Beta basis functions; Simultaneous approximation

Abstract In the present paper, we consider the generalized Baskakov operators having the weight functions of Beta basis functions. We study the rate of convergence for functions having derivatives of bounded variation. ª 2011 King Saud University. Production and hosting by Elsevier B.V. All rights reserved.

Introduction To approximate Lebesgue integrable functions on the interval [0, 1), Gupta [6] introduced the integral modification of the well known Baskakov operators by taking the weight functions of Beta basis functions. It was observed in [6] that by taking weights of Beta basis functions, one can have better approximation than the usual Baskakov–Durrmeyer operators [7]. In [6] the author has estimated an asymptotic formula and error estimation in simultaneous approximation for the Baskakov–Beta operators. In recent years a lot of work has been done on such * Corresponding author. Tel.: +91 9999015326. E-mail addresses: [email protected] (V. Gupta), [email protected] (R. Yadav). 1319-5166 ª 2011 King Saud University. Production and hosting by Elsevier B.V. All rights reserved. Peer review under responsibility of King Saud University. doi:10.1016/j.ajmsc.2011.08.001

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40

V. Gupta, R. Yadav

operators, we refer to some of the important papers on the recent developments on similar types of operators (see [1–5,8], etc.). We can define the Baskakov–Beta operators in generalized form as: Z 1 1 ðn þ r 1Þ!ðn r 1Þ! X p ðxÞ bnr;kþr ðtÞfðtÞdt; ð1:1Þ Vn;r ðf; xÞ ¼ nþr;k ððn 1Þ!Þ2 0 k¼0 where n 2 N, r 2 N0, n > r and the Baskakov and Beta basis functions are respectively defined as nþk1 xk 1 tk ; b ðtÞ ¼ ; pn;k ðxÞ ¼ n;k Bðk þ 1; nÞ ð1 þ tÞnþkþ1 k ð1 þ xÞnþk and Bðm; nÞ ¼ ðm1Þ!ðn1Þ! . ðnþm1Þ! The rate of convergence for certain Durrmeyer type operators and their Be´zier variants is one of the important areas of research in recent years. Zeng and collaborators have done commendable work in this direction and they estimated the rate of convergence for bounded/bounded variation functions (see [9–11]). In the present article, we extend the studies and here we estimate the rate of convergence for functions having derivatives of bounded variation. Auxiliary results In the sequel, we need the following results: Lemma 1 [6]. Let the m 2 N0, x 2 [0, 1), and suppose that Tn;r;m ðxÞ ¼

1 X k¼0

pnþr;k ðxÞ

Z

1

m

bnr;kþr ðtÞðt xÞ dt: 0

Then, Tn;r;0 ðxÞ ¼ 1;

Tn;r;1 ðxÞ ¼

ð1 þ rÞ þ xð1 þ 2rÞ ; nr1

n>rþ1

and Tn;r;2 ðxÞ ¼

2ð2r2 þ 4r þ n þ 1Þx2 þ 2ð2r2 þ 5r þ n þ 2Þx þ r2 þ 3r þ 2 ; ðn r 1Þðn r 2Þ

n > r þ 2: Also for n > m + r + 1, there holds the recurrence relation: h i ðn m r 1ÞTn;r;mþ1 ðxÞ ¼ xð1 þ xÞ T0n;r;m ðxÞ þ 2mTn;r;m1 ðxÞ þ ½ðm þ r þ 1Þð1 þ 2xÞ xTn;r;m ðxÞ:

41

Rate of convergence for generalized Baskakov operators

Consequently for all x 2 [0, 1), we have Tn;r;m ðxÞ ¼ Oðn½ðmþ1Þ=2 Þ; where [a] denotes the integral part of a. Remark 1. From Lemma 1, taking n to be sufficiently large, x 2 (0, 1), we observe that 2xð1 þ xÞ Cxð1 þ xÞ 6 Tn;r;2 ðxÞ 6 ; nr2 nr2

for ðC > 2Þ:

Remark 2. Applying the Cauchy–Schwarz inequality and keeping the same conditions as in Remark 1 for x, n and C, we derive from Lemma 1, that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 1 X 1 Cxð1 þ xÞ pnþr;k ðxÞ bnr;kþr ðtÞjt xjdt 6 ½Tn;r;2 ðxÞ2 6 : nr2 0 k¼0 Lemma 2. Suppose that x 2 (0, 1) and C > 2, then for sufficiently large n, we have kn;r ðx; yÞ ¼

1 X

pnþr;k ðxÞ

y

bnr;kþr ðtÞdt 6

0

k¼0

1 kn;r ðx; zÞ ¼

Z

1 X

pnþr;k ðxÞ

Z

Cxð1 þ xÞ ðn r 2Þðx yÞ2

1

bnr;kþr ðtÞdt 6 z

k¼0

;

Cxð1 þ xÞ ðn r 2Þðz xÞ2

06y max{q, r + s + 1}, we have

42

V. Gupta, R. Yadav

Ds Vn;r ðf; xÞ ¼ Vn;rþs ðDs f; xÞ;

D

d : dx

Proof. First nþk1

D½pn;k ðxÞ ¼

k nþk1

¼

k

! D

xk ð1 þ xÞnþk

!" k

xk1 ð1 þ xÞnþk

ðn þ kÞ

#

xk ð1 þ xÞnþkþ1

ðn þ k 1Þ! xk1 ðn þ k 1Þ! xk ðn þ kÞ ðk 1Þ!ðn 1Þ! ð1 þ xÞnþk k!ðn 1Þ! ð1 þ xÞnþkþ1 ! ! ðn þ 1Þ þ ðk 1Þ 1 nþk xk1 xk ¼n : nþk ð1 þ xÞ ð1 þ xÞnþkþ1 k1 k ¼

Thus D½pn;k ðxÞ ¼ n½pnþ1;k1 ðxÞ pnþ1;k ðxÞ:

ð2:1Þ

Proceeding along similar lines, we have D½bn;k ðxÞ ¼ n½bnþ1;k1 ðxÞ bnþ1;k ðxÞ

ð2:2Þ

The identities (2.1) and (2.2), are true even for the case k = 0, as we observe that bn+1,negative(x) = 0 and pn+1,negative(x) = 0. We shall prove the result by using the principle of mathematical induction. Using (2.1) and (2.2), we have

D½Vn;r ðf; xÞ ¼ ¼

1 ðn þ r 1Þ!ðn r 1Þ! X

ððn 1Þ!Þ2

Dpnþr;k ðxÞ

1

bnr;kþr ðtÞfðtÞdt

0

k¼0

1 ðn þ r 1Þ!ðn r 1Þ! X ðn þ rÞ½pnþrþ1;k1 ðxÞ pnþrþ1;k ðxÞ ððn 1Þ!Þ2 k¼0 Z 1 bnr;kþr ðtÞfðtÞdt 0

¼

Z

1 ðn þ rÞ!ðn r 1Þ! X

ððn 1Þ!Þ2 bnr;kþr ðtÞfðtÞdt:

k¼0

pnþrþ1;k ðxÞ

Z

1

½bnr;kþrþ1 ðtÞ 0

43


Using (2.2), and integrating by parts we have DVn;r ðf; xÞ ¼ ¼

1 ðn þ rÞ!ðn r 1Þ! X 2

ððn 1Þ!Þ

pnþrþ1;k ðxÞ

ððn 1Þ!Þ2

1

0

k¼0

1 ðn þ rÞ!ðn r 2Þ! X

Z

pnþrþ1;k ðxÞ

Z

D½bnr1;kþrþ1 ðtÞ fðtÞdt nr1

1

bnr1;kþrþ1 ðtÞfð1Þ ðtÞdt 0

k¼0

¼ Vn;rþ1 ðDf; xÞ; which means that the identity is satisfied for s = 1. Let us suppose that the result holds for s = l, i.e., Dl Vn;r ðf; xÞ ¼ Vn;rþl ðDl f; xÞ ¼

1 ðn þ r þ l 1Þ!ðn r l 1Þ! X

ððn 1Þ!Þ2

Z

pnþrþl;k ðxÞ

k¼0

1

bnrl;kþrþl ðtÞDðlÞ fðtÞdt:

0

Now, Dlþ1 Vn;r ðf; xÞ ¼


ððn 1Þ!Þ2

Z

Dpnþrþl;k ðxÞ

k¼0

1

bnrl;kþrþl ðtÞDl fðtÞdt 0

¼


ðn þ r þ lÞ½pnþrþlþ1;k1 ðxÞ ððn 1Þ!Þ2 k¼0 Z 1 bnrl;kþrþl ðtÞDl fðtÞdt pnþrþlþ1;k ðxÞ 0

¼

1 ðn þ r þ lÞ!ðn r l 1Þ! X

ððn 1Þ!Þ2

Z

pnþrþlþ1;k ðxÞ

k¼0

1

½bnrl;kþrþlþ1 ðtÞ bnrl;kþrþl ðtÞDl fðtÞdt 0

¼


ððn 1Þ!Þ2

Z

1

0

pnþrþlþ1;k ðxÞ

k¼0

D½bnrl1;kþrþlþ1 ðtÞ l D fðtÞdt: nrl1

44

V. Gupta, R. Yadav

Integrating by parts the last integral, we get Dlþ1 Vn;r ðf; xÞ ¼


Z

ððn 1Þ!Þ2

pnþrþlþ1;k ðxÞ

k¼0

1

bnrl1;kþrþlþ1 ðtÞDlþ1 fðtÞdt:

0

Therefore, Dlþ1 Vn;r ðf; xÞ ¼ Vn;rþlþ1 ðDlþ1 fðxÞÞ: Thus the result is true for s = l + 1, hence by mathematical induction, proof of the lemma is complete. h Rate of convergence The class of absolutely continuous functions f defined on (0, 1) is defined by Bq(0, 1), q > 0 and satisfying: (i) |f(t)| 6 C1tq, C1 > 0, (ii) having a derivative f0 on the interval (0, 1) which coincide a.e. with a function which is of bounded variation on every finite sub-interval of (0, 1). It can be observed that for all functions f 2 Bq(0, 1) possess for each C > 0 the representation Z x wðtÞdt; x P c: fðxÞ ¼ fðcÞ þ c

Theorem 1. Let f 2 Bq(0, 1), q > 0 and x 2 (0, 1). Then for C > 2 and n sufficiently large, we have ððn 1Þ!Þ2 V ðf; xÞ fðxÞ ðn þ r 1Þ!ðn r 1Þ! n;r pffiffi pffiffi ½ n xþx=k xþx= n Cð1 þ xÞ X _ 0 x _ Cð1 þ xÞ 6 ððf Þx Þ þ pffiffiffi ððf0 Þx Þ þ ðjfð2xÞ fðxÞ n r 2 k¼1 xx=k ðn r 2Þx n xx=pffiffin rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Cxð1 þ xÞ 0 þ 0 þ q 0 þ Cð1 þ xÞ þ xf ðx Þj þ jfðxÞjÞ þ Oðn Þ þ jf ðx Þj jf ðx Þ nr2 2 nr2 1 ð1 þ rÞ þ xð1 þ 2rÞ f0 ðx Þj þ jf0 ðxþ Þ þ f0 ðx Þj ; 2 nr1 Wb where a fðxÞ denotes the total variation of fx on [a, b], and the auxiliary function fx is defined by

45


8 > < fðtÞ fðx Þ; fx ðtÞ ¼ 0; > : fðtÞ fðxþ Þ;

0 6 t < x; t ¼ x; x < t < 1:

Proof. On applying the mean value theorem, we get ððn 1Þ!Þ2 Vn;r ðf; xÞ fðxÞ ðn þ r 1Þ!ðn r 1Þ! Z 1 1 X 6 pnþr;k ðxÞ bnr;kþr ðtÞjfðtÞ fðxÞjdt 0

k¼0

1 1 Z t X ¼ pnþr;k ðxÞbnr;kþr ðtÞf0 ðuÞdudt: 0 x k¼0 Z

Also, using the identity f0 ðuÞ ¼

f0 ðxþ Þ þ f0 ðx Þ f0 ðxþ Þ f0 ðx Þ þ ðf0 Þx ðuÞ þ sgnðu xÞ 2 2 f0 ðxþ Þ þ f0 ðx Þ þ f0 ðxÞ vx ðuÞ; 2

where

vx ðuÞ ¼

1; u ¼ x; 0; u – x:

We can see that 1 X k¼0

pnþr;k ðxÞ

Z 0

1

Z t x

f0 ðxÞ

f0 ðxþ Þ þ f0 ðx Þ vx ðuÞdu bnr;kþr ðtÞdt ¼ 0: 2

Now, by using the above identities, we have ððn 1Þ!Þ2 Vn;r ðf; xÞ fðxÞ ðn þ r 1Þ!ðn r 1Þ! Z 0 þ ! 1 Z tX 1 f ðx Þ þ f0 ðx Þ 0 6 pnþr;k ðxÞbnr;kþr ðtÞ þ ðf Þx ðuÞ du dt 0 2 x k¼0 Z ! 1 Z tX 1 0 þ 0 ½f ðx Þ f ðx Þ þ sgnðu xÞdu dt: pnþr;k ðxÞbnr;kþr ðtÞ 0 2 x k¼0 ð3:1Þ

46

V. Gupta, R. Yadav

Also, we have Z Z X 1 1 t ½f0 ðxþ Þ f0 ðx Þ p ðxÞb ðtÞdt sgnðu xÞdu nr;kþr nþr;k 0 2 x k¼0 6

jf0 ðxþ Þ f0 ðx Þj ½Tn;r;2 ðxÞ1=2 2

1

Z

ð3:2Þ

and Z 0

t x

¼

X 1 ½f0 ðxþ Þ þ f0 ðx Þ pnþr;k ðxÞbnr;kþr ðtÞdt du 2 k¼0

½f0 ðxþ Þ þ f0 ðx Þ Tn;r;1 ðxÞ: 2

ð3:3Þ

Combining (3.1)–(3.3), we have ððn 1Þ!Þ2 V ðf; xÞ fðxÞ ðn þ r 1Þ!ðn r 1Þ! n;r Z Z X 1 1 t 0 6 ðf Þx ðuÞdu pnþr;k ðxÞbnr;kþr ðtÞdt x x k¼0 þ

Z 0

x

Z x

t

X 1 0 ðf Þx ðuÞdu pnþr;k ðxÞbnr;kþr ðtÞdt k¼0

jf0 ðxþ Þ f0 ðx Þj jf0 ðxþ Þ þ f0 ðx Þj 1=2 ½Tn;r;2 ðxÞ þ Tn;r;1 ðxÞ 2 2 jf0 ðxþ Þ f0 ðx Þj 1=2 ½Tn;r;2 ðxÞ ¼ jAn;r ðf; xÞ þ Bn;r ðf; xÞj þ 2 jf0 ðxþ Þ þ f0 ðx Þj þ Tn;r;1 ðxÞ: 2 þ

Applying Remark 2 and Lemma 1, in (3.4), we have 2 ððn 1Þ!Þ Vn;r ðf; xÞ fðxÞ ðn þ r 1Þ!ðn r 1Þ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jf0 ðxþ Þ f0 ðx Þj Cxð1 þ xÞ 6 jAn;r ðf; xÞj þ jBn;r ðf; xÞj þ 2 nr2 0 þ 0 jf ðx Þ þ f ðx Þj ð1 þ rÞ þ xð1 þ 2rÞ þ : 2 nr1

ð3:4Þ

ð3:5Þ

47


The estimation of the terms An,r(f, x) and Bn,r(f, x) will lead to proof of the theorem. First, Z Z X 1 1 t ðf0 Þx ðuÞdu pnþr;k ðxÞbnr;kþr ðtÞdt jAn;r ðf; xÞj ¼ x x k¼0 Z Z 1 1 t X ¼ ðf0 Þx ðuÞdu pnþr;k ðxÞbnr;kþr ðtÞdt 2x x k¼0 Z 2x Z t 0 þ ðf Þx ðuÞdu dt ð1 kn;r ðx; tÞÞ x x Z 1 X 1 6 pnþr;k ðxÞ ðfðtÞ fðxÞÞbnr;kþr ðtÞdt k¼0 2x Z 1 X 1 0 þ þ jf ðx Þj p ðxÞ bnr;kþr ðtÞðt xÞdt k¼0 nþr;k 2x Z 2x þ ðf0 Þx ðuÞduj1 kn;r ðx; 2xÞj Z

x 2x

jðf0 Þx ðtÞjj1 kn;r ðx; tÞjdt x Z 1 1 X pnþr;k ðxÞ bnr;kþr ðtÞC1 t2q dt 6 þ

2x

k¼0

þ

jfðxÞj x2

1 X

pnþr;k ðxÞ

þ jf0 ðxþ Þj

1

bnr;kþr ðtÞðt xÞ2 dt

0

k¼0

Z

Z

1 2x

1 X

pnþr;k ðxÞbnr;kþr ðtÞjt xjdt

k¼0

Cð1 þ xÞ jfð2xÞ fðxÞ xf0 ðxþ Þj ðn r 2Þx pffiffi x xþpxffi ½ n xþ Cð1 þ xÞ X _k 0 x _n 0 þ ððf Þx Þ þ pffiffiffi ððf Þx Þ: ð3:6Þ n r 2 k¼1 x n x R1 P 2q For estimating the integral ðn r 1Þ 1 k¼0 pnþr;k ðxÞ 2x bnr;kþr ðtÞC1 t dt in (3.6) above, we proceed as follows: Obviously t P 2x implies that t 6 2(t x) and it follows from Lemma 1, that þ

1 X k¼0

pnþr;k ðxÞ

Z

1 2q

bnr;kþr ðtÞt dt 6 C1 2

2q

2x

1 X k¼0

pnþr;k ðxÞ

Z

1

bnr;kþr ðtÞðt xÞ2q dt

0

¼ C1 2 Tn;r;2q ðxÞ ¼ Oðnq Þðn ! 1Þ: 2q

48

V. Gupta, R. Yadav

To estimate the second term, we use Remark 1, thus Z 1 1 jfðxÞj X jfðxÞj Cxð1 þ xÞ pnþr;k ðxÞ bnr;kþr ðtÞðt xÞ2 dt ¼ 2 : : 2 x k¼0 x nr2 0 Applying Schwarz inequality and Remark 1, third term in right hand side of (3.6) is estimated as follows: Z 1 1 X pnþr;k ðxÞ bnr;kþr ðtÞjt xjdt jf0 ðxþ Þj 2x

k¼0

Z 1 1 jf ðx Þj X Cð1 þ xÞ 2 : 6 pnþr;k ðxÞ bnr;kþr ðtÞðt xÞ dt ¼ jf0 ðxþ Þj x nr2 0 k¼0 0

þ

By collecting the estimates, we have Cð1 þ xÞ Cð1 þ xÞ þ ðjfð2xÞ fðxÞ n r 2 ðn r 2Þx pffiffi x xþpxffin ½ n xþ k _ X _ Cð1 þ xÞ x ððf0 Þx Þ þ pffiffiffi ððf0 Þx Þ: xf0 ðxþ Þj þ jfðxÞjÞ þ n r 2 k¼1 x n x

jAn;r ðf; xÞj 6 Oðnq Þ þ jf0 ðxþ Þj

ð3:7Þ On applying, Lemma 2 with y ¼ x pxffiffin, and integrating by parts, we have Z x Z x Z t 0 jBn;r ðf; xÞj ¼ ðf Þx ðuÞdu dt ðkn;r ðx; tÞÞ ¼ kn;r ðx; tÞðf0 Þx ðtÞdt 0 x 0 Z y Z x 6 þ jðf0 Þx ðtÞjjkn;r ðx; tÞjdt 0

6

6

¼

y

Cxð1 þ xÞ nr2 Cxð1 þ xÞ nr2 Cxð1 þ xÞ nr2

Z

y

x _ ððf0 Þx Þ

0

Z

t y

x _

ððf0 Þx Þ

0

t pffiffi n

Z 1

1 ðx tÞ

dt þ 2

Z

x y

x _ ððf0 Þx Þdt t

x x _ p ffiffi ffi dt þ ððf0 Þx Þ 2 n xpxffi ðx tÞ n

1

x _

x x _ ððf0 Þx Þdu þ pffiffiffi ððf0 Þx Þ n xxu xpxffin

pffiffi ½ n x x Cð1 þ xÞ X _ x _ 6 ððf0 Þx Þ þ pffiffiffi ððf0 Þx Þ; n r 2 k¼1 xx n xpxffi k n

x where u ¼ xt . The required result is obtained on combining (3.5), (3.7) and (3.8).

ð3:8Þ

h


49

As a consequence of Lemma 3, we have the following corollary: Corollary 1. Let f(s) 2 DBq(0, 1), q > 0 and x 2 (0, 1). Then for C > 2 and for n sufficiently large, we have ððn 1Þ!Þ2 Ds Vn;r ðf; xÞ fðsÞ ðxÞ ðn þ r 1Þ!ðn r 1Þ! pffiffi pffiffi ½ n xþx=k xþx= n Cð1 þ xÞ X _ x _ Cð1 þ xÞ sþ1 6 ððD fÞx Þ þ pffiffiffi ððDsþ1 fÞx Þ þ n r 2 k¼1 xx=k xðn r 2Þ n xx=pffiffin ðjDs fð2xÞ Ds fðxÞ xDsþ1 fðxþ Þj þ jDs fðxÞjÞ þ Oðnq Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cð1 þ xÞ sþ1 þ 1 Cxð1 þ xÞ sþ1 þ jD fðx Þj þ jD fðx Þ Dsþ1 fðx Þj þ nr2 2 n 1 ð1 þ rÞ þ xð1 þ 2rÞ ; þ jDsþ1 fðxþ Þ þ Dsþ1 fðx Þj 2 nr1 where

Wb

a fðxÞ

denotes the total variation of fx on [a, b], and fx is defined by 8 sþ1 sþ1 > < D fðtÞ D fðx Þ; 0 6 t < x; Dsþ1 fx ðtÞ ¼ 0; t ¼ x; > : sþ1 sþ1 þ D fðtÞ D fðx Þ; x < t < 1:

Acknowledgement The authors are thankful to the referee for making valuable suggestions, leading to a better presentation of the paper. References [1] P.N. Agrawal, A.R. Gairola, On Lp-inverse theorem for a linear combination of Sza´sz–Beta operators, Thai J Math 8 (3) (2010) 11–20. [2] N. Deo, Pointwise estimate for modified Baskakov type operators, Lobachevskii J Math 31 (1) (2010) 36–42. [3] N. Deo, N. Bhardwaj, On the degree of approximation by modified Baskakov operators, Lobachevskii J Math 32 (1) (2011) 16–22. [4] N. Deo, S.P. Singh, On the degree of approximation by new Durrmeyer type operators, Gen Math 18 (2) (2010) 195–209. [5] A.R. Gairola, On certain Baskakov–Durrmeyer type operators, Surv Math Appl 5 (2010) 123–124. [6] V. Gupta, A note on modified Baskakov type operators, Approx Theory Appl 10 (3) (1994) 75–78. [7] A. Sahai, G. Prasad, On simultaneous approximation by modified Lupas operators, J Approx Theory 45 (1985) 122–128. [8] Z. Walczak, Baskakov type operator, Rocky Mountain, J Math 39 (3) (2009) 981–993. [9] X.M. Zeng, On the rate of convergence of the generalized Sza´sz type operators for functions of bounded variation, J Math Anal Appl 226 (1998) 309–325.

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[10] X.M. Zeng, W. Tao, Rate of convergence of the integral type Lupas–Be´zier operators, Kyungpook Math J 43 (2003) 593–604. [11] X.M. Zeng, X. Cheng, Pointwise approximation by the modified Sza´sz–Mirakyan operators, J Comput Anal Appl 9 (4) (2007) 421–430.