An extension based on qR-integral for a sequence of operators

Applied Mathematics and Computation 218 (2011) 140–147

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

An extension based on qR-integral for a sequence of operators Octavian Agratini ⇑, Cristina Radu Babesß–Bolyai University, Faculty of Mathematics and Computer Science, 1 Koga˘lniceanu St., 400084 Cluj-Napoca, Romania

a r t i c l e

i n f o

a b s t r a c t The paper deals with a sequence of linear positive operators introduced via q-Calculus. We give a generalization in Kantorovich sense of its involving qR-integrals. Both for discrete operators and for integral operators we study the error of approximation for bounded functions and for functions having a polynomial growth. The main tools consist of the K-functional in Peetre sense and different moduli of smoothness. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Linear positive operator q-Integers Moduli of smoothness K-functional Weighted space

1. Introduction During the last two decades, q-Calculus was intensively used for the construction of different classical approximation processes of positive type. The breakthrough came in this research field with Lupasß [8] and Phillips [13] who proposed q-variants of the original Bernstein operators. These primary results are the bases of many research papers, and the comprehensive survey of Ostrovska [11] gives a good perspective of the subsequent achievements. Recently, in [14] the second author introduced a q-analogue of Baskakov–Mastroianni operators investigating their statistical approximation properties in weighted spaces. Further on, these operators have been more thoroughly investigated in [1], proving new properties. This class has the degree of exactness null. In this paper we have two goals. Firstly, we give an improved variant of this sequence which enjoys the property to preserve the polynomials of first degree. By using the Peetre’s K-functional, for this new class of operators we establish the rate of convergence. Secondly, we construct an integral version in Kantorovich sense of our discrete class. By examining two of the integrals introduced in q-Calculus, we select the one that gives an additional property of our construction. The central result consists in obtaining the rate of convergence of this extension both for bounded functions and for functions with polynomial growth. In connection with q-Calculus, for the reader’s convenience, we recall the following definitions and notation, see, e.g., [3,6]. Let q > 0. For any n 2 N0 ¼ f0g [ N, the q-integer [n]q and the q-factorial [n]q! are defined as follows

½nq ¼

n1 X

qj ;

½nq ! ¼

j¼0

n Y ½jq ;

n2N

j¼1

and [0]q = 0, [0]q! = 1. For integers k 2 {0, 1, . . . , n}, the q-binomial coefficients are denoted by

  n k

q

¼

½nq ! : ½kq !½n  kq !

For q = 1, we have [n]1 = n, [n]1! = n! and

  n and are defined by k q

    n n represents , the ordinary binomial coefficient. k 1 k

⇑ Corresponding author. E-mail addresses: [email protected] (O. Agratini), [email protected] (C. Radu). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.05.073

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

141

Throughout the paper we consider q 2 (0, 1). The q-derivative of a function f :¼ R ! R is defined by

Dq f ðxÞ ¼

f ðxÞ  f ðqxÞ ; ð1  qÞx

x – 0; D0q f

and the higher q-derivatives The following product rule

Dq f ð0Þ ¼ lim Dq f ðxÞ x!0

Dnq f

¼ f;



 ¼ Dq Dn1 f ; n 2 N. q

Dq ðf ðxÞgðxÞÞ ¼ Dq ðf ðxÞÞgðxÞ þ f ðqxÞDq ðgðxÞÞ is valid. There are different types of q-integrals associated to a function. The most known q-analogue of integration for the function f : Rþ ! R is given by

Iq ðf ; 0; aÞ ¼

Z

a

f ðtÞdq t ¼ ð1  qÞa

0

1 X

qj f ðaqj Þ;

a > 0;

ð1:1Þ

j¼0

provided that the series of the right hand side is convergent. Over a general interval [a, b], 0 < a < b, one defines

Iq ðf ; a; bÞ ¼ Iq ðf ; 0; bÞ  Iq ðf ; 0; aÞ:

ð1:2Þ

Remark 1.1. Some properties of the Riemann integral hold for the q-integral, others are not preserved. In this respect the Rb positivity property does not occur, this meaning that relation h P 0 does not imply 0 hðtÞdq t P 0, as it can be seen from the following example.  pffiffiffi  Let q 2 1=2; 2=2 be fixed. Let hðtÞ ¼ ð2  tÞv½1;2 ðtÞ; t P 0, where vI stands for the characteristic function of the interval R2 I. One has h P 0 and 1 hðtÞdq t < 0. Indeed,

Z

2

hðtÞdq t ¼ 2ð1  qÞðhð2Þ þ qhð2qÞÞ;

Z

0

1

hðtÞdq t ¼ ð1  qÞhð1Þ 0

and consequently,

Z

2

hðtÞdq t ¼ ð1  qÞð2q  1Þ2 :

1

Remark 1.2. Examining relations (1.1) and (1.2), we easily deduce: if h is q-integrable, non-negative and satisfies the property

khðktÞ P hðtÞ;

k > 1;

t P 0;

ð1:3Þ

then Iq(h; a, b) P 0 for any 0 < a < b (we choose k :¼ b/a and t :¼ at), Among the functions satisfying relation (1.3) the nondecreasing functions are included but it is not limited to this class only. In [15] the authors have introduced a q-integral of the Riemann-type as follows

Rq ðf ; a; bÞ ¼

Z a

b

R

f ðtÞdq t ¼ ðb  aÞð1  qÞ

1 X

f ða þ ðb  aÞqj Þqj :

ð1:4Þ

j¼0

A function f is qR-integrable on [a, b] if the series in (1.4) converges. The following correlation holds

Rq ðf ; a; bÞ ¼ ðb  aÞIq ð^f ; 0; 1Þ;

where ^f ðxÞ ¼ f ða þ ðb  aÞxÞ;

see [9, Lemma 2.1]. We also mention, for this integral the positivity property takes place. 2. Vn,q operators and their integral extension Inspired by a general class of operators introduced by Baskakov [4] and developed by Mastroianni [10], in [14] the author presented a q-analogue of it. We recall this construction. Let (/n)nP1 be a sequence of real functions on R which are continuous infinitely q-differentiable on R, satisfying the following conditions. (P1) /n ð0Þ ¼ 1; n 2 N, (P2) ð1Þk Dkq /n ðxÞ P 0; n 2 N; k 2 N0 ; x P 0, i.e., each /n is q-total monotonicity, sometimes also called q-complete monotonicity. (P3) For every ðn; kÞ 2 N  N0 there exists a positive integer ik, 0 6 ik 6 k, such that

142

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147 ik þ1 kik Dkþ1 Dq /n ðqik þ1 xÞbn;k;ik ;q ðxÞ; q /n ðxÞ ¼ ð1Þ

ð2:1Þ

where

lim n

bn;k;ik ;q ð0Þ

¼ 1:

½niqk þ1 qkik

ð2:2Þ

We define the operators

ðV n;q f ÞðxÞ ¼

! 1 X ½kq ðan xÞk kðk1Þ k q 2 Dq /n ðan xÞf k1 ; ½kq ! q ½nq k¼0

x P 0;

ð2:3Þ

where

an ¼ 

½nq : Dq /n ð0Þ

ð2:4Þ

For each n 2 N; V n;q is a linear positive operator satisfying the interpolation property (Vn,qf)(0) = f(0). Examining (2.1) and (2.2) with k = 0 and consequently ik = 0, it is clear that limnan = 1. On the other hand, taking into account the operators Tn(f; q; ), see [14, Eq. (3.8)], we have

ðV n;q f ÞðxÞ ¼ T n ðf ; q; an xÞ;

x P 0:

ð2:5Þ

This slight modification guarantees that Vn,q has the degree of exactness one. The result established in [14, Lemma 1] turns into the following Lemma 2.1. For all n 2 N and 0 < q < 1, one has

V n;q e0 ¼ e0 ;

V n;q e1 ¼ e1 ;

V n;q e2 ¼

D2q /n ð0Þ qðDq /n ð0ÞÞ2

e2 þ

1 e1 ; ½nq

ð2:6Þ

where ej stands for the monomial of j degree, 0 6 j 6 2. Moreover, on the basis of both (2.4) and relations (2.1) and (2.2), we get

lim n

D2q /n ð0Þ 2

ðDq /n ð0ÞÞ

¼ lim

D2q /n ð0Þ

n

½n2q

an ¼ q1i1 ;

ð2:7Þ

where i1 could be 0 or 1. R In order to present an integral extension of these operators, theoretically we can use dqt or dq t. Since it is useful to keep the positivity property of the operators, we use the second variant, see Remark 1.1. In what follows, we denote by DqR;loc ðRþ Þ the linear space of all real-valued functions qR-integrable on any interval [a,b], 0 6 a < b. For each n 2 N and f 2 DqR;loc ðRþ Þ we consider

ðK n;q f ÞðxÞ ¼

1 X ðan xÞk kðk1Þ k q 2 Dq /n ðan xÞkn;k ðf Þ; ½kq k¼0

x P 0;

ð2:8Þ

where

kn;k ðf Þ ¼ ½nq Rq ðf ; q1 an;k ; an;kþ1 Þ with an;k ¼

½kq qk1 ½nq

ð2:9Þ

and an is given at (2.4). The operators Kn,q, n 2 N, are linear and positive. For each m 2 N0 , the monomial em belongs to DqR;loc ðRþ Þ. On the basis of (1.4) and (2.9), by a straightforward calculation, we obtain

kn;k ðe0 Þ ¼ 1; kn;k ðe2 Þ ¼

kn;k ðe1 Þ ¼

1 1 ; an;k þ q ½2q ½nq

1 2 2 1 a þ an;k þ : q2 n;k q½2q ½nq ½3q ½n2q

Using these three identities in (2.8) and taking into account relation (2.6), we state Lemma 2.2. The operators K n;q ; n 2 N, defined by (2.8) satisfy the following identities

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

K n;q e0 ¼ e0 ; 1 1 ; K n;q e1 ¼ e1 þ q ½2q ½nq K n;q e2 ¼

D2q /n ð0Þ q3 ðDq /n ð0ÞÞ2

143

ð2:10Þ ð2:11Þ

e2 þ

3q þ 1 e1 1 þ : ½2q q2 ½nq ½3q ½n2q

ð2:12Þ

Introducing the second order central moment of the operators Kn,q, that is

Xn;q ðxÞ ¼ ðK n;q u2x ÞðxÞ;

where ux ðtÞ ¼ t  x;

ðt; xÞ 2 Rþ  Rþ

ð2:13Þ

and by using Lemma 2.1, we get

Xn;q ðxÞ ¼

D2q /n ð0Þ

 q3 ðDq /n ð0ÞÞ2

! 2  q 2 3q þ 1  2q2 x 1 x þ þ 6 q q2 ð1 þ qÞ ½nq ½3q ½n2q

! 3  1 x2 þ 2 ðx þ 1Þ; q ½nq q3 ðDq /n ð0ÞÞ2 D2q /n ð0Þ

x P 0:

Setting

(

sn ðqÞ ¼ max

) 3 ;  1; q2 ½nq q3 ðDq /n ð0ÞÞ2 D2q /n ð0Þ

ð2:14Þ

one obtains

Xn;q ðxÞ 6 sn ðqÞðx2 þ x þ 1Þ;

x P 0:

ð2:15Þ

Knowing (2.7), clearly (sn(q))n is a bounded sequence for each given q 2 (0, 1). Remark 2.1. Since limn[n]q = (1  q)1, relation (2.11) implies

lim K n;q e1 – e1 : n

Consequently, (Kn,q)nP1 does not form an approximation process. To acquire this property, for each n 2 N, the constant q will be replaced by a number qn 2 (0, 1) such that limnqn = 1. T We set Dloc ðRþ Þ ¼ 0
limðK n;qn f ÞðxÞ ¼ f ðxÞ; n

uniformly in x 2 K:

ð2:16Þ

Proof. We apply the universal Korovkin-type property with respect to positive linear operators, see the monograph of Altomare and Campiti [2, Theorem 4.1.4(vi)]. In our case we use the lattice homomorphism T : CðRþ Þ ! CðKÞ defined by T ðf Þ ¼ f jK . Since limn ½nqn ¼ 1 and, by virtue of (2.7), limn

D2qn /n ð0Þ

q3n ðDqn /n ð0ÞÞ2

¼ 1 holds. Taking the advantage of (2.10)–(2.12), we deduce

T ðK n;qn ej Þ ¼ T ðej Þ uniformly on K, where j = 0, 1, 2. The mentioned Korovkin-type theorem implies (2.16) and the proof is concluded. h

3. On the rate of convergence In what follows, C B ðRþ Þ represents the space of all real-valued continuous bounded functions on Rþ endowed with the usual sup-norm kk, kfk = supxP0jf(x)j. Set C 2B ðRþ Þ ¼ ff 2 C B ðRþ Þ : f 0 ; f 00 2 C B ðRþ Þg. The Peetre’s K-functional [12] is defined by

K 2 ðf ; dÞ ¼

inf fkf  gk þ dkg 00 kg;

g2C 2B ðRþ Þ

d > 0:

By using this functional, for Vn,q operators a general estimation of the error will be read as follows. Theorem 3.1. Let the operators Vn,q, n 2 N, be defined by (2.3). For any f 2 C B ðRþ Þ, one has

jðV n;q f ÞðxÞ  f ðxÞj 6 2K 2 ðf ; dn;q ðxÞÞ;

x P 0;

where

dn;q ðxÞ ¼ cn;q x2 þ

x ½nq

and cn;q ¼

D2q /n ð0Þ qðDq /n ð0ÞÞ2

 1:

ð3:1Þ

144

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

Proof. For x = 0 the inequality is evident. Let x > 0 and g 2 C 2B ðRþ Þ be arbitrarily fixed. Using the Taylor’s formula

Z

gðtÞ ¼ gðxÞ þ ðt  xÞg 0 ðxÞ þ

t

ðt  uÞg 00 ðuÞdu;

t P 0;

x

on the basis of identities (2.6), we obtain

ðV n;q gÞðxÞ  gðxÞ ¼ V n;q

Z

:

 ð  uÞg 00 ðuÞdu; x :

ð3:2Þ

x

Since g 00 2 C B ðRþ Þ, we can write

Z t Z t Z t Z t ðt  uÞg 00 ðuÞdu 6 jt  ujjg 00 ðuÞjdu 6 kg 00 k jt  ujdu 6 kg 00 kjt  xj du ¼ ðt  xÞ2 kg 00 k: x

x

x

x

2

Taking into account Lemma 2.1, one has Vn,q((  x) ; x) = dn,q(x), where dn,q(x) is given at (3.1). Returning at (3.2) and by using (2.6), we get

jðV n;q gÞðxÞ  gðxÞj 6 kg 00 kdn;q ðxÞ: Also, for any h 2 C B ðRþ Þ the Vn,q operator is non-expansive, i.e.,

jðV n;q hÞðxÞj 6 khkðV n;q e0 ÞðxÞ ¼ khk;

x P 0:

Since Vn,q is a linear positive operator, the inequalities established above allow us to write

 1 jðV n;q f ÞðxÞ  f ðxÞj 6 jV n;q ðf  g; xÞj þ jðV n;q gÞðxÞ  gðxÞj þ jgðxÞ  f ðxÞj 6 2 kf  gk þ dn;q ðxÞkg 00 k : 2

Applying inf g2C 2 ðRþ Þ , we obtain the desired result. h B

For any function f 2 C B ðRþ Þ, the first and the second order moduli of smoothness are defined as

x1 ðf ; dÞ ¼ sup jf ðx þ hÞ  f ðxÞj; 0
x2 ðf ; dÞ ¼ sup jf ðx þ 2hÞ  2f ðx þ hÞ þ f ðxÞj; 0
respectively. Corollary 3.1. Let the operators V n;q ; n 2 N, be defined by (2.3). For any f 2 C B ðRþ Þ one has

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðV n;q f ÞðxÞ  f ðxÞj 6 M x2 f ; dn;q ðxÞ ;

x P 0;

where M is a constant independent of f and dn,q(x) is given at (3.1). The proof is based on the connection between K2 functional and x2 modulus of smoothness. To certify this see the classical result due to Johnen [5, Proposition 6.1], where we choose X ¼ C B ðRþ Þ; r ¼ 2 and c2 = M. Theorem 3.2. If Kn,q is defined by (2.8), then for each f 2 DqR;loc ðRþ Þ \ CðRþ Þ the following inequality

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jðK n;q f ÞðxÞ  f ðxÞj 6 2x1 f ; sn ðqÞðx2 þ x þ 1Þ ;

x P 0;

holds, where sn(q) is given at (2.14). Proof. At first step we recall that the qR-integral possesses the following properties: (i) if f 6 g then Rq(f; a, b) 6 Rq(g; a, b), (ii) jRq(f; a, b)j 6 Rq(jfj; a, b), where 0 < a < b. Since (Kn,qe0)(x) = 1, we can write

jðK n;q f ÞðxÞ  f ðxÞj 6 ½nq

Z an;kþ1 1 X ðan xÞk kðk1Þ k R q 2 Dq /n ðan xÞ jf ðtÞ  f ðxÞjdq t: ½kq q1 an;k k¼0

On the other hand, the properties of x1 guarantee

jf ðtÞ  f ðxÞj 6 x1 ðf ; jt  xjÞ 6 ð1 þ d2 ðt  xÞ2 Þx1 ðf ; dÞ; This way, relations (3.3), (2.10) and (2.13) imply

d > 0:

ð3:3Þ

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

145

jðK n;q f ÞðxÞ  f ðxÞj 6 ð1 þ d2 Xn;q ðxÞÞx1 ðf ; dÞ: Taking into account (2.15) and choosing d ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sn ðqÞðx2 þ x þ 1Þ, the proof is complete. h

Further on, we focus on evaluating the local rate of approximation in the frame of weighted spaces. We consider the weight wN, wN(x) = (1 + xN)1, x P 0, where N 2 N, N P 2, is fixed. The spaces

BN ðRþ Þ ¼ ff : Rþ ! Rþ jwN f is boundedg; C N ðRþ Þ ¼ BN ðRþ Þ \ CðRþ Þ are endowed with the usual norm kkN, i.e.,

kf kN ¼ sup wN ðxÞjf ðxÞj: xP0

Remark 3.1. According to [1, Lemma 5], Tn,q operators map C N ðRþ Þ into C N ðRþ Þ. On the basis of (2.5) we deduce that the same property is inherited by Vn,q operators. Lemma 3.1. The operators K n;q ; n 2 N, defined by (2.8) map C N ðRþ Þ into C N ðRþ Þ. Proof. By using the elementary inequality (x + y)N 6 2N1(xN + yN), x P 0, y P 0, we get

jRq ðeN ; q1 an;k ; an;kþ1 Þj 6

! 2N1 aNn;k 1 : þ ½nq qN ½N þ 1q ½nNq

Consequently, kn;k ðeN Þ 6 2N1 ðqN aNn;k þ 1Þ and

jðK n;q eN ÞðxÞj 6 2N1 ðqN ðV n;q eN ÞðxÞ þ 1Þ: Since V n;q eN 2 C N ðRþ Þ, the conclusion follows.

ð3:4Þ h

Let hx 2 C N ðRþ Þ; x P 0, be defined as follows

hx ðtÞ ¼ 1 þ ðx þ jt  xjÞN ;

t P 0:

ð3:5Þ

Lemma 2.2 guarantees that the constants ci(n, N, q), i = 1 and i = 2, exist such that

(

ðK n;q hix ÞðxÞ 6 ci ðn; N; qÞð1 þ xN Þi ; ci ðn; N; qÞ ¼ Oð1Þðn ! 1Þ;

ð3:6Þ

i ¼ 1; 2:

The latter statement is implied by [1, Lemma 6] and relation (3.4). Motivated by López-Moreno’s paper [7], for weighted estimates we use the following modulus

XN ðf ; dÞ ¼ sup wN ðx þ hÞjf ðx þ hÞ  f ðxÞj;

d > 0;

ð3:7Þ

xP0 0
where f 2 C N ðRþ Þ. Among its properties, we recall

XN ðf ; adÞ 6 ða þ 1ÞXN ðf ; dÞ;

d > 0;

a > 0:

ð3:8Þ

Theorem 3.3. Let K n;q ; n 2 N, be defined by (2.8). For every f 2 DqR;loc ðRþ Þ \ C N ðRþ Þ; N P 2, one has

 pffiffiffiffiffiffiffiffiffiffiffi jðK n;q f ÞðxÞ  f ðxÞj 6 cðn; N; qÞð1 þ xNþ1 ÞXN f ; sn ðqÞ ;

x P 0;

ð3:9Þ

where sn(q) is given at (2.14). Here c(n, N, q) is a constant depending only on the quantities indicated in the brackets and bounded with respect to n. Proof. The reasoning follows closely the course of [1, Theorem 3]. In the interests of completeness, we indicate the main steps. Let n 2 N and f 2 DqR;loc ðRþ Þ \ C N ðRþ Þ be given. For t P 0 and d > 0, on the basis of (3.7) and (3.8) with a :¼ jt  xjd1, we get

    jt  xj 1 jf ðtÞ  f ðxÞj 6 ð1 þ ðx þ jt  xjÞN Þ 1 þ XN ðf ; dÞ ¼ hx ðtÞ þ hx ðtÞjt  xj XN ðf ; dÞ; d d

146

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

where hx was introduced at (3.5). Further on, we write

  1 jðK n;q f ÞðxÞ  f ðxÞj 6 K n;q ðjf  f ðxÞj; xÞ 6 K n;q hx þ hx jux j; x XN ðf ; dÞ d  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðn; N; qÞ K n;q ðu2x ; xÞ ð1 þ xN ÞXN ðf ; dÞ: 6 c1 ðn; N; qÞ þ d

Taking into account (2.14), we choose d ¼

pffiffiffiffiffiffiffiffiffiffiffi sn ðqÞ. Since

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ xN Þ 1 þ x þ x2 6 2ð1 þ xNþ1 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi setting cðn; N; qÞ ¼ 2 max c1 ðn; N; qÞ; c2 ðn; N; qÞ , the claimed result follows. Moreover, cðn; N; qÞ ¼ Oð1Þðn ! 1Þ is a consequence of the identities (3.6). h Corollary 3.2. Under the assumptions of Theorem 3.2, the inequality (3.9) implies the following global estimate

 pffiffiffiffiffiffiffiffiffiffiffi kK n;q f  f kNþ1 6 ~cðN; qÞXN f ; sn ðqÞ ; where ~cðN; qÞ is a constant independent of f and n. For the particular case N = 2 we present the error of local approximation by Kn,q operators using a truncated modulus of continuity. For k > 0, setting Ik = [0, k] and Ik ¼ ðk; 1Þ, for any function f 2 CðRþ Þ we can associate modulus of continuity on Ikdefined by

xIk ðf ; dÞ ¼ supfjf ðtÞ  f ðxÞj : jt  xj 6 d; t; x 2 Ik g; d P 0: Theorem 3.4. Let Kn,q, n 2 N, be defined by (2.8). For every f 2 DqR;loc ðRþ Þ \ C 2 ðRþ Þ and x 2 Ik one has

 pffiffiffiffiffiffiffiffiffiffiffi  jðK n;q f ÞðxÞ  f ðxÞj 6 cðf ; kÞsn ðqÞ þ 2xIkþ1 f ; sn ðqÞðk þ 1Þ ;

where sn(q) is given at (2.14) and c(f, k) is a constant depending on f and k. Proof. Let x 2 Ikbe arbitrarily fixed and t 2 Rþ . Case 1. t 2 Ik+1. We get

  jt  xj jf ðtÞ  f ðxÞj 6 xIkþ1 ðf ; jt  xjÞ 6 1 þ xIkþ1 ðf ; dÞ: d

Case 2. t 2 Ikþ1 . Since f 2 C 2 ðRþ Þ and 1 < (t  x)2, we have

jf ðtÞ  f ðxÞj 6 M f ð2 þ x2 þ t2 Þ 6 M f ð2 þ 3x2 þ 2ðt  xÞ2 Þ 6 M f ð4 þ 3k2 Þðt  xÞ2 : Taking into account the above two cases,  for any t2 Rþ we can write

jf ðtÞ  f ðxÞj 6 M f ð4 þ 3k2 Þðt  xÞ2 þ 1 þ

jt  xj xIkþ1 ðf ; dÞ: d

Further on, we deduce

  ðK n;q jux jÞðxÞ jðK n;q f ÞðxÞ  f ðxÞj 6 K n;q ðjf  f ðxÞj; xÞ 6 M f ð4 þ 3k2 ÞXn;q ðxÞ þ 1 þ xIkþ1 ðf ; dÞ d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 sn ðqÞðk2 þ k þ 1Þ AxI ðf ; dÞ; 6 cðf ; kÞsn ðqÞ þ @1 þ kþ1 d where c(f, k) = Mf(4 + 3k2)(k2 + k + 1). In the above we also used (2.15). Choosing

d¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sn ðqÞðk2 þ k þ 1Þ

and knowing that xIkþ1 ðf ; Þ is increasing, we arrive at (3.10). h Remark 3.2. The inequality (3.10) implies the following global estimate

 pffiffiffiffiffiffiffiffiffiffiffi  kK n;q f  f kCð½0;kÞ 6 cðf ; kÞsn ðqÞ þ 2xIkþ1 f ; sn ðqÞðk þ 1Þ for every f 2 DqR;loc ðRþ Þ \ C 2 ðRþ Þ.

ð3:10Þ

O. Agratini, C. Radu / Applied Mathematics and Computation 218 (2011) 140–147

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