Fractional Korovkin theory

Chaos, Solitons and Fractals 42 (2009) 2080–2094

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

Fractional Korovkin theory George A. Anastassiou Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

a r t i c l e

i n f o

a b s t r a c t

Article history: Accepted 30 March 2009

In this article we study quantitatively with rates the weak convergence of a sequence of finite positive measures to the unit measure. Equivalently we study quantitatively the pointwise convergence of sequence of positive linear operators to the unit operator, all acting on continuous functions. From there we derive with rates the corresponding uniform convergence of the last. Our inequalities for all of the above in their right hand sides contain the moduli of continuity of the right and left Caputo fractional derivatives of the involved function. From our uniform Shisha–Mond type inequality we derive the first fractional Korovkin type theorem regarding the uniform convergence of positive linear operators to the unit. We give applications, especially to Bernstein polynomials for which we establish fractional quantitative results. In the background we establish several fractional calculus results useful to approximation theory and not only. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction ln this paper among others we are motivated by the following results. Theorem 1 [10], p. 14. Let ½a; b be a closed interval in R and ðLn Þn2N be a sequence of positive linear operators mapping Cð½a; bÞ into itself. Suppose that ðLn f Þ converges uniformly to f for the three test functions f ¼ 1; x; x2 . Then ðLn f Þ converges uniformly to f on ½a; b for all functions f 2 Cð½a; bÞ. Let f 2 Cð½a; bÞ and 0 6 h 6 b  a. The first modulus of continuity of f at h is given by

x1 ðf ; hÞ ¼ sup jf ðxÞ  f ðyÞj: x;y2½a;b jxyj6h

If h > b  a, then we define x1 ðf ; hÞ ¼ x1 ðf ; b  aÞ. Another motivation is the following. Theorem 2 [15]. Let ½a; b  R a closed interval. Let fLn gn2N be a sequence of positive linear operators acting on Cð½a; bÞ into itself. For n ¼ 1; . . ., suppose Ln (1) is bounded. Let f 2 Cð½a; bÞ. Then for n ¼ 1; 2; . . ., we have

  kLn f  f k1 6 k f k1 kLn 1  1k1 þ kLn 1 þ 1k1 x1 f ; ln ;

where

 



1

ln ¼ Ln ðt  xÞ2 ðxÞ

2

1

and k  k1 stands for the sup-norm over ½a; b. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.183

ð1Þ

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

2081

One can easily see, for n ¼ 1; 2; . . .

 





l2n 6 Ln t2 ; x  x2 1 þ 2ckLn ðt; xÞ  xk1 þ c2 kLn ð1; xÞ  1k1 ; where c ¼ maxðj a j; j b jÞ. Thus, given the Korovkin assumptions (see Theorem 1) as n ! 1 we get ln ! 0, and by (1) that kLn f  f k1 ! 0 for any f 2 Cð½a; bÞ. That is one derives the Korovkin conclusion in a quantitative way and with rates of convergence. One more motivation follows. Theorem 3 (see Corollary 7.2.2, p. 219 [1]). Consider the positive linear operator

L : C n ð½a; bÞ ! Cð½a; bÞ; n 2 N: Let

ck ðxÞ ¼ Lððt  xÞk ; xÞ; k ¼ 0; 1; . . . ; n;

 1 ba dn ðxÞ ¼ Lðjt  xjn ; xÞ n ; cðxÞ ¼ max ðx  a; b  xÞ cðxÞ P : 2 Let f 2 C n ð½a; bÞ such that x1 ðf ðnÞ ; hÞ 6 w, where w; h are fixed positive numbers, 0 < h < b  a. Then we have the following upper bound:

jLðf ; xÞ  f ðxÞj 6 jf ðxÞjjc0 ðxÞ  1j þ

  n  ðkÞ X f ðxÞ jck ðxÞj þ Rn : k! k¼1

ð2Þ

Here

Rn ¼ w/n ðcðxÞÞ



n

dn ðxÞ w h n ¼ hn d ðxÞ; cðxÞ n! cðxÞ n

where hn



h ¼ n!/n ðuÞ=un ; u

with

/n ðxÞ ¼

Z 0

jxj

t ðjxj  t Þn1 dt h ðn  1Þ!

ðx 2 RÞ:

de is the ceiling of the number. Inequality (2) is sharp. It is approximately attained by w/n ððt  xÞþ Þ and a measure lx supported by fx; bg when x  a 6 b  x, also approximately attained by w/n ððx  tÞþ Þ and a measure lx supported by fx; ag when x  a P b  x : in each  n  n n ðxÞ n ðxÞ and dcðxÞ , respectively. case with masses c0 ðxÞ  dcðxÞ Using the last method and its refinements one derives nice and simple results for specific operators. For example from Corollary 7.3.4, p. 230 [1], we obtain: let f 2 C 1 ð½0; 1Þ and consider the Bernstein polynomials

ðBn f ÞðtÞ ¼



n X n k k t ð1  tÞnk ; f n k k¼0 l

t 2 ½0; 1;

1 ffiffi pffiffi x1 ðf 0 ; p then kBn f  f k1 6 0:78125 Þ: So Bn f ! f as n ! 1 with rates. 4 n n In this article we study quantitatively the rate of weak convergence of a sequence of finite positive measures to the unit measure given the existence and presence of the left and right Caputo fractional derivatives of the involved function. That is in the right hand sides of the derived inequalities appear the first moduli of continuity of the above mentioned fractional derivatives, see Theorem 25 and Corollary 26. Then via the Riesz representation theorem we transfer Theorem 25 into the language of quantitative pointwise convergence of a sequence of positive linear operators to the unit operator, all operators acting from Cð½a; bÞ into itself, see Theorem 27, Corollary 28 and Theorem 30. From there we derive quantitative results with respect to the sup-norm k  k1 , regarding the uniform convergence of positive linear operators to the unit. Again in the right hand side of our inequalities we have moduli of continuity with respect to right and left Caputo derivatives of the engaged function. For the last see Theorem 32, a Shisha–Mond type result. From there we derive the first Korovkin type convergence theorem at the fractional level, see Theorem 33. We give plenty of applications of our fractional Shisha–Mond and Korovkin theory, see Corollaries 35–38. In the background section we present many interesting fractional results which by themselves have their own merit and appear for the first time.

2082

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

This is the first in literature article studying the quantitative convergence of positive linear operators to the unit at the fractional level. In approximation theory the involvement of fractional derivatives is very rare, almost nothing exists. The only fractional articles that exist so far are of Dzyadyk [5] of 1959, Nasibov [11] of 1962, Demjanovic [3] of 1975, and of Jaskolski [9] of 1989, all regarding estimates to best approximation of functions by algebraic and trigonometric polynomials [8].

2. Background We give Definition 4. Let v P 0, n ¼ dv e (de is the ceiling of the number), f 2 AC n ð½a; bÞ (space of functions f with f ðn1Þ 2 ACð½a; bÞ, absolutely continuous functions). We call left Caputo fractional derivative (see [4], p. 38 [7], [14]) the function

Dva f ðxÞ ¼

1 Cðn  v Þ

Z

x

ðx  tÞnv 1 f ðnÞ ðtÞdt

ð3Þ

a

8x 2 ½a; b, where C is the gamma function Cðv Þ ¼ We set

D0a f ðxÞ

R1 0

et t v 1 dt,

v > 0.

¼ f ðxÞ; 8x 2 ½a; b.

v ¼ 12, then n ¼ 1 and f ðtÞ ¼ tb 2 Cð½0; 1Þ, 0 < b 6 12, t 2 ½0; 1. See that f 0 ðtÞ ¼ btb1 2 L1 ð½0; 1Þ.

Example 5. Take

We observe that 1 b D20 f ðxÞ ¼ 1

C

2

Z

x

1

ðx  tÞ2 tb1 dt:

ð4Þ

0

By setting t ¼ xs, dt ¼ x ds, 1 b D20 f ðxÞ ¼ 1

C

2

Z

1

1

ðx  xsÞ2 xb1 sb1 xds ¼

0

Cðb þ 1Þ b12  x : C b þ 12

1

Let 0 < b < 12, then D20 f ð0Þ ¼ þ1: Let b ¼ 12, then

pffiffiffiffi

1

p

D20 f ð0Þ ¼

2

>0

ð5Þ

a positive real number! Conclusion: In general for Dva f ðaÞ we do not know what it is, it could be infinite, or finite non-zero, or zero! (see next). Lemma 6. Let

v > 0, v R N, n ¼ dv e, f

2 C n1 ð½a; bÞ and f ðnÞ 2 L1 ð½a; bÞ. Then Dva f ðaÞ ¼ 0.

Proof. By (3) we get

Z

 v  D f ðxÞ 6 a

1 Cðn  v Þ

 v  D f ðxÞ 6 a

 ðnÞ  f  1 ðx  aÞnv ; Cðn  v þ 1Þ

x

 f

nv 1  ðnÞ

ðx  tÞ

 ðtÞdt 6

a

 ðnÞ  f  1 ðx  aÞnv ; Cðn  v þ 1Þ

i.e.

That is Dva f ðaÞ ¼ 0.

8x 2 ½a; b:

ð6Þ



We need Definition 7. (see also [7,6,2]) Let f 2 AC m ð½a; bÞ, m ¼ dae, a > 0. The right Caputo fractional derivative of order a > 0 is given by

Dab f ðxÞ ¼

ð1Þm Cðm  aÞ

Z

b

ðf  xÞma1 f ðmÞ ðfÞdf

x

8x 2 ½a; b. We set D0b f ðxÞ ¼ f ðxÞ. Lemma 8. Let f 2 C m1 ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae, a > 0. Then Dab f ðbÞ ¼ 0. Proof. As in Lemma 6.



ð7Þ

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

Lemma 9. Let f 2 AC m ð½a; bÞ, m ¼ dae, a > 0;

2083

l is a positive finite measure on the Borel r-algebra of ½a; b, x0 2 ½a; b. Then

Z

Z m1 ðkÞ X f ðx0 Þ Ex0 :¼ f ðxÞdlðxÞ  ðx  x0 Þk dlðxÞ k! ½a;b ½a;b k¼0 (Z Z x0  

1 ðf  xÞa1 Dax0  f ðfÞ  Dax0  f ðx0 Þ df dlðxÞ ¼ CðaÞ ½a;x0  x ) Z x Z  

a1 a a þ ðx  fÞ Dx0 f ðfÞ  Dx0 f ðx0 Þ df dlðxÞ :

ð8Þ

x0

ðx0 ;b

Proof. From [4], p. 40, we get by left Caputo fractional Taylor formula that

f ðxÞ ¼

Z x m1 ðkÞ X f ðx0 Þ 1 ðx  fÞa1 Dax0 f ðfÞdf ðx  x0 Þk þ k! C ð a Þ x 0 k¼0

ð9Þ

for all x0 < x 6 b. Also from [2], using the right Caputo fractional Taylor formula we get

f ðxÞ ¼

Z x0 m1 ðkÞ X f ðx0 Þ 1 ðf  xÞa1 Dax0  f ðfÞdf ðx  x0 Þk þ k! C ð a Þ x k¼0

ð10Þ

for all a 6 x 6 x0 . Consequently we find

Z

f ðxÞdlðxÞ ¼

½a;b

Z

f ðxÞdlðxÞ þ

Z

½a;x0 

¼

f ðxÞdlðxÞ

ðx0 ;b

Z m1 ðkÞ X f ðx0 Þ ðx  x0 Þk dlðxÞ k! ½a;b k¼0 (Z Z x0

1 þ ðf  xÞa1 Dax0  f ðfÞdf dlðxÞ CðaÞ ½a;x0  x ) Z x

Z a1 a þ ðx  fÞ Dx0 f ðfÞdf dlðxÞ : ðx0 ;b

ð11Þ

x0

Notice also that Dax0  f ðx0 Þ ¼ Dax0 f ðx0 Þ ¼ 0. The proof of (8) is now complete.  Convention 10. We assume that

Dax0 f ðxÞ ¼ 0 for x < x0

ð12Þ

Dax0  f ðxÞ ¼ 0 for x > x0

ð13Þ

and

for all x; x0 2 ½a; b. We mention Proposition 11. Let f 2 C n ð½a; bÞ, n ¼ dv e; v > 0. Then Dva f ðxÞ is continuous in x 2 ½a; b. Proof. We notice that

Dva f ðxÞ ¼

1 Cðn  v Þ

Dva f ðyÞ ¼

1 Cðn  v Þ

Z

xa

znv 1 f ðnÞ ðx  zÞdz

0

and

Z

ya

znv 1 f ðnÞ ðy  zÞdz:

0

Here a 6 x 6 y 6 b and 0 6 x  a 6 y  a. Hence it holds

ð14Þ

2084

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

Dva f ðyÞ  Dva f ðxÞ ¼

1 Cðn  v Þ

Z

xa

  znv 1 f ðnÞ ðy  zÞ  f ðnÞ ðx  zÞ dz þ

0

Z

ya

 znv 1 f ðnÞ ðy  zÞdz :

ð15Þ

xa

We have that

 v  D f ðyÞ  Dv f ðxÞ 6 a a

" #  ðnÞ  f    1 ðx  aÞnv 1 x1 ðf ðnÞ ; jy  xjÞ þ ðy  aÞnv  ðx  aÞnv Cðn  v Þ ðn  v Þ ðn  v Þ #  ðnÞ   nv f    1 ðb  aÞ 1 6 x1 ðf ðnÞ ; jy  xjÞ þ ðy  aÞnv  ðx  aÞnv : Cðn  v Þ ðn  v Þ ðn  v Þ

So as y ! x the last expression goes to zero. As a result,

Dva f ðyÞ ! Dva f ðxÞ; proving the claim.

ð16Þ



Proposition 12. Let f 2 C m ð½a; bÞ, m ¼ dae, a > 0. Then Dab f ðxÞ is continuous in x 2 ½a; b. Proof. As in Proposition 11.



We also mention Proposition 13. Let f 2 C m1 ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae, a > 0 and

Dax0 f ðxÞ ¼

Z

1 Cðm  aÞ

x

ðx  tÞma1 f ðmÞ ðtÞdt

ð17Þ

x0

for all x; x0 2 ½a; b : x P x0 . Then Dax0 f ðxÞ is continuous in x0 . Proof. Fix x : x P y0 P x0 ; x; x0 ; y0 2 ½a; b. Then

  Z y0   1 ma1 ðmÞ   ðx  tÞ f ðtÞdt   Cðm  aÞ x0  ðmÞ  Z

y0 f  1 6 ðx  tÞma1 dt Cðm  aÞ x0  ðmÞ  f    1 ¼ ðx  y0 Þma  ðx  x0 Þma Cðm  a þ 1Þ

   a  a Dx0 f ðxÞ  Dy0 f ðxÞ ¼

! 0, as y0 ! x0 , proving continuity of Dax0 f in x0 2 ½a; b.

ð18Þ



Proposition 14. Let f 2 C m1 ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae, a > 0 and

Dax0  f ðxÞ ¼

ð1Þm Cðm  aÞ

Z

x0

ðf  xÞma1 f ðmÞ ðfÞdf

ð19Þ

x

for all x; x0 2 ½a; b : x0 P x. Then Dax0  f ðxÞ is continuous in x0 . Proof. As in Proposition 13.



We need Proposition 15. Let g 2 Cð½a; bÞ, 0 < c < 1, x; x0 2 ½a; b. Define

Lðx; x0 Þ ¼

Z

x

ðx  tÞc1 gðtÞdt

for x P x0 ;

ð20Þ

x0

and Lðx; x0 Þ ¼ 0, for x < x0 . Then L is jointly continuous in ðx; x0 Þ on ½a; b2 . Proof. We notice that Lðx0 ; x0 Þ ¼ 0. Assume x P x0 , then

Lðx; x0 Þ ¼

Z 0

xx0

zc1 gðx  zÞdz ¼

Z 0

ba

v½0;xx0  ðzÞzc1 gðx  zÞdz;

ð21Þ

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

2085

where v is the characteristic function. Let xN ! x; x0N ! x0 ; N 2 N and assume without loss of generality that xN P x0N . So we have again

Z

LðxN ; x0N Þ ¼

xN x0N

zc1 gðxN  zÞdz ¼

Z

ba

0

0

v½0;xN x0N  ðzÞzc1 gðxN  zÞdz:

ð22Þ

We have that

v½0;xN x0N  ðzÞ ! v½0;xx0  ðzÞ a:e:

ð23Þ

v½0;xN x0N  ðzÞzc1 gðxN  zÞ ! v½0;xx0  ðzÞzc1 gðx  zÞ a:e:

ð24Þ

and

Notice that

v½0;xN x0N  ðzÞzc1 jgðxN  zÞj 6 zc1 kg k1 ;

ð25Þ

which is an integrable function. Thus by Dominated Convergence theorem we obtain

LðxN ; x0N Þ ! Lðx; x0 Þ; asN ! 1:

ð26Þ 2

Clearly now Lðx; x0 Þ is jointly continuous on ½a; b .



We mention Proposition 16. Let g 2 Cð½a; bÞ, 0 < c < 1, x; x0 2 ½a; b. Define

Kðx; x0 Þ ¼

Z

x0

ðf  xÞc1 gðfÞdf for x 6 x0 ;

ð27Þ

x

and Kðx; x0 Þ ¼ 0, for x > x0 . Then Kðx; x0 Þ is jointly continuous from ½a; b2 into R. Proof. As in Proposition 15.



Based on Propositions 15 and 16 we derive Corollary 17. Let f 2 C m ð½a; bÞ, m ¼ dae, a > 0, x; x0 2 ½a; b. Then Dax0 f ðxÞ, Dax0  f ðxÞ are jointly continuous functions in ðx; x0 Þ from ½a; b2 into R. We need Theorem 18. Let f : ½a; b2 ! R be jointly continuous. Consider

GðxÞ ¼ x1 ðf ð; xÞ; d; ½x; bÞ; d > 0, x 2 ½a; b. Then G is continuous on ½a; b. Proof. (i) Let xn ! x, a 6 xn 6 x, and 0 < d 6 b  x first (The case when xn ! x with xn P x is similar). Then we can write

Gðxn Þ ¼ maxðA; B; CÞ; where

A ¼ sup fjf ðu; xn Þ  f ðv ; xn Þj; u; v 2 ½x; b; ju  v j 6 dg;

0 < d 6 b  x;

B ¼ sup fjf ðu; xn Þ  f ðv ; xn Þj; u 2 ½xn ; x; v 2 ½x; b; ju  v j 6 dg; C ¼ sup fjf ðu; xn Þ  f ðv ; xn Þj; u; v 2 ½xn ; x; ju  v j 6 dg: Now, when xn ! x, then A ! GðxÞ; B ! KðxÞ 6 GðxÞ; C ! 0 (since also u converges to v). In conclusion, Gðxn Þ ! maxfGðxÞ; KðxÞ; 0g ¼ GðxÞ. (ii) If d > b  x, then x1 ðf ð; xÞ; d; ½x; bÞ ¼ x1 ðf ð; xÞ; b  x; ½x; bÞ, a case covered by (i). That is proving the claim.



Theorem 19. Let f : ½a; b2 ! R be jointly continuous. Then

2086

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

HðxÞ ¼ x1 ðf ð; xÞ; d; ½a; xÞ; x 2 ½a; b, is continuous in x 2 ½a; b; d > 0. Proof. As in Theorem 18.



We make Remark 20. Let

Z

l be a finite positive measure on Borel r-algebra of ½a; b. Let a > 0, then by Hölder’s inequality we obtain a !ðaþ1Þ

Z

a

ðx0  xÞ dlðxÞ 6

½a;x0 

ðx0  xÞ

aþ1

dlðxÞ

ð28Þ

½a;x0 

and

Z

1

lð½a; x0 Þðaþ1Þ ; a !ðaþ1Þ

Z

ðx  x0 Þa dlðxÞ 6

ðx0 ;b

ðx  x0 Þaþ1 dlðxÞ

1

lððx0 ; bÞðaþ1Þ :

ð29Þ

ðx0 ;b

Let now m ¼ dae; a R N, a > 0, k ¼ 1; . . . ; m  1. Then by applying again Hölder’s inequality we obtain

Z

k

jx  x0 j dlðxÞ 6

Z

½a;b

k !ðaþ1Þ

j x  x0 j

aþ1

dlðxÞ

aþ1k

lð½a; bÞ ðaþ1Þ :

ð30Þ

½a;b

Terminology 21. Let LN : Cð½a; bÞ ! Cð½a; bÞ, N 2 N, be a sequence of positive linear operators. By Riesz representation theorem (see [13], p. 304) we have

LN ðf ; x0 Þ ¼

Z

½a;b

f ðtÞdlNx0 ðtÞ

ð31Þ

8x0 2 ½a; b, where lNx0 is a unique positive finite measure on rBorel algebra of ½a; b. Call LN ð1; x0 Þ ¼ lNx0 ð½a; bÞ ¼ M Nx0 :

ð32Þ

We make Remark 22. Let f 2 C n1 ð½a; bÞ, f ðnÞ 2 L1 ð½a; bÞ, n ¼ dv e; v > 0; v R N. Then as in the proof of Lemma 6, we have

 v  D f ðxÞ 6 a

 ðnÞ  f  1 ðx  aÞnv Cðn  v þ 1Þ

8x 2 ½a; b:

ð33Þ

Thus we observe

  x1 ðDva f ; dÞ ¼ sup Dva f ðxÞ  Dva f ðyÞ x;y2½a;b jxyj6d

6 sup x;y2½a;b jxyj6d

6 Consequently

x1 ðDva f ; dÞ 6

!  ðnÞ   ðnÞ  f  f  nv nv 1 1 þ ðx  aÞ ðy  aÞ Cðn  v þ 1Þ Cðn  v þ 1Þ

  2f ðnÞ 1 ðb  aÞnv : Cðn  v þ 1Þ

  2f ðnÞ 1 ðb  aÞnv : Cðn  v þ 1Þ

ð34Þ

ð35Þ

ð36Þ

Similarly, let f 2 C m1 ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae; a > 0; a R N, then

  2f ðmÞ 1 x1 ðDb f ; dÞ 6 ðb  aÞma : Cðm  a þ 1Þ a

ð37Þ

So for f 2 C m1 ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae; a > 0; a R N, we find

  2f ðmÞ 1 ðb  aÞma Cðm  a þ 1Þ

ð38Þ

  2f ðmÞ 1 6 ðb  aÞma : Cðm  a þ 1Þ

ð39Þ

sup x1 ðDax0 f ; dÞ½x0 ;b 6

x0 2½a;b

and a

sup x1 ðDx0  f ; dÞ½a;x0 

x0 2½a;b

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

2087

We also make Remark 23. Let LN : Cð½a; bÞ ! Cð½a; bÞ, N 2 N, be a sequence of positive linear operators. Using (31) and Hölder’s inequality we obtain ðx 2 ½a; b; k ¼ 1; . . . ; m  1; m ¼ dae; a R N; a > 0Þ for k ¼ 1; . . . ; m  1 that

    k ðaþ1kÞ    ðaþ1Þ k aþ1 LN ðj  xj ; xÞ 6 LN ðj  xj ; xÞ kLN 1k1 aþ1 : 1

ð40Þ

1

Also we observe that

Cð½a; bÞ 3 j  xjaþ1 v½a;x ðÞ 6 j  xjaþ1

8x 2 ½a; b

ð41Þ

Cð½a; bÞ 3 j  xjaþ1 v½x;b ðÞ 6 j  xjaþ1

8x 2 ½a; b:

ð42Þ

and

By positivity of LN we obtain

        aþ1 aþ1 LN ðj  xj v½a;x ðÞ; xÞ 6 LN ðj  xj ; xÞ 1

ð43Þ

1

and

        aþ1 aþ1 LN ðj  xj v½x;b ðÞ; xÞ 6 LN ðj  xj ; xÞ : 1

ð44Þ

1

So if the right hand side of each of (43,44) tends to zero, so do the left hand sides of these. We also make Remark 24. Let a > 0; a R N. Take a 6 x 6 x0 , then

ðx0  xÞaþ1 6 ðx0  xÞaþ1 1 þ 0: Similarly, for x0 6 x 6 b, we get

ðx  x0 Þaþ1 6 0 þ ðx  x0 Þaþ1  1: So we have

j  xjaþ1 6 j  xjaþ1 v½a;x ðÞ þ j  xjaþ1 v½x;b ðÞ; 8x 2 ½a; b:

ð45Þ

Thus, by positivity of LN , we obtain

               aþ1 aþ1 aþ1 LN j  xj ; x  6 LN j  xj v½a;x ðÞ; x  þ LN j  xj v½x;b ðÞ; x  : 1

1

1

ð46Þ

So if both kLN ðj   xjaþ1 v½a;x ðÞ; xÞk1 , kLN ðj   xjaþ1 v½x;b ðÞ; xÞk1 ! 0, as N ! 1, then kLN ðj   xjaþ1 ; xÞk1 ! 0. 3. Main results We present our first main result Theorem 25. Let f 2 AC m ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ; m ¼ dae; a R N; a > 0; r1 ; r2 > 0; algebra of ½a; b; x0 2 ½a; b. Then

l is a positive finite measure on the Borel r-

 Z Z   m 1 ðkÞ X f ðx0 Þ   k f ðxÞdlðxÞ  ðx  x0 Þ dlðxÞ    ½a;b k! ½a;b  k¼0  1 1 1 6 lð½a; x0 Þaþ1 þ ða þ 1Þr 1 Cða þ01Þ 1 1 !ðaþ1Þ Z a þ1 a A ðx0  xÞ dlðxÞ  x1 @Dx0  f ; r 1 ½a;x0 



Z

a !ðaþ1 Þ

ðx0  xÞaþ1 dlðxÞ

½a;x0 

0

 x1 @Dax0 f ; r2



Z ðx0 ;b

ðaþ1Þ

ðx  x0 Þaþ1 dlðxÞ

9 a !ðaþ1 Þ= aþ1 : ðx  x0 Þ dlðxÞ ; ðx0 ;b

Z

½a;x0 

  1 1 þ ðlððx0 ; bÞÞðaþ1Þ þ ða þ 1Þr 2 ! 1 1 A ½x0 ;b

ð47Þ

2088

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

Proof. By (8) we obtain

(Z  ! Z x0  1 a1 a a ð f  xÞ j Dx0  f ðfÞ  Dx0  f ðx0 Þdf dlðxÞ CðaÞ ½a;x0  x ) Z x Z  

 a1  a a þ ðx  fÞ Dx0 f ðfÞ  Dx0 f ðx0 Þdf dlðxÞ ¼ ðÞ:

  Ex  6 0

ð48Þ

x0

ðx0 ;b

Let h1 ; h2 > 0, then

("Z Z x0



  1 x0  f df dlðxÞx1 Dax0  f ; h1 ðf  xÞa1 1 þ ½a;x0  h1 CðaÞ ½a;x0  x ) "Z # Z x



  f  x0 df dlðxÞ x1 Dax0 f ; h2 ; þ ðx  fÞa1 1 þ ½x0 ;b h2 x0 ðx0 ;b

ðÞ 6

ð49Þ

i.e.

("Z #

Z x0   1 ðx0  xÞa 1 21 a1 þ ðx0  fÞ ðf  xÞ df dlðxÞ x1 Dax0  f ; h1 ½a;x0  h CðaÞ a 1 x ½a;x0  ) "Z #

Z x   ðx  x0 Þa 1 þ þ ðx  fÞa1 ðf  x0 Þ21 df dlðxÞ x1 Dax0 f ; h2 ½x0 ;b h2 x0 a ðx0 ;b ! ("Z # a aþ1   1 ðx0  xÞ 1 ðx0  xÞ dlðxÞ x1 Dax0  f ; h1 ¼ þ ½a;x0  h1 aða þ 1Þ CðaÞ a ½a;x0  ! "Z #  a aþ1   ðx  x0 Þ 1 ðx  x0 Þ dlðxÞ x1 Dax0 f ; h2 : þ þ ½x0 ;b h2 aða þ 1Þ a ðx0 ;b

  Ex  6 0

ð50Þ

Therefore

(" Z # Z   1 1 1 a aþ1 ðx0  xÞ dlðxÞþ ðx0  xÞ dlðxÞ x1 Dax0  f ; h1 ½a;x0  h1 aða þ 1Þ ½a;x0  CðaÞ a ½a;x0  ) " Z # Z   1 1 : þ ðx  x0 Þa dlðxÞ þ ðx  x0 Þaþ1 dlðxÞ x1 Dax0 f ; h2 ½x0 ;b h2 aða þ 1Þ ðx0 ;b a ðx0 ;b

  Ex  6 0

ð51Þ

Momentarily we assume positive choices of

h1 ¼ r 1

Z

1 !ðaþ1Þ

aþ1

ðx0  xÞ

dlðxÞ

>0

ð52Þ

> 0:

ð53Þ

½a;x0 

and

h2 ¼ r 2

Z

1 !ðaþ1Þ

ðx  x0 Þaþ1 dlðxÞ

ðx0 ;b

Consequently we obtain

  Ex  6 0

  a nh  1 1 1 h1 ðlð½a; x0 ÞÞðaþ1Þ þ x1 Dax0  f ; h1 ½a;x0  r 1 ða þ 1Þr 1 Cða þ 1Þ   a   h 1 1 h2 þ ðlððx0 ; bÞÞðaþ1Þ þ x1 Dax0 f ; h2 ; ½x0 ;b r 2 r 2 ða þ 1Þ

ð54Þ

proving (47). R Next we examine special cases. If ðx0 ;b ðx  x0 Þaþ1 dlðxÞ ¼ 0, then ðx  x0 Þ ¼ 0, a.e. on ðx0 ; b, that is x ¼ x0 a.e. on ðx0 ; b, more precisely lfx 2 ðx0 ; b : x–x0 g ¼ 0, hence lðx0 ; b ¼ 0. Therefore l concentrates on ½a; x0 . In that case inequality (47) is written and holds as

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

 Z  m1  X f ðkÞ ðx0 Þ Z   k f ðxÞd l ðxÞ  ðx  x Þ d l ðxÞ   0   ½a;x0  k! ½a;x  0 k¼0   1 1 1 ðlð½a; x0 ÞÞðaþ1Þ þ 6 ða þ 1Þr 1 Cða þ 1Þ 0 1 1 !ðaþ1Þ Z a þ1 a A  x1 @D f ; r 1 ðx0  xÞ dlðxÞ x0 

½a;x0 

Z

ðx0  xÞ

aþ1

½a;x0 

½a;x0 

9 a !ðaþ1Þ = : dlðxÞ ;

2089

ð55Þ

Since ðb; b ¼ ; and lð;Þ ¼ 0, in the case of x0 ¼ b, we get again (55) written for x0 ¼ b. So inequality (55) is a valid inequalR ity when ½a;x0  ðx0  xÞaþ1 dlðxÞ–0. R If additionally we assume that ½a;x0  ðx0  xÞaþ1 dlðxÞ ¼ 0, then ðx0  xÞ ¼ 0, a.e. on ½a; x0 , that is x ¼ x0 a.e. on ½a; x0 , which means lfx 2 ½a; x0  : x–x0 g ¼ 0. Hence l ¼ dx0 M, where dx0 is the unit Dirac measure and M ¼ lð½a; bÞ > 0. In the last case we obtain that L:H:S ð55Þ ¼ R:H:S: ð55Þ ¼ 0, that is (55) is valid trivially. R Finally let us go the other way around. Let us assume that ½a;x0  ðx0  xÞaþ1 dlðxÞ ¼ 0, then reasoning similarly as before we get that l over ½a; x0  concentrates at x0 . That is l ¼ dx0 lð½a; x0 Þ, on ½a; x0 . In the last case (47) is written and it holds as

 Z   m1 X f ðkÞ ðx0 Þ Z   k f ðxÞdlðxÞ  ðx  x0 Þ dlðxÞ    ðx0 ;b k! ðx0 ;b k¼0   1 1 1 6 ðlððx0 ; bÞÞðaþ1Þ þ ða þ 1Þr 2 Cða þ 1Þ 0 1 1 !ðaþ1Þ Z aþ1 a @ A  x1 Dx0 f ; r2 ðx  x0 Þ dlðxÞ ðx0 ;b

Z ½x0 ;b

ðx  x0 Þ

aþ1

ðx0 ;b

9 a !ðaþ1Þ = : dlðxÞ ;

ð56Þ

If x0 ¼ a then (56) can be redone and rewritten, just replace ðx0 ; b by ½a; b all over. So inequality (56) is valid when

Z

ðx  x0 Þaþ1 dlðxÞ–0:

ðx0 ;b

R If additionally we assume that ðx0 ;b ðx  x0 Þaþ1 dlðxÞ ¼ 0, then as before L:H:S: ð56Þ ¼ R:H:S: ð56Þ ¼ 0. The proof of (47) now has completed in all possible cases. 

lðx0 ; b ¼ 0. Hence (56) is trivially true, in fact

We continue in a special case. In the assumptions of Theorem 25, when r ¼ r1 ¼ r2 > 0, and by calling M ¼ lð½a; bÞ P lð½a; x0 Þ; lððx0 ; bÞ, we get Corollary 26. It holds

 Z   m1 X f ðkÞ ðx0 Þ Z   k f ðxÞd l ðxÞ  ðx  x Þ d l ðxÞ   0   ½a;b k! ½a;b k¼0 2 0 1 1 !ðaþ1Þ   Z 1 1 1 6 @ a aþ1 A a þ1Þ ð 6 þ ðx0  xÞ dlðxÞ M 4x1 Dx0  f ; r ða þ 1Þr Cða þ 1Þ ½a;x0  

Z ½a;x0 



Z

0

a !ðaþ1 Þ

ðx0  xÞaþ1 dlðxÞ

þ x1 @Dax0 f ; r

Z ½x0 ;b

a 3 !ðaþ1 Þ aþ1 5: ðx  x0 Þ dlðxÞ

½a;x0 

1 1 !ðaþ1Þ aþ1 A ðx  x0 Þ dlðxÞ ½x0 ;b

ð57Þ

½x0 ;b

Based on Theorem 25, Corollary 26 and (31), we obtain Theorem 27. Let f 2 AC m ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae; a R N; a > 0; r > 0, and LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, a sequence of positive linear operators, x0 2 ½a; b. Then

2090

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

   m1  X f ðkÞ ðx0 Þ   k L ðf ; x Þ  ððx  x Þ ; x Þ L  N 0 N 0 0    k! k¼0   1 1 1 6 ðLN ð1; x0 ÞÞðaþ1Þ þ ða þ 1Þr Cð"a þ 1Þ

1   ðaþ1Þ  x1 Dax0  f ; r LN jx  x0 jaþ1 v½a;x0  ðxÞ; x0

½a;x0 

a   ðaþ1 Þ  LN jx  x0 jaþ1 v½a;x0  ðxÞ; x0

1   ðaþ1Þ þ x1 Dax0 f ; r LN jx  x0 jaþ1 v½x0 ;b ðxÞ; x0 ½x0 ;b a    ðaþ1 Þ aþ1  LN jx  x0 j v½x0 ;b ðxÞ; x0 :

ð58Þ

Corollary 28 (to Theorem 27). It holds

    m 1 ðkÞ X f ðx0 Þ   k LN ððx  x0 Þ ; x0 Þ LN ðf ; x0 Þ    k! k¼0   1 1 1 6 ðLN ð1; x0 ÞÞðaþ1Þ þ ða þ 1Þr Cð"a þ 1Þ

1   ðaþ1Þ a  x1 Dx0  f ; r LN jx  x0 jaþ1 ; x0

½a;x0 

# 



1   ðaþ1Þ þx1 Dax0 f ; r LN jx  x0 jaþ1 ; x0

a  ðaþ1 Þ LN jx  x0 jaþ1 ; x0 :

ð59Þ

½x0 ;b

We make Remark 29. Let f 2 ACð½a; bÞ, f 0 2 L1 ð½a; bÞ, 0 < a < 1; x0 2 ½a; b; LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, sequence of positive linear operators. Then by Theorem 27 and

jLN ðf ; x0 Þ  f ðx0 Þj 6 jLN ðf ; x0 Þ  f ðx0 ÞLN ð1; x0 Þj þ jf ðx0 ÞjjLN ð1; x0 Þ  1j;

ð60Þ

we obtain Theorem 30. Let f 2 ACð½a; bÞ, f 0 2 L1 ð½a; bÞ, 0 < a < 1; x0 2 ½a; b; LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, sequence of positive linear operators. Then

  1 1 1 ðLN ð1; x0 ÞÞðaþ1Þ þ jLN ðf ; x0 Þ  f ðx0 Þj 6jf ðx0 ÞjjLN ð1; x0 Þ  1j þ ða þ 1Þr Cða þ 1Þ "

1   ðaþ1Þ aþ1 a  x1 Dx0  f ; r LN jx  x0 j v½a;x0  ðxÞ; x0 ½a;x0 

a   ðaþ1 Þ  LN jx  x0 jaþ1 v½a;x0  ðxÞ; x0

1   ðaþ1Þ þ x1 Dax0 f ; r LN jx  x0 jaþ1 v½x0 ;b ðxÞ; x0 ½x0 ;b a    ðaþ1 Þ aþ1  LN jx  x0 j v½x0 ;b ðxÞ; x0 :

ð61Þ

We make Remark 31. We observe that

R:H:S: ð58Þ 6

  1 1 1 kLN ð1Þkð1aþ1Þ þ ða þ 1Þr Cð"a þ 1Þ

  1 ðaþ1Þ   a  sup x1 Dx f ; r LN j  xjaþ1 v½a;x ðÞ; x  1

x2½a;b

  a ðaþ1   Þ  LN j  xjaþ1 v½a;x ðÞ; x  1

  1 ðaþ1Þ   a þ sup x1 Dx f ; r LN j  xjaþ1 v½x;b ðÞ; x  x2½a;b

  a  ðaþ1   Þ   LN j  xjaþ1 v½x;b ðÞ; x  ¼ H: 1

1

½a;x

½x;b

ð62Þ

2091

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

So that

    m1 X f ðkÞ ðxÞ   Z :¼ LN ðf ; xÞ  LN ðð  xÞk ; xÞ 6 H:   k! k¼0

ð63Þ

1

We further observe that

 ðkÞ    f ðxÞ   k LN ðð  xÞ ; xÞ k! k¼1   m1  X f ðkÞ ðxÞ   k 6 jf ðxÞjjLN ð1; xÞ  1j þ LN ðð  xÞ ; xÞ þ H: k! k¼1

jLN ðf ; xÞ  f ðxÞj 6 Z þ jf ðxÞjjLN ð1; xÞ  1j þ

m 1 X

ð64Þ

We have proved the main result, a Shisha–Mond type inequality at the fractional level. Theorem 32. Let f 2 AC m ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ, m ¼ dae; a R N; a > 0; r > 0, and LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, a sequence of positive linear operators, x 2 ½a; b. Then

kLN f  f k1 6k f k1 kLN 1  1k1 þ

m1 X k¼1

 ðkÞ    f   1  k 1 LN ðð  xÞ ; xÞ þ k! 1 Cða þ 1Þ

1 1  kLN ð1Þkð1aþ1Þ þ ða þ 1Þr "   1

 ðaþ1Þ  sup x1 Dax f ; rLN ðj  xjaþ1 v½a;x ðÞ; xÞ

  a  ðaþ1Þ aþ1 LN ðj  xj v½a;x ðÞ; xÞ

  1

 ðaþ1Þ þ sup x1 Dax f ; r LN ðj  xjaþ1 v½x;b ðÞ; xÞ

  a   ðaþ1Þ aþ1 : LN ðj  xj v½x;b ðÞ; xÞ

1

x2½a;b

1

x2½a;b

½a;x

½x;b

1

1

ð65Þ

Next we derive the following Korovkin type convergence result at fractional level. Theorem 33. Let a R N; a > 0; m ¼ dae, and LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, a sequence of positive linear operators. Assume u u LN 1 ! 1 (uniformly), and kLN ðj   xjaþ1 ; xÞk1 ! 0, as N ! 1: Then LN f ! f ; 8f 2 AC m ð½a; bÞ, f ðmÞ 2 L1 ð½a; bÞ. (The second condiu aþ1 tion means ðLN ðj   xj ÞÞðxÞ ! 0; x 2 ½a; b.) Proof. Since kLN 1  1k1 ! 0 we get kLN 1  1k1 6 K, for some K > 0. We write LN 1 ¼ LN 1  1 þ 1, hence

kLN 1k1 6 kLN 1  1k1 þ k1k1 6 K þ 1 8N 2 N: That is kLN 1k1 is bounded. So we are using inequality (65). By assumption kLN ðj   xjaþ1 ; xÞk1 ! 0 and (40) we get kLN ðj   xjk ; xÞk1 ! 0, for all k ¼ 1; . . . ; m  1. Also by (43) and (44) we obtain that

        aþ1 aþ1 LN ðj  xj v½a;x ðÞ; xÞ ! 0; andLN ðj  xj v½x;b ðÞ; xÞ ! 0; 1

1

as N ! 1. Additionally by (38) and (39) we get that

sup x1 ðDax f ; Þ½a;x ; sup x1 ðDax f ; Þ½x;b 6

x2½a;b

x2½a;b

  2f ðmÞ 1 ðb  aÞma ; Cðm  a þ 1Þ

so they are bounded. Thus based on the above, from (65), we derive that kLN f  f k1 ! 0, proving the claim.

ð66Þ



We make Remark 34. Based on Corollary Theorems17–19, given that f 2 C m ð½a; bÞ, we obtain that

ðiÞ

  1

 ðaþ1Þ sup x1 Dax f ; rLN ðj  xjaþ1 v½a;x ðÞ; xÞ 1

x2½a;b

½a;x

  1

 ðaþ1Þ ¼ x1 Dax1  f ; rLN ðj  xjaþ1 v½a;x ðÞ; xÞ 1

½a;x 

1     ! 0; asLN ðj  xjaþ1 ; xÞ ! 0; as N ! 1

1

ð67Þ

2092

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

for some x1 2 ½a; b. Similarly

ðiiÞ

  1

 ðaþ1Þ sup x1 Dax f ; r LN ðj  xjaþ1 v½x;b ðÞ; xÞ 1

x2½a;b

½x;b

  1

 ðaþ1Þ ¼ x1 Dax2 f ; rLN ðj  xjaþ1 v½x;b ðÞ; xÞ 1

½x ;b

2     aþ1 ! 0; as LN ðj  xj ; xÞ ! 0; as N ! 1

ð68Þ

1

for some x2 2 ½a; b. We give Corollary 35. Here LN : Cð½a; bÞ ! Cð½a; bÞ; N 2 N, positive linear operators. Let 0 < a < 1; r > 0; f 2 ACð½a; bÞ, f 0 2 L1 ð½a; bÞ. Then



1 1 1 ðaþ1Þ kLN f  f k1 6k f k1 kLN 1  1k1 þ kLN ð1Þk1 þ ða þ 1Þr Cða þ 1Þ "( )   1

 ðaþ1Þ  sup x1 Dax f ; r LN ðj  xjaþ1 v½a;x ðÞ; xÞ 1

x2½a;b

  a  ðaþ1Þ  LN ðj  xjaþ1 v½a;x ðÞ; xÞ 1 (   1

 ðaþ1Þ þ sup x1 Dax f ; r LN ðj  xjaþ1 v½x;b ðÞ; xÞ 1

x2½a;b

  a   ðaþ1Þ  LN ðj  xjaþ1 v½x;b ðÞ; xÞ :

½a;x

)

½x;b

ð69Þ

1

4. Application Consider f 2 Cð½0; 1Þ and the Bernstein polynomials ðBN f ÞðtÞ ¼ We have BN 1 ¼ 1 and BN are positive linear operators. Here let 0 < a < 1; r > 0 and take f 2 ACð½0; 1Þ, f 0 2 L1 ð½0; 1Þ. Applying Corollary 35 we get

PN



k k¼0 f ðN Þ

N k t ð1  tÞNk ; 8t 2 ½0; 1; N 2 N. k

Corollary 36. It holds



"   1

1 1  ðaþ1Þ 1þ sup x1 Dax f ; r BN ðj  xjaþ1 v½0;x ðÞ; xÞ kBN f  f k1 6 ða þ 1Þr x2½0;1 Cða þ 1Þ 1 ½0;x   a   1

ð a þ1Þ    ðaþ1Þ þ sup x1 Dax f ; r BN ðj  xjaþ1 v½x;1 ðÞ; xÞ  BN ðj  xjaþ1 v½0;x ðÞ; xÞ 1

  a   ðaþ1Þ  BN ðj  xjaþ1 v½x;1 ðÞ; xÞ

x2½0;1

1

ð70Þ

1

8N 2 N. 1 , that is r ¼ 23. Notice C Next let a ¼ 12, and r ¼ aþ1

½x;1

3 2

pffiffiffi ¼ 2p.

Corollary 37. Let f 2 ACð½0; 1Þ, f 0 2 L1 ð½0; 1Þ, N 2 N. Then

" 2

1 3 4 2  3 kBN f  f k1 6 pffiffiffiffi sup x1 D2x f ; BN ðj  xj2 v½0;x ðÞ; xÞ 1 ½0;x 3 p x2½0;1  1  2

1 3 3 2 3    3  BN ðj  xj2 v½0;x ðÞ; xÞ þ sup x1 D2x f ; BN ðj  xj2 v½x;1 ðÞ; xÞ 1 1 ½x;1 3 x2½0;1    1 3  3  BN ðj  xj2 v½x;1 ðÞ; xÞ : 1

ð71Þ

Here we have

( j t  xj

3=2

v½0;x ðtÞ ¼

ðx  t Þ3=2

for 0 6 t 6 x

0

for x < t 6 1

ð72Þ

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

2093

and

( jt  xj3=2 v½x;1 ðtÞ ¼

ðt  xÞ3=2

for x 6 t 6 1;

0

for 0 6 t < x:

ð73Þ

Consequently for x 2 ½0; 1, we find

3=2

½xN    X N k 3 k BN j  xj2 v½0;x ðÞ ðxÞ ¼ x x ð1  xÞNk N k k¼0

ð74Þ



3=2

N   X N k 3 k BN j  xj2 v½x;1 ðÞ ðxÞ ¼ x x ð1  xÞNk : N k k¼dxNe

ð75Þ

and

One further has

    3 3 BN j  xj2 v½0;x ðÞ ðxÞ; BN j  xj2 v½x;1 ðÞ ðxÞ 3=2

N    X  N k 3 x  k  6 BN j  xj2 ðxÞ ¼ x ð1  xÞNk   N k k¼0 € lder’s inequalityÞ ðby discrete Ho !34

2

N X N k k Nk x x ð1  xÞ 6 N k k¼0

34 1 1 8x 2 ½0; 1: ¼ xð1  xÞ 6 N ð4NÞ3=4

ð76Þ ð77Þ

ð78Þ ð79Þ

We have proved. Corollary 38. Let f 2 ACð½0; 1Þ, f 0 2 L1 ð½0; 1Þ, N 2 N. Then

"



# 3 1 1 22 1 1 2 2 ffiffiffi ffi p ffiffiffi ffi p ffiffiffi ffi : x D f ; þ sup x D f ; sup kBN f  f k1 6 pffiffiffiffip 1 1 x x p 4 N x2½0;1 3 N ½0;x x2½0;1 3 N ½x;1

ð80Þ

3=2

Notice 2pffiffipffi  1:59. u So as N ! 1 we derive again that BN f ! f with rates. Discussion 39. From (80), Corollary 17, and Theorems 18, 19 we get that

" #



3 1 1 22 1 1 2 2 ffiffiffi ffi p ffiffiffiffi p ffiffiffi ffi þ x D f ; x D f ; kBN f  f k1 6 pffiffiffiffip 1 1 x1  x2 p4N 3 N ½0;x1  3 N ½x2 ;1

ð81Þ

for some x1 ; x2 2 ½0; 1; f 2 C 1 ð½0; 1Þ. That is

"



# 3 1 1 22 1 1 2 2 ffiffiffiffi x1 Dx1  f ; pffiffiffiffi þ x1 Dx2 f ; pffiffiffiffi : kBN f  f k1 6 pffiffiffiffip 3 N ½0;1 3 N ½0;1 p4N 1

ð82Þ

1

Further we assume that D2x1  f and D2x2 f are Lipschitz functions of order 1, that is

 1  1  2  Dx1  f ðxÞ  D2x1  f ðyÞ 6 K 1 jx  yj

ð83Þ

 1  1  2  Dx2 f ðxÞ  D2x2 f ðyÞ 6 K 2 jx  yj 8x; y 2 ½0; 1 and K 1 ; K 2 > 0:

ð84Þ

and

Then from (82) we get 3

kBN f  f k1 6

22 ðK 1 þ K 2 Þ pffiffiffiffi 3 : 3 pN 4

Assume next that f 0 is a Lipschitz function of order 1, that is

ð85Þ

2094

G.A. Anastassiou / Chaos, Solitons and Fractals 42 (2009) 2080–2094

jf 0 ðxÞ  f 0 ðyÞj 6 K 3 jx  yj 8x; y 2 ½0; 1 and K 3 > 0:

ð86Þ

Then from Section 1, we get

kBN f  f k1 6

0:1953125K 3 N

N 2 N:

ð87Þ

In [12], Popoviciu for f 2 Cð½0; 1Þ proved that

kBN f  f k1 6





5 1 1 x1 f ; pffiffiffiffi ¼ 1:25x1 f ; pffiffiffiffi : 4 N N

ð88Þ

If f is a Lipschitz function of order 1, that is

jf ðxÞ  f ðyÞj 6 K 4 jx  yj

ð89Þ

8x; y 2 ½0; 1 and K 4 > 0, then we have kBN f  f k1 6

1:25K 4 pffiffiffiffi : N

ð90Þ

We also notice that

1 1 1 < < N N 34 N 12

for N 2 N n f1g:

ð91Þ

So looking at (87, 85) and (90), we observe that as the used in the estimates differentiability of f increases so the resulting speed of convergence of BN f to f increases, in fact at the used 12-derivative the speed is in between the corresponding speeds 1

1

for f and f 0 . Of course in the last argument we assumed that f, D2x1  f ; D2x2 f and f 0 are all Lipschitz functions. If f 0 is a Lipschitz 1

1

function or just f 2 C 1 ð½0; 1Þ, not necessarily D2x1  f ; D2x2 f are Lipschitz ones. References [1] Anastassiou GA. Moments in probability and approximation theory. Pitman research notes in math, vol. 287. Harlow (UK): Longman Sci. & Tech.; 1993. [2] Anastassiou G. On right fractional calculus. Chaos, Solitons and Fractals [accepted for publication]. [3] Dem’janovicˇ J. Approximation by local functions in a space with fractional derivatives. (Lithuanian, English summaries), Differencial’nye Uravnenija i Primenen, Trudy Sem. Processy 1975;103:35–49. [4] Diethelm K. Fractional differential equations. Available from:http://www.tu-bs.de/~diethelm/lehre/f-dgl02/fde-skript.ps.gz. [5] Dzyadyk VK. On the best trigonometric approximation in the L metric of some functions. Dokl Akad Nauk SSSR (Russian) 1959;129:19–22. [6] El-Sayed AMA, Gaber M. On the finite Caputo and finite Riesz derivatives. Electron J Theoret Phys 2006;3(12):81–95. [7] Frederico GS, Torres DFM. Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int Math Forum 2008;3(10):479–93. [8] Gorenflo R, Mainardi F. Essentials of fractional calculus. Maphysto Center; 2000. Available from: http://www.maphysto.dk/oldpages/events/ LevyCAC2000/MainardiNotes/fm2k0a.ps. [9] Jaskólski Miroslaw. Contributions to fractional calculus and approximation theory on the square. Funct Approx Comment Math 1989;18:77–89. [10] Korovkin PP. Linear operators and approximation theory. Delhi (India): Hindustan Publ. Corp.; 1960. [11] Nasibov FG. On the degree of best approximation of functions having a fractional derivative in the Riemann–Liouville sense. (Russian–Azerbaijani summary), Izv Akad Nauk Azerbai˘dzˇan. SSR Ser Fiz-Mat Tehn Nauk 1962;(3):51–7. [12] Popoviciu T. Sur l’approximation des fonctions convexes d’ordre superieur. Mathematica (Cluj) 1935;10:49–54. [13] Royden HL. Real analysis. 2nd ed. New York: Macmillan; 1968. [14] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives, theory and applications. Amsterdam: Gordon & Breach; 1993 [English translation from the Russian, Integrals and derivatives of fractional order and some of their applications. Minsk: Nauka i Tekhnika; 1987]. [15] Shisha O, Mond B. The degree of convergence of sequences of linear positive operators. Natl Acad Sci US 1968;60:1196–200.