11_voichita cleciu. 80KB Jun 04 2011 12:07:25 AM

ON SOME CLASSES OF BERNSTEIN TYPE OPERATORS

WHICH PRESERVE THE GLOBAL SMOOTHNESS IN THE

CASE OF UNIVARIATE FUNCTIONS

by

Voichita Cleciu

Abstract. The aim of this paper is to describe the linear and positive operators who have the property of preservation of the global smoothness. We insist on the Bernstein and Stancu operators. We consider the case of univariate function.

In order to show the preservation of global smoothness, we use the notions of modulus of continuity and K- functionals.

1. Introduction

First of all we give some notation: • Let be the space

( ) {

I = f I →R f continuous bounded onI I ≠

θ

}

C : : & , .

• Let be f∈C(I), I a real interval and δ∈I. The application

( )

δ

{

( ) ( )

δ

}

ω

ω

:[0,∞)→R, :=sup f x − f x' :x,x'∈I, x−x' ≤ is the modulus of continuity.

• If f∈C(I), I⊂R and k∈N, δ∈I then we can define the modulus of smoothness of order k

( )

δ

{

δ

}

ω

ω

_f

=

_k

f

,

:

=

sup

∆

k_h

f

(

x

)

:

x

,

x

+

kh

∈

I

,

h

≤

where

(

)

∑

₌ −

+









−

=

∆

k

i

i k k

h

f

x

ih

i

k

x

f

0

)

1

(

)

(

is the divided differences.

• The Pochhammer symbol is defined by

∏

−

=

−

=

1

0

)

(

:

)

(

s i

s

n

i

n

,

∏

∏

−

−

=

−

=

1

=

1

0

}

0

{

:

,

1

:

)

(

i

i

n

.

• Let be the compacts intervals I and I’, I’⊆I. For r≥1 an operator

( )

( )

'

:

C

I

C

I

L



→

is almost convex of order r-1, if the following holds:

Let KI,i:={f∈C(I): [x0,x1,…,xi;f] ≥0 for any x0<x1<…<xi∈I}. There exist p≥0,

integers ij , 1≤i≤p, satisfying 0≤i1<…<ip<r such that

r I r

I p

j i

I

K

Lf

K

K

f

j , ,'

1

,

→

∈









∈

=

I

I

.

For p=0 we put

K

C

( )

I

p j

i I _j

=

=

:

1 ,

I

. In this case KI,r is mapped by L into KI’,r and

L is called “convex” of order r-1≥0.

2. The operators

We introduce here the well-known Bernstein and Stancu operators. ♦ The Bernstein operators are defined for f: [0,1]→R as

( )

(

)

(

)

;

_,

(

)

( )

1

,

0

,

,

[ ]

0

,

1

0

,



−

=

∈







=













=

−

=

∑

x

x

k

m

x

k

m

x

p

m

k

f

x

p

x

f

B

_mk k m k

m k

k m m

♦ The Stancu operators.

In paper [13] D.D. Stancu has introduced and investigated a linear operator Sm(α):C[0,1]→C[0,1] defined by

(

)

(

) (

)

(

α

)(

α

) (

α

)

µα

να

α

ω

α

ω

µ ν

α

) 1 ( 1 ... 2 1 1

1 )

; (

, ) ; ( )

(

1

0 1

0 ,

0 , )

(

− + +

+

+ − +

    =

      =

∏

∏

∑

− −

= −

= =

m x x

k m x

m k f x x

f S

k m k

k m

m k

k m m

,

α being a parameter which may depend only on the natural number m. If α≥0, then these operators preserve the positivity of the function f. For α=0, Pm(α) coincides with

3. The preservation of the global smoothness

Even since 1951 the mathematicians have been showing their interest for the problem of the preservation of the global smoothness preservation by linear operators. In this year, in his paper [14], S.B. Steckin give as the following

Theorem 1. For fixed s,n∈N and

f

∈

C

_2πlet tn be a trigonometric polynomial of

degree ≤n such that













⋅

≤

−

n

f

c

t

f

_{n 1}

ω

_s

;

1

.

Then for all

δ

>0one has

( )

δ

(

)

ω

( )

δ

ω

1

2

,

2

1

sin

,

c

₁

f

t

s

s s n

s



+

⋅











≤

− .

On 1965 a result on smoothness preservation by the Bernstein operators Bm on

C[0,1] was given by Hajek [8] :

Theorem 2. Let f ∈LipM

(

1;[0,1]

)

. Then

B

m

f

∈

Lip

M

(

1

;

[

0

,

1

]

)

.

A few years later, this result was generalized by Lindvall [9] and Brown, Elliott & Paget [3]. They showed that we could replace the statement Lip_M

(

1;[0,1]

)

by

(

α

;[0,1]

)

,

α

∈[0,1]

M

Lip . This means that, if global smoothness of a function

[ ]

0,1

C

f ∈ is represented by stating that it satisfied a certain Lipschitz condition, then the same is true for its approximant Bnf.

Regarding Stancu operators, in 1987, in paper [5] B. Della Vecchia was proved:

Theorem 3. If f ∈Lip_M

(

1;[0,1]

)

thenS_m(α) f ∈Lip_M

(

1;[0,1]

)

for m∈N, α≥0. Anastassiou, Cottin and Gonska was generalized the second theorem in 1991 [2], so that we have:

Theorem 4. For ∀f ∈C

[ ]

0,1 and

δ

≥0, for the Bernstein operator Bm

Here

ϖ

₁

( )

f;⋅ denotes the least concave majorant of

ω

₁

( )

f;⋅ . The constants 1 and 2 are best possible.

Another important fact is that Theorem 4 can be generalized by express global smoothness preservation in terms of K- functionals of order s, as Cottin & Gonska showed in [4]:

Theorem 5. For the operators Bm, one has for ∀s∈N, f ∈C

[ ]

0,1,

δ

≥0the

inequalities

(

;

)

1

;

(

)

δ

K

( )

f

;

δ

n

n

f

K

s

f

B

K

s s

s s

n

s



≤











⋅

⋅

≤

.

For the case s=1, Theorem 5 implies Theorem 4. Next we have a generalized version of Theorem 5 for a certain class of operators which includes those of Bernstein as special cases:

Theorem 6. Let k≥0, s∈N * and let I:=[a,b] and I’:=[c,d]⊂[a,b] be compact intervals with non-empty interior (a≠b, c≠d). If L is a linear operator satisfying

( )

( )

( )

( )

I C f all for

a f

a Lf

that such

I C I

C L

k

l k I k l k I k k

k

∈

≠ ⋅

≤ →

 ' , 0

: _, ( ) _,

' ) (

as well as

( )

( )

( )

( )

I C g all for

g b Lg

that such

I C I

C L

s k

I s k l s k I s k s

k s

k

+

+ +

+ +

∈

⋅ ≤ →

 ' ,

: _{, ,} ( )

' ) (

then for all f ∈Ck

[ ]

I and

δ

>0one has

( )

(

)

I l k

l s k k s l k I k s

a b f K a Lf

K

  

 ⋅

≤

δ

δ

, , , ) ( ,

' ) (

;

; .

Theorem 7. [4] Let k≥0, s∈N * and I, I’ be given as above. Let

( )

( )

'

:

C

I

C

I

L

k



→

k be a linear operator having the following properties: (1) L is almost convex of orders k-1 and k+s-1

(2) L maps Ck+s(I) into Ck+s(I’) (3) L(Πk-1)⊆Πk-1 and L(Πk+s-1)⊆Πk+s-1

(4) L(Ck(I))⊄Πk-1

Then for ∀f ∈Ck

[ ]

I and

δ

>0we have

( )

(

)

( )

(

( )

)

I k

k s k s k s k

s k k I

k s

Le Le s

k f K Le

k Lf

K

    

  

⋅ + ⋅

⋅ ≤

+ +

δ

δ

( )

) ( )

( )

( '

) (

) (

1 ; !

1 ;

with ek=xk .

Proof. We show that the assumptions of Theorem 6 are satisfied. Let l∈{k, k+s} and

( )

( )

'

:

C

I

C

I

L

p



→

p which is almost convex of order l-1, satisfying L(Πl-1)⊆Πl-1 .

ƒ For l=0, L is positive, maps C0(I) into C0(I’), and the third assumption is satisfied. For such operators we know that Lf _I' ≤ Le_{0 I'}⋅ f _I.

ƒ For l≥1, define

I

_l

:

C

( )

I



→

C

l

( )

I

by

( )

f t dt

l t x x

f I

x

a

l

l ( )

)! 1 (

) ( ;

1

⋅ − −

=

∫

− .

Since L is almost convex of order l-1, the operator Ql given by Ql:=(L Il)(l) is

linear and positive. The assumption L(Πl-1)⊆Πl-1 implies Ql f(l)=(Lf)(l) for f C

[ ]

I

l

∈

∀ .

Hence

( )

Lf

(l)

=

Q

_l

f

(l)

≤

Q

_l

⋅

f

(l)

for

∀

f

∈

C

l

(

I

)

Since Ql is positive, we have

( )

) ( 0

!

1

l

l l

l

Le

l

e

Q

Putting now

( )

(

)

( )

, , )

( ,

)! (

1 : !

1

: _k k _{ksl k s} k s

l

k Le

s k b

and Le

k

a ₊ +

+ =

= yields two

nonnegative constants for which the assumption of Theorem 6 are satisfied.

All that we still have to do is to proof that ak,l≠0. Suppose that

( )

Le

Q

( )

Lf

for

f

C

( )

I

k

k k

l k

k

=

=

0

⇒

=

0

∀

∈

!

1

( ) ( )

or

( )



→

∏

₋₁

:

k p

I

C

L

. But this contradicts condition (4) and the proof is complete.

Now, we apply the general result to the classical Bernstein operators

∏

→



n

n

C

B

:

[

0

,

1

]

.

Proposition 8. [4] Let k≥0 and s∈N * be fixed. Then for ∀ n≥k+s,

∀

f

∈

C

k

[ ]

0

,

1

and ∀

δ

≥0 the following inequality holds:

( )

(

)

( )

] 1 , 0 [ )

( ]

1 , 0 [ )

( ( )

;

; 

  



 − _⋅

≤

δ

δ

s

s k

s k

k k

n s

n k n f K n n f

B K

Proof.

ƒ We have the next representation

( )

k k

( )

n l k

l n l

n k l

l l

n

f

C

x

x

n

l

k

n

k

l

n

n

x

f

B

₋ −−

−

=



⋅

⋅

⋅

−







+

⋅

⋅

=

(

)

!

∑

,...,

;

1

)

(

0 )

(

due to Lorentz [10] which show as the fact that condition (1) is satisfied (Bn is almost

convex for ∀ l-1≥-1).

ƒ Since Bn is a polynomial operator, the general assumption and condition (2)

from Theorem 7 are satisfied.

ƒ Since Bn maps a polynomial of degree l into a polynomial of degree

min{n,l}, condition (3) is also satisfied for ∀n∈N.

ƒ We consider the k-th monomial ek∈Ck[0,1]. From the assumption that n≥k+s

Gonska gives the representation

( )

() ( ) l!

n n e

B l _ll l

n = ⋅ , in paper [6] and plugging

into the inequality of Theorem 7 yields our claim.

The main result of this paper regards the case of Stancu operator. Proposition 9. Let k≥0 and s∈N * be fixed. Then for ∀ n≥k+s, k

[ ]

0

,

1

C

f

∈

∀

and ∀

0

≥

δ

the following inequality holds:

(

)

(

)

] 1 , 0 [ , , ) ( , ] 1 , 0 [ ) ( ) (

;

;









⋅

⋅

≤

+

δ

β

β

β

δ

α α_α α k m s k m k s k m k n

s

S

f

K

f

K

Proof. The four condition and general assumption from Theorem 7 are again satisfied.

The representation of (Bmαf)(l) is due to the Mastroianni and Occorsio [11]:

( )

( )

( )

∏

∑ ∑

∏

∑

= − ∈ = = − = −       ∆ + +     + + − + − − = l j u j I h i i h

l m l

i i l m l m m i f u x a f m l i m i m i x m m l f S j p l l 1 } 0 { 0 , 1 0 , ) ( ) ( 1 , ; ,..., 1 , , ) ( 1 1 1 ! ν ν ν α

α

ω

α

ω

α

ν

ν

Given the notation :

( )

( )

∑

∏

∑

∏

− ∈ = = =

=













∆

=

−

+

−

−

=

} 0 { , , 1 0 , , 1 ,

)

,

(

,

1

,

)

,

(

)

(

1

1

1

p j l l I h l h l m l j u j i i h l h l l m

x

x

m

i

f

u

x

a

x

m

m

α

λ

α

ϕ

α

ω

α

λ

α

ν

ν

β

ν ν ν α

we can write

( )

( )

f

( )

x

m

l

i

m

i

m

i

x

l

f

S

_mk

l m i i l m l m l

m

,...,

;

,

Finally we have the representation of the quantities (Sm(α)el)(l), l∈{k,k+s} as

(

α

)

β

α

l m l

l

m e l

S ,

) ( ) (

!

= and plugging into the inequality of Theorem 8 our claim yields.

We now consider two special cases of s≥1, which are of particular interest. • The first case is s=1:

Proposition 10. Let k≥0 be a fixed integer. Then for ∀ n≥k+1,

∀

f

∈

C

k

[ ]

0

,

1

and

0

≥

δ

we have:

( )

(

δ

)

ϖ

δ

ϖ

( )

δ

ω

₁ ( )

;

(

)

₁ ( )

;

1

₁

f

;

n

k

n

f

n

n

f

B

k _kk k

n



≤

⋅











−

⋅

⋅

≤

.

The left inequality is best possible , means that for ek+1 both sides are equal .

Proof. Proposition 8 gives the particular case

( )

(

)

] 1 , 0 [ )

( 1 ]

1 , 0 [ ) (

1

;

)

(

;













−

⋅

≤

δ

δ

n

k

n

f

K

n

n

f

B

K

k _kk k

n .

For K-functionals K1 we have Brudnyî’ representation:

( )

δ

ϖ

(

;

2

δ

)

2

1

;

_{[0,1] 1}

1

f

f

K

=

. Using this in both sides of the inequalities which involves Kl leads to the first condition fron Proposition 10. Furthermore, for the

function ek+1(x)=xk+1 it can easily be verified that, for n≥k+1, both sides in the left part

of the inequality in Proposition 10 equal

(

)

₁1

⋅

( )

+

1

!

≥

0

,

>

0

.

++

k

δ

for

δ

n

n

k k

ƒ The case s=2.

( )

(

;

δ

)

(

)

;

(

)(

1

)

δ

₂

(

( )

;

δ

)

2 )

( 2 )

( 2

k k

k k k

n

K

f

n

k

n

k

n

f

K

n

n

f

B

K



≤











−

−

−

⋅

⋅

≤

.

Proposition 10 was derived from Proposition 8 by an representation of K-functionals K1 using modulus of continuity ω1. We present only the case s=2 because

only in this situation we know some constants involving in this relations. We use, for δ∈[0,1/2] :

( )

δ

δ

ω

( )

δ

ω

;

8

9

4

1

;

;

4

1

2 2

2

2

f

K

f



≤

f











≤

.

Similar statements involving ω2(f(k);δ) are obtained if one starts with

ω2((Bnf)(k);δ). As the result we have:

Proposition 12. For fixed integer k≥0, f∈Ck[0,1] and all δ≥0 the Bm operator satisfy

the inequality

( )

(

δ

)

ω

(

δ

)

ω

;

2

)

1

)(

(

1

)

(

3

;

_{2 2} ( )

) ( 2

k k

k k

n

f

n

k

n

k

n

n

n

f

B

⋅









+

−

−

−

≤

.

In particular, for k=0 we have

(

δ

)

ω

( )

δ

ω

( )

δ

ω

;

4

,

5

;

2

1

1

3

;

_{2 2}

2

f

f

n

n

f

B

_n

≤









+

−

≤

.

References.

[1] Agratini O., “Aproximare prin operatori liniari”, Presa Univ. Clujeana, 2000, p.354 [2] Anastassiou G.A., Cottin C., Gonska H., “Global Smoothness of Approximatting

Functions”, Analysis, 11 (1991), 43 – 57

[3] Brown B.M., Elliott D., Paget D.F., “Lipschitz Constants for the Bernstein Polynomials of a Lipschitz Continuous Function”, J. of Approx. Theory 49 (1987), 196 – 199

[5] Della Vecchia B., “On the Preservation of Lipschitz Constants for Some Linear Operators”, Boll. Un. Mat. Ital. B. (7), 3 (1989), 125 – 136

[6] Gonska H.H., “Quantitative korovkin – type Theorems on Simultaneous Approximation”, Math. Z., 186 (1984), 419 – 433

[7] Gonska H.H., “Simultaneous Approximation by Algebraic Blending Functions”, Alfred Haar Memorial Conference (Proc. Int. Conference Budapest 1985), Colloq. Soc. Janos Bolyai 49, Amsterdam-Oxford-New York:Nord Holland (1987), 363 - 382

[8] Hajek O., “Uniform polynomial Approximation”, Amer. Math. Monthly, 72 (1965), 681

[9] Lindvall T., “Bernstein Polynomials and the Law of Large Numbers”, Math. Scient, 7 (1982), 127 – 139

[10] Lorentz G.G., “Bernstein Polynomials”, Toronto, 1953

[11] Mastroianni G., Occorsio M., “Sulle derivate dei polinomi di Stancu”, Rend. Accad. Sci. Fis. Mat., Napoli, IV, vol. 45, 1978, 273 - 281

[12] Stancu D.D., “Asupra unor polinoame de tip Bernstein”, Stud. Cerc. Mat. Iasi, 11 (1960), 221 – 233

[13] Stancu D.D., “Approximation of functions by a new class of linear polynomial operators”, Rev. Roumaine Math. Pures Appl, 13, 8 (1968), 1173 - 1194

[14] Steckin S.B., “On the Order of Best Approximation of Periodic Functions” , Izv. Akad. Nauk URSS, 15 (1951), 219 – 242

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