Copositive Approximation by Using Whitney Theorem in

(1)

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ToCite ThisArticle:Saheb K.AL-Saidy and Nagham Ali Hussen ,1≤p<∞. Aust. J. Basic & Appl. Sci., 10(8): 13

Copositive Approximation by Using Whitney Theorem in

∞

1

Saheb K.AL-Saidy and 2Nagham Ali Hussen

1

Al-Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq

2_{And Tikrit University, College of Education, Department of Mathematics,Tikrit, Iraq}

Address For Correspondence:

Saheb K.AL-Saidy,Al-Mustansiriyah University, E-mail:[email protected]

A R T I C L E I N F O Article history:

Received 12 February 2016 Accepted 18 March 2016 Available online 20 April 2016

Keywords:

Hasslerwhitney is one of the scientist who accomplished Whitney theorem in approximation theory, which

provides the following : (for f_∈

1 −

≤

k

,such that

k b a L

k

c

f

b

a

q

f

p

[

,

(

] , [

1

≤

−

ω

Many reasercher proved this theorem, Burkill Whitney(Burkill(1952),Whitney.(1959))

when (0<p<1), and K.A.kopotun proved the Whitney theorem of type k

In(2003) E.S. Bhaya(2003) proved in theorem (A) the Whitney theory of interpolators type for k functions for K.A.kopotun

Theorem A:

for

m

,

k

∈

N

,

m

<

k

and

f

n n

p

∈

Π

such that :

j j k p j n j

f

k

p

c

p

f

( )

−

( )

≤

(

,

)

ω

ϕ₋

(

The classical Whitney theorem establishes the

p r

(

f

,

I

,

I

)

ω

and the error of best approximation of degree 1 in

L

p

,

1 ≤

p

<

∞

APPLIED SCIENCES

ISSN:1991-8178 EISSN: 2309-8414 Journal home page: www.ajbasweb.com

This work is licensed under the Creative Commons Attribution International License (CC

http://creativecommons.org/licenses/by/4.0/

Saidy and Nagham Ali Hussen.,Copositive Approximation by Using Whitney Theorem in

133-142, 2016

Copositive Approximation by Using Whitney Theorem in

Nagham Ali Hussen

Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq And Tikrit University, College of Education, Department of Mathematics,Tikrit, Iraq

Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq

A B S T R A C T

In this paper, the relationship between the degree of best approximation and an averaged module of smoothness of order in the space

⊆ and , have been discussed. The estimation between the degree of best approximation by algebraic polynomial of degree

smoothness of degree to the function ∈ ∩ , where

∆ and 1 ∞ have been found

INTRODUCTION

∞ ≤ < p b a

L_p[ , ],0 , then there exists

q

k−1

∈

Π

k−1,a polynomial of degree

p

b

a

,

])

[

Many reasercher proved this theorem, Burkill(1952) when ( .(1959)) when (

p

=

∞

), Brudnyi(1964) when (₁_≤ _p_<

), and K.A.kopotun proved the Whitney theorem of type k-monotone functions . proved in theorem (A) the Whitney theory of interpolators type for k

)

(I

W

f

∈

∆

k

∩

_pm .Then for any,

n

≥

k

−

1

, there exists a polynomial

p j

I

n

,

)

,

1

) −

for

j

=

1 ,...,

m

.

The classical Whitney theorem establishes the equivalence between the modulus of smoothness

and the error of best approximation _!of a function

f

:

I

→

R

by algebraic polynomials

∞

Copositive Approximation by Using Whitney Theorem in 〖 L〗_(p,μ) (I)

$,%

&

,

' $

College of Science, Department of Mathematics, Baghdad, Iraq.

In this paper, the relationship between the degree of best approximation and an

!,( , ,1 ∞,where estimation between the degree of best

1, and modulus of , where _!,( and

,a polynomial of degree

when (k₌2,p₌_∞),

∞),Storozhenko(1978) monotone functions .

proved in theorem (A) the Whitney theory of interpolators type for k-monotone

, there exists a polynomial

equivalence between the modulus of smoothness

(2)

Proposition(B) (Kassim, N.M.,):

Let ) be an arbitrary polynomials and let *_+,- ) be a polynomial of degree . 1 interpolating ) at . points inside , , then for every 0 < ≤∞

‖*+,-())‖! 1,2 ≤ 3‖)(. )‖! 1,2

Lemma (C) (Anton , H.,2002):

Let be a continuous function on = , , then

5 (6)76 =

2

1

5 (6 − 8-) 29:;

19:;

76<= 5 (6)76 =

2

1

5 (6 + 8-) 2,:;

1,:;

76

8- is absolute

2-% −Best Approximation:

The space _!,(( ) of all ? −measurable function , where 1 ≤ <∞_, ⊆ _, = , _{, b is a positive}

integer and ?: ⟶ 9∪ C D, D E is a measurable function can be defined as

!,(( ) = F , : , ⊆ ⟶ : G5| (I)|!7?(6) J

K

; L

<∞M

NℎP=P 1 ≤ <∞

And the (quasi) norm ‖ ‖_R_L,S(T)<∞

‖ ‖RL,S(T)= UV | (I)|J !7?(I)W

; L

, I ∈ (2.1)

For a function f in the space _!,(( ),1 ≤ <∞_{and the measurable positive function}?_{we have}

(I)7?(I) ≥ 0 for every (I) ≥ 0 and I ∈

The polynomials used in our work differ in the form and according to the degree of what we want to achieve in the proof . for = ≥ 0 and let = CY_ZE_Z[- be the collection of point , so that a= < _-< ⋯ < Y <

Y 9-= , we set *_](I) = ∏ (I − Y_Z[- _-) . and we let = ∆ ( ) be the set of functions f which change their sign exactly at the point Y_Z∈ and we will write ∈ ∆ . This assumption is equivalent to (I) ∏(I, ) ≥ 0

≤ I ≤ .

3- Modulus of Smoothness:

The moduli of smoothness are intended for mathematicians working in approximation theory, numerical analysis and real analysis. Measuring the smoothness of a function by differentiability is too crude for many purposes in approximation theory. More subtle measurement are provided by modulus of smoothness. We will use modulus of smoothness which are connected with difference of higher orders.

The k-th symmetric difference of the function f(Sendov, B. and opove, V. A., 1983)is defined by

∆_`( , I, )(=

∆_`( , I)(= Fa U

k iW

d

e[

(−1)d,e _{fI −} ℎ

2 + hℎi 0 , <IℎP=Nh P

D ,I ∓ ℎ_{2 ∈}

Where

k`

Zl =Z!(`,Z)!`! , is the binomial coefficient

The k-th local module of smoothness (Abdul Naby, Z. E., 2008) of a function ∈ _{!,( (J)}, is defined by

n`( , o, )!,µ = pq r_st

u∆_`( , . )u_R_L,S(T) ,o > 0 (3.1)

n`( , o, )!,µ = pq

r_stwa UkiW d

e[

(−1)d,e _{fI −} ℎ

2 + hℎiw

RL,S(T)

The so called

τ

−

modulus (or sendov-popov modulus) (Ditzian, Z. and V. Totik, 1987), an averaged modulus of smoothness, defined for bounded measurable function

f

on an interval

I

by.

x`( , o, )!,µ = ‖n`( , . , o)‖R_L,S(T) Where

n` ( , I, o)(= p yz∆_`( , {)z: { ∓|}~ ∈ I −|}~, I +|}~ ∩ • is the k-th local modulus of smoothness of

(3)

x` , o, )µ,∞ = n

A new way of measuring smoothness was introduced by Ditzian–Totik (Ditzian, Z. and V. Totik, 1987).The Ditzian - Totik modulus of smoothness of ∈ _!,((J), 1 ≤ <∞_{which is define for such an as}

follows.

n`€ ( , o, )!,µ = pq |_|rt

u∆_€(.)` ( , . )u_R_L,S(T) , = , (3.2)

4- Chebyshev Partition:

We have used in this paper the following notations and facts also the partition of period ℓ_Z_{, therefore we}

found Chebyshev partition, which takes the form:

n j a

X_j = cos

π

,

a

=

positive integer such that

1 ≤

a

<

∞

,

0 ≤

j

≤

n

,

to an interval

I

. Now we denote ]

, [ _j₁ _j

j X X

I = ₊

ℎ• is a length of _Z that is ℎ_•=z_•z = _•− _•9- ,0 ≤ Y ≤ ‚

and

(

)

(

)

1

₂

n

t

n

=

+

∆

ϕ

, and

c

₁

∆

_n

(

t

)

≤

h

_j

≤

c

₂

∆

_n

(

t

)

for

t

∈

I

_j and

c

₁

, c

₂ are constant.

Let = CY_-, … . . , Y ∶ Y = < Y_-< ⋯ < Y < = Y_9-E we denote by ∆ ( ) the set of all functions

∈ ∩ has 0 ≤ = <∞_{change sign} −_{times in} _{(Leviatan, D., 1996),in particular if}= = 0_{, then}∆ =

∆ ( ) denotes the set of all nonnegative functions on , . For

o = .h‚|YZ9-− YZ| whereY = ‚7Y _9-= .

If Y_Z∈ ( _•(Z)9-, _•(Z))h = 1, … . , = then it is convenient to denote Y_Z(…)≤ _•(Z)9- and Y_Z(`,-)≥ _•(Z) , > 1 such that

YZ′ < YZ"< ⋯ < YZ(`,-) that is YZ∈ (YZ(…), YZ(`,-))‡ = 1,2, … , − 2

ℓ_Z= Y_Z(…), Y_Z(`,-)_and_Z= ˆ‰Š‹‰Š

(Œ) |•; ,‰Š‹‰Š

(|•;) |•; Ž then

j i

i

j

k

J

c

h

c

₁

≤

l

=

(

−

1 )

≤

₂ _{, where}

_c

_,

_i

₌

₁

_,

₂

i ,….,r positive number.

and therefore , we get the following facts which we used to prove many results

)

(t

h

J

_i _j _n

i

≈

∆

l

_also

n

(

t

)

(

t

)

n

i

≈

∆

≈

ϕ

l

_for_{I ∈}_ℓ

Z (4.1)

We would like to point out that the symbol ℓ_Z_,‡ = 1, … , − 1_{not represent a derivatives but a symbols}

of a set of points that exist between _•(Z) and _•(Z)9-,meaning within an period ℓ_Z_{, and} = ⋃_Z[-ℓ_Z_{, we proved}

many results andtheories on the period ℓ_Z_{, and the fact that the periods}ℓ_Z_,h = 1, … , =_{isomorphic and have the}

same properties .So is the proof of these results is true on the aggregate period .

In (Bhaya, E.S., 2003), recall that for any continuous function on , there exist an algebraic polynomial _`,- of degree ≤ − 1 interpolating inside , ,such that

‖ − `,-‖RL1,2 ≤ 3( )n`( , − , , )! (4.2)

5-Auxiliary Results:

We need the following lemmas to prove the main result

Lemma (5.1):

Let ∈ _•,(( )be unbounded function, 1 ≤ * <∞_{, let} = _-, _- ⊂

Define _-= _-, _-+ ℎ , where ℎ ∈ ’D0,2;,1;

` WD let = 1,2, ….If -⊂ then

‖ (. )‖•,J;,( ≤ 8n`( , ℎ, -)•,(+ 8‖ (. )‖•,–,( (5.1.1)

Proof:

Since ∆_` (6) = a UhW sZs`

(−1)`,Z _{(6 + hℎ)}

(4)

∆_` (6) = (6 + ℎ) + a (−1)`,Z

sZs`,- UhW (6 + hℎ)

(6 + ℎ) = ∆_` (6) − a (−1)`,Z

sZs`,- UhW (6 + hℎ)

If we take the absolute value for two sides of equation, we get

| (6 + ℎ)| = š∆_` (6) − a (−1)`,ZUhW (6 + hℎ)

sZs`,-š

≤ z∆_` (6)z + š a

sZs`,- UhW (6 + hℎ)š

Integrate of both sides of equation, we get

G 5 | (6 + ℎ)|• 2;,(`,-)_

2;,`_

7?(6)K

; ›

≤ G 5 z∆_` (6)z• 2;,(`,-)_

2;,`_

7?(6)K

; ›

+ G 5 a | (6 + hℎ)|•

sZs`,-2;,(`,-)_

2;,`_

7?(6)K

; ›

Since _-− ℎ, _-− ( − 1)ℎ ⊂ _-, then

≤ G5z∆_` (6)z• J;

7?(6)K

; ›

+ G a 5 | (6 + hℎ)|• 2;,(`,-)_

2;,`_

sZs`,-7?(6)K

; ›

≤ 8 G5z∆_` (6)z• J;

7?(6)K

; ›

+ 8 G a 5 | (6 + hℎ)|• 2;,(`,-)_

2;,`_

sZs`,-7?(6)K

; ›

Using Lemma (C), we get

G 5 | (6)|• 2;,(`,-)_9`_

2;,`_9`_

7?(6)K

; ›

≤ 8 G5z∆_` (6)z• J;

7?(6)K

; ›

+ 8 G a 5 | (6)|• 2;,(`,-)_

2;,`_

sZs`,-7?(6)K

; ›

G 5 | (6)|• 2;9_

2;

7?(6)K

; ›

≤ 8 G5z∆_` (6)z• J;

7?(6)K

; ›

+ 8 G a 5 | (6)|• 2;,(`,-)_9Z_

2;,(`,Z)_

sZs`,-7?(6)K

; ›

Since _-, _-+ ℎ ⊂ _- and _-− ( − h)ℎ, _-− ( − 1)ℎ + hℎ ⊂ then

‖ (. )‖•,J;,( ≤ 8n`( , ℎ, -)•,(+ 8‖ (. )‖•,–,( In the same way we get the same result when the interval _-= _-− ℎ,

-Corollary (5.2):

Let ∈ _•,(( ), 1 ≤ * <∞, _let = 3, 7 ⊂

Let _-⊂ be such that ⊂ _- then

‖ (. )‖•,J;,( ≤ 8n`( , |-|, -)•,(+ 8‖ (. )‖•,–,( (5.2.1) Where 8 depends on the ratio |_-|/| |.

Lemma 5.3:

Let _Z be subset of ℓ_Z_and ∈ _!,((ℓ_Z) ∩ ∆ (ℓ_Z)_,1 ≤ <∞_{,and there exists a polynomial} _`,-( ) ∈

∏ ∩`,- ∆ (ℓ_Z)_{interpolate at the point} − 1_inside_Z

Then for any constant 7 > 0, we have two cases. Case(1): For •_-=•Š9•Š

(|•;)

`,- + 7|Z| < YZ(`,-) ,

‖*`,-( )‖ RL,S‰Š‹‰Š

(Œ) |•; ,ž;

≤ 3( , 7)‖ ‖

RL,S‰Š‹‰Š (Œ) |•; ,ž;

Case (2): For •_Ÿ=•Š9•Š

(Œ)

`,- − 7|Z| > YZ (…)_,

‖*`,-( )‖

RL,S ~,‰Š‹‰Š (|•;) |•;

≤ 3( , 7)‖ ‖

(5)

Proof cases (1):

For _Z⊆ ℓ_Z , and let *_`,-( ) = ∑_Z[- (Y_Z) ∏ (I_Z[- _•− I_Z), 0 ≤ Y ≤ ‚

be a linear polynomial of degree ≤ − 1 interpolating inside _Z and belong to∆ (ℓ_Z) Since (Y_Z) ≥ 0 ,∀_Z= 1, … , = ,and that *_`,-( ) is nondecreasing for Y_Z> I_•, and hence *_`,-( ) ≥ 0 for Y_Z> I_• [since (Y_Z) ≥ 0]

Thus − _`,-( ) ≥ 0 changes sign in side ℓ_Z. In particular − _`,-( ) ≥ 0

for

YZ(`,-)>•Š9•Š

(|•;)

`,- , hence `,-( ) ≤

for •Š9•Š (|•;)

`,- <

•Š9•Š(|•;)

`,- + 7|Z| < YZ(`,-) ,

then for any constant 7 > 0 such that :

•-=•Š9•Š

(|•;)

`,- + 7|Z| < YZ(`,-) , and from proposition (B) ,we get

‖*`,-( )‖ RL,S‰Š‹‰Š

(|•;) |•; ,ž;

= ¤5_{‰Š‹‰Š}(|•;) |*`,-( )|!

|•; ,ž;

7μ(6)¦

; L

≤ 3( , 7) ¤5 | |!

‰Š‹‰Š(|•;) |•; ,ž;

7μ(6)¦

; L

= 3( , 7)‖( )‖

RL,S‰Š‹‰Š (|•;) |•; ,ž;

Since |_Z| ‚7 §ˆ‰Š‹‰Š_|•;(|•;), •_-Ž§ are equivalence then we get

¤5_{‰Š‹‰Š}(Œ) |*`,-( )|!

|•; ,ž;

7μ(6)¦

; L

≤ 3( , 7) ¤5 | |!

‰Š‹‰Š(Œ) |•; ,ž;

7μ(6)¦

; L

Hence

‖*`,-( )‖ RL,S‰Š‹‰Š

(Œ) |•; ,ž;

≤ 3( , 7)‖ ‖

RL,S‰Š‹‰Š (Œ) |•; ,ž;

(5.4)

Case (2):

By the same method in case (1) and in particular

− `,-( ) ≥ 0

for Y_Z(…)<•Š9•Š (Œ)

`,- , hence `,-( ) ≤ for YZ

(…)_<•Š9•Š(Œ)

`,- + 7|Z| < •Š9•Š(Œ)

`,- ,then for any constant 7 > 0,such

that

•Ÿ=•Š9•Š

(Œ)

`,- − 7| Z| > YZ(…) we get

‖*`,-( )‖

RL,S ~,‰Š‹‰Š (Œ) |•;

= ¤5 |*`,-( )|!

~,‰Š‹‰Š_|•;(Œ) 7μ(6)¦

; L

≤ 3( , 7) ¤V | |!

~,‰Š‹‰Š

(Œ) |•;

7μ(6)¦

; L

=3( , 7)‖ ‖

RL,S ~,‰Š‹‰Š (Œ) |•;

Since |_Z| ‚7 ¨©ž~,•Š9•Š (Œ)

`,- ª¨ are equivalence then we get

¤5 |*`,-( )|!

~,‰Š‹‰Š_|•;(|•;) 7μ(6)¦

; L

≤ 3( , 7) ¤5 | |!

~,‰Š‹‰Š_|•;(|•;) 7μ(6)¦

; L

Hence

‖*`,-( )‖

RL,S ~,‰Š‹‰Š (|•;) |•;

≤ 3( , 7)‖ ‖

RL,S ~,‰Š‹‰Š (|•;) |•;

(5.5)

From lemma (5.3) and since _« super in the interpolate set of _Z and by case (1) and case (2) we get

‖*`,-( )‖RL,S(–¬)≤ 3( , 7)‖ ‖R_L,S(-¬)

(6)

For ∈ _!,((ℓ_Z) ∩ ∆ (ℓ_Z), then there exists _`,-( ) ∈ ∏ ∩_`,- ∆ (ℓ_Z) interpolate at − 1 points inside

ℓZ, such that

‖ − *`,-( )‖RL,S(ℓŠ)≤ 8( , )n`€ ( , |ℓZ|, ℓZ)!,®(5.7)

Proof:

For an interval _Z , such that

Z= ©•Š9•Š

(Œ)

`,- ,

•Š9•Š(|•;)

`,- ª, we have|Z| =

•_Š(|•;),•_Š(Œ) `,- ,

`,- ) ∪ ( •Š9•Š(|•;)

`,- , DYZ(`,-)±D² .

Since ℓ_Z= ( − 1)_Z , that is ℓ_Z consists of − 1 interval _Z with ( − 1)|_Z| = |ℓ_Z|, ≥ 4, let

(Œ)

`,- ²D¨ and |Z| ≈ ¨°D •Š9•Š(|•;)

`,- , YZ(`,-)ªD¨ (5.8)

From (4.1) suppose that _Z⊂ ( − 1)_Z= ℓ_Z, ≥ 4

Since ∈ _!,((ℓ_Z) ∩ ∆ (ℓ_Z), then by lemma (5.3) there exists _`,-( )interpolate at points inside _Z, hence from

‖*`,-( )‖RL,S(–Š)≤ 3( , 7)‖ ‖R_L,S(-Š)

And since (5.8) are satisfy then we get, where

(Œ)

`,- ²D and Zµµ= °D •Š9•Š(|•;)

`,- , YZ(`,-)ªD , that

‖*`,-( )‖RL,S(–Š¶)= °5 |*_– `,-( )|! Š

¶ 7μ(6)² ; L

Suppose that ) ∈ ∏_`,- , ) interpolates at − 1 points inside_Zµ Then *_`,-( ) = *_`,-())

We take the integral for both sided, we get

‖*`,-( )‖RL,S(–Š¶)= °5 |*_– `,-( )|! Š

¶ 7μ(6)² ; L

=UV |*_– _`,-())|!

Š¶ 7μ(6)W ; L

By Proposition (B) we get

°5 |*`,-( )|! –_Š¶ 7μ(6)²

; L

≤ C(p, d) °5 |)(6)|! –_Š¶ 7μ(6)²

; L

Since ) = )(6) − (6) + (6), then

= C(p, d) °5 |)(6) − (6) + (6)|! –_Š¶ 7μ(6)²

; L

≤ C(p, d) G5 |)(6) − (6)|!

–_Š¶ 7μ(6)K

; L

+ C(p, d) °5 | (6)|! –_Š¶ 7μ(6)²

; L

≤ C(p, d)‖)(. ) − (. )‖!,(+ C(p, d) °5 | (6)|! –_Š¶ 7μ(6)²

; L

By (3.1)n_`( , 6, )_!,®= p u∆_{_}`( , . , )u_!,( , ∈ 1, ∞) we get

°5 |*`,-( )|! –_Š¶ 7μ(6)²

; L

≤ C(p, d)n`( , |Zµ|, Zµ)!,®+ C(p, d) °5 | (6)|! –_Š¶ 7μ(6)²

; L

by using (5.1.1) we get

‖*`,-( )‖RL,S(–Š¶)≤ 8( , 7)‖ ‖RL,S(–Š¶) , and also

‖*`,-( )‖RL,S(–Š¶¶)≤ 8( , 7)‖ ‖RL,S(–Š¶¶)

`,- ) ∪ ( •Š9•Š(|•;)

`,- , DYZ

(`,-)_±D²_{we have}

(7)

Now applied the same relation in (4.2) for an interval _Z, we get

‖ *`,- ‖RL,S(–Š)= °5 | − *`,-( )|!

–Š

7μ(6)²

; L

=UV | − º_– ∗+ º∗− *_`,-( )|!

Š 7μ(6)W

; L

≤ °5 | − º∗_|! –Š

7μ(6)²

; L

+ °5 |º∗_{− *}

`,-( )|! –Š

7μ(6)²

; L

≤ G5 | − º∗_|! –Š

7μ(6)K

; L

+ 8( ) °5 |*`,-(º∗− )|! –Š

7μ(6)²

; L

≤ ‖ − º∗_‖

RL,S(–Š)+ 8( )‖º∗− ‖RL,S(–Š) = 8( )n_`( , |_Z|, _Z)_!,®

Hence we get

‖ − *`,-( )‖RL,S(–Š)≤ 8( )n`( , |Z|, Z)!,® Since |_Z| → 0, then we get

‖ − *`,-( )‖RL,S(–Š)≤ 8( )n`€( , |Z|, Z)!,® Since _Z⊂ ( − 1)_Z= ℓ_Zthen we get

‖ − *`,-( )‖RL,S(ℓŠ)≤ 8( , )n`€( , |ℓZ|, ℓZ)!,®

Lemma 5.9:

For ∈ _!,((ℓ_Z) ∩ ∆ (ℓ_Z), 1 ≤ < ∞, then there exists½_`,-( ) ∈ ∏ ∩_`,- ∆ (ℓ_Z) interpolate at − 1 points inside ℓ_Z, such that

‖ − ½`,-( )‖RL,S(ℓŠ)≤ 8( , )x`( , |ℓZ|, ℓZ)!,®

Proof:

From Lemma (5.6), there exist a polynomial ½_`,- of degree ≤ − 1 copositive with in ℓ_Z and interpolate at the points inside ℓ_Z, hence from Lemma (5.6) we get

‖ − ½`,-( )‖RL,S(ℓŠ)≤ 8( , )n`€( , |ℓZ|, ℓZ)!,®

| − ½`,-| ≤ 8( ) G5 | − ½`,-|! ℓŠ

7μ(6)K

; L

=8( )‖ − ½_`,-( )‖_R_L,S_(ℓ_Š₎ Then we get

| − ½`,-| ≤ 8( , )n`€( , |ℓZ|, ℓZ)!,®

Now by taking the _!,((ℓ_Z) -norm for both sides we get

‖ − ½`,-( )‖RL,S(ℓŠ)≤ 8( , )un`€( , |ℓZ|, ℓZ)uRL,S(ℓŠ) By x -modulus (or Sendove Popov modulus) with measure ?, for on ℓ_Z, we get

‖ − ½`,-( )‖RL,S(ℓŠ)≤ 8( , )x`( , |ℓZ|, ℓZ)!,®

6-Main Results:

Theorem 6.1:

For ∈ _!,(( ) ∩ ∆ ( ),1 ≤ p < ∞ and let º_`,-( ) ∈ ∏ ∩_`,- ∆ ( ), > 1 interpolate at − 1 points in side _« where _«= + 7| |, − 7| | , then

‖ − º`,-( )‖RL,S(J)≤ 8( , )n`€( , | |, )!,®

Proof:

Let d > 0 be a fixed and let ℓ_e , i = 1, … , r be an interval of length |ℓ_Z| = Y_Z(`,-)− Y_Z(…) , > 1, ‡ =

1, … , − 2 in the center of = , , that is dis(ℓ_e, a) = dis(ℓ_e, b) then by Lemma(5.6) there exist linear apolynomial½_`,-∗ ( ) ∈ ∏_`,- of best approximation, copositive and interpolate at points in side ℓ_Z∩ _« , hence we get

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Also by lemma (5.6) there exist a linear polynomial ℎ`,- ℎ`,- ∈ ∏_`,- of best approximation, copositive and interpolate at points in side _Â =Ã − 7| |, −;_~7| |± ,7 <

-Ÿ and −;~7| | ≤ hence

‖ − ℎ`,-( )‖RL,S(J)= °5 | − ℎ`,-( )|!

J 7μ(6)²

; L

= °5 | − ½`,-∗ + ½`,-∗ − ℎ`,-( )|!

J 7μ(6)²

; L

≤ 8( ) G5 | − ½`,-∗ |!

J 7μ(6)K

; L

+8( ) °5 | |½`,-∗ − ℎ`,-( )!

J 7μ(6)²

; L

≤ 8( ) G5 | − ½`,-∗ |!

J 7μ(6)K

; L

+8( ) °5 |ℎ`,-( − ½`,-∗ )|!

J 7μ(6)²

; L

≤ 8( ) G5 | − ½`,-∗ |!

J 7μ(6)K

; L

+8( ) °5 |ℎ`,-( − ½`,-∗ )|! –Ä 7μ(6)²

; L

Where _Å= − 7| |, and |_Â| ≈ |_Å|

since we have from an interval _Â and _Å that − 7| | ≤ b −;_~d| | hence

°5 | − ℎ`,-( )|! J 7μ(6)²

; L

≤ 8( ) G5 | − ½`,-∗ |!

J 7μ(6)K

; L

+8( ) °5 |ℎ`,-( − ½`,-∗ )|! –Æ

7μ(6)²

; L

Then by lemma (5.3) we get

°5 | − ℎ`,-( )|! J 7μ(6)²

; L

≤ 8( ) G5 | − ½`,-∗ |!

J 7μ(6)K

; L

+8( ) °5 | − ½`,-∗ |! –Æ 7μ(6)²

; L

That is

‖ − ℎ`,-( )‖RL,S(J)≤ 8( )‖ − ½`,-∗ ‖RL,S(J)+ 8( )‖ − ½`,-∗ ‖RL,S(–Æ)

Then by lemma (5.9) and inequality (6.2) we get

°5 | − ℎ`,-( )|! J 7μ(6)²

; L

≤ 8( , )n_`€( , | |, )!,®

Therefore

‖ − ℎ`,-( )‖RL,S(J)≤ 8( , )n`€( , | |, )!,® (6.3)

Also there exist a linear polynomial º_`,-( ) ∈ ∏_`,- ,copositive and interpolate at points in side _« where + 7| | ≥ ,

also − 7| | ≤ hence

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Hence

‖ º`,-‖R_L_,S(J)= °5 | − º`,-( )|! J 7μ(6)²

; L

= °5 | − ℎ`,-( ) + ℎ`,-( )−º`,-( )|!

J 7μ(6)²

; L

≤ 8( ) G5 | −ℎ`,-|!

J 7μ(6)K

; L

+8( ) G5 |º`,-−ℎ`,-( ) |!

J 7μ(6)K

; L

≤ 8( ) G5 | −ℎ`,-|!

J 7μ(6)K

; L

+8( ) G5 |º`,- ( −ℎ`,-) |!

J 7μ(6)K

; L

≤ 8( ) G5 | −ℎ`,-|!

J 7μ(6)K

; L

+8( ) G5 |º`,- ( −ℎ`,-) |!

–| 7μ(6)K ; L

Where _` = + 7| |, , and |_«| ≈ |_`|, since we have from an interval _« and _` that + 7| | ≤ −

7| | , hence

G5 | − º`,-( )|!

J 7μ(6)K

; L

≤ 8( ) G5 | −ℎ`,-|!

J 7μ(6)K

; L

+8( ) G5 |º`,- ( −ℎ`,-) |! –¬

7μ(6)K

; L

Then by lemma (5.3), we get

G5 | − º`,-( )|!

J 7μ(6)K

; L

≤ 8( ) G5 | −ℎ`,-|!

J 7μ(6)K

; L

+8( ) G5 | −ℎ`,-|!

–¬ 7μ(6)K ; L

Then by lemma (5.9) and inequality (6.3) we get

°5 | − º`,-|! J 7μ(6)²

; L

≤ 8( , )n_`€( , | |, )!,®

Therefore

‖ − º`,-( ) ‖RL,S(J)≤ 8( , )n`€( , | |, )!,® (6.4)

(6.4) means there exist a polynomial copositive and interpolate in an interval _« , such that + 7| | ≤ will satisfy the Whitney theorem .

And by the same method in the above we can get the same result for an interval _« such that ≤ − 7| | . Hence the result is true for . If 7 = 0 then the inequality (6.4) is not true.

Theorem 6.5:

For ∈ _!,(( ) ∩ ∆ ( ),1 ≤ < ∞then there exist a polynomial _`,-( ) ∈ ∏ ∩_`,- ∆ ( ), > 1, such that

‖ − `,-‖RL,S(J)≤ 8( , )x`( , | |, )!,®

Proof:

By lemma (5.6) there exists polynomial º∗∈ ∏ ∩_d,- ∆ ( ) of degree ≤ − 1 and º∗best approximation to on = ,

‖ − `,-( )‖RL,S(J)= ‖ − º∗+ º∗− `,-( )‖RL,S(J) Hence

‖ − `,-( )‖RL,S(J)= fV | − ºJ ∗+ º∗− `,-( ) |!7μ(6)i

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G5 | º∗_|!

J 7μ 6 K

; L

+ G5 | º∗ _`,- _|!

J 7μ 6 K

; L

G5 | º∗_|!

J 7μ 6 K

; L

+8 G5 | `,- º∗ |!

J 7μ 6 K

; L

‖ º∗_‖R

L,S(J)

REFERENCES

Abdul Naby, Z.E., 2008. "Integral average modulus with applications in approximation of functions in the

!,® spaces", M.Sc. thesis Education college, Al-Mustansiriy university, Iraq.

Bhaya, E.S., 2003. "On constrained and unconstrained approximation."Ph.D dissertation, Baghdad University, College of education Ibn Al-Haitham.

Brudnyi, YU.A., 1964. "On a theorem concerning local best approximation."Uchen.Zap.Kazansk.Univ.,124: 43-49.(Russian).

Burkill, H., 1952. " Cesaro-Perron almost periodic functions."Proc.LondonMath.Soc.,3:150-174. Carthers, N.L., 2000. "A short course in Banach space theory ". Bowling Green State University , USA. Ditzian, Z. and V.Totik, 1987. "Moduli of Smoothness" , Springer Series in Computational Mathematics , vol. 9 ,Springer –Verlage ,New York.

Kassim, N.M.,"On the Monotone and Co-monotone Approximation", M.Sc. thesis, kufa University mathematical

Kopotun, K.A., 2001. "Whitney theorem of interpolators type for k-monotone functions."Constr.Approx, 17: 307-317.

Leviatan, D., 1996. "Shape preserving approximation by polynomial and splines ", in : J . Szabados , K . Tandori (Eds.) approximation, and function Series , Bolyai Society , Budapest , Hungary,63-84.

Sendov, B.,V.A.Popov, 1983. " The Averaged Moduli ofSmoothness" , Sofia .

Sendov, B.l. and V.A.Popov, 1988. "The averaged modulus of smoothness", Publ. nouse of Bulg. Acad. of Science.

Storozhenko, E.A. and P.Oswald, 1978. "Jackson theorem in the spaces L (R ),0<p<1 k

p _{."Sib.Mat.Zh.,19(4):} 888-901.

Whitney, H., 1959. "On bounded functions with bounded n-th differences."Proc.Amer.Math.Soc.,10,480-481.

Zygmand, A., 2002. "Trigonometric series", Third edition, volumes I&II Combined with a forword by Robert Feffernan, Combridge university press.