Error Estimate of Unbounded Function for Spline Approximation in Space ��, �� , �� � 0 , 0 � �

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To Cite This Article: Saheb K. Al-Saidy, Hussein Ali AL for Spline Approximation in Space , ,

Error Estimate of Unbounded Function for Spline Approximation in Space

,

,

,

0 , 0

1_{Saheb K. Al-Saidy,} 2_{Hussein Ali AL-Juboori}

1_{(1st Affiliation) Department of Mathematics} 2_{(2nd Affiliation) Department of Mathematics}

3_{(3rd Affiliation) Department of Mathematics, College of Education,University of AL}

Address For Correspondence:

Saheb K. Al-Saidy, (1st Affiliation) Department of Mathematics, College of Science ,Universitya of Al

A R T I C L E I N F O Article history: Received 3 April 2016 Accepted 21 May 2016 Published 2 June 2016

Keywords:

Besove space, The generalized

steklove function, spline

approximation

A spline is a numerical function that consists smoothness at the places (Chen, Wai

We shall prove a new theorem about the convergence to unbounded functions by spline polynomial in terms of K-functional of J. Peeter generated in space

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Saidy, Hussein Ali AL-Juboori, Abdulsattar Ali AL-Dulaimi., Error Estimate of Unbounded Function

, 0 , 0 1. Aust. J. Basic & Appl. Sci., 10(10): 6-15, 2016

Error Estimate of Unbounded Function for Spline Approximation in Space

1

Juboori, 3_{Abdulsattar Ali AL-Dulaimi}

of Mathematics, College of Science ,Universitya of Al-Mustansiry, Baghdad, Iraq. of Mathematics, College of Science,Universitya of Al-Mustansiry, Baghdad, Iraq.

Mathematics, College of Education,University of AL-Anbar, AL-Anbar, Iraq.

(1st Affiliation) Department of Mathematics, College of Science ,Universitya of Al-Mustansiry, Baghdad, Iraq.

A B S T R A C T

Background: during the last six decades Spline Approximation has been rapidly increasing in importance in many fields, such as numerical analysis, statistics, engineering, computer science, maneuvering analysis,and others. In 1946 (Ahlberg, J. H., E. N. Nilson, and J. L. Walsh, 1965),Schoenberg introduced the terminology (spline function). As far as we can determine" the only other papers mentioning splines explicitly prior to 1960 were those by Curry and Schoenberg [1947]" Schoenberg and Whitney [1949, 1953], Schoenberg [1958], and Maclaren [1958].see (Larryl Schumaker, 2007)The theory of spline functions had a rather modest development up until 1960. The important impetus for the intense interest in splines in the early 1960 seems to have been provided primarily by the fact that (in addition to I. J. Schoenberg) a number of researchers realized that spline functions were a way to mathematically model the physical process of drawing a smooth curve with a mechanical spline. Despite the feverish activity in splines during the past 30 years, there are relatively few authors on the subject. Work was also being done at Pratt & Whitney Aircraft, where two of the authors of the first book-length treatment of splines

1967) were employed; and the David Taylor Model Basin, by

work at General Motors is detailed nicely in Birkhoff (1990) and Young (1997). (Young 1997) Davis (1997) summarizes some of this material. In (2010) Hawraa A. Fadhil estimated approximation a K-monotone function by free knot spline (Hawraa A. Fadhil 2010) Objective: the main objective for our paper is to study the best approximation of unbounded measurable functions by spline polynomials in weighted space L ,α ,α 0, 0 1, we introduced direct and converse estimate for the degree

of best approximation of this function by means the K-functional. results of this paper is to Approximation error of unbounde

quasi-norm such that we proved a direct and converse theory and some result. Conclusion: there are main conclusion for our paper. we prove that for the unbounded function f ∈ B_σα, ,α, ,α 0 , 0 p 1, the error estimate of spline

quasi-norm. we proved pairs of adjusted inequalities of Jackson and Bernstein type, we using this prove to obtain complete direct and converse theorems in approximation theory.

INTRODUCTION

A spline is a numerical function that consists of piecewise polynomial which is has a high degree of (Chen, Wai-Kai 2009), ( Judd, Kenneth L.1998).

We shall prove a new theorem about the convergence to unbounded functions by spline polynomial in terms functional of J. Peeter generated in space L _,α 0,1 .

Error Estimate of Unbounded Function , 2016

Error Estimate of Unbounded Function for Spline Approximation in Space

Mustansiry, Baghdad, Iraq.

: during the last six decades Spline Approximation has been rapidly increasing in importance in many fields, such as numerical analysis, statistics, engineering, computer science, maneuvering analysis,and others. In 1946 (Ahlberg, J. J. L. Walsh, 1965),Schoenberg introduced the terminology (spline function). As far as we can determine" the only other papers mentioning splines explicitly prior to 1960 were those by Curry and Schoenberg [1947]" Schoenberg and berg [1958], and Maclaren [1958].see (Larryl Schumaker, 2007)The theory of spline functions had a rather modest development up until 1960. The important impetus for the intense interest in splines in the early 1960 the fact that (in addition to I. J. Schoenberg) a number of researchers realized that spline functions were a way to mathematically model the physical process of drawing a smooth curve with a mechanical spline. ing the past 30 years, there are relatively few authors on the subject. Work was also being done at Pratt & Whitney Aircraft, where length treatment of splines (Ahlberg; Nilson; Walsh , by Feodor Theilheimer. The is detailed nicely in Birkhoff (1990) and Young (1997). me of this material. In (2010) Hawraa A. monotone function by free knot spline (Hawraa A. the main objective for our paper is to study the best line polynomials in weighted , we introduced direct and converse estimate for the degree functional. Results: the main results of this paper is to Approximation error of unbounded measurable functions in norm such that we proved a direct and converse theory and some result. Conclusion: there are main conclusion for our paper. we prove that for the unbounded , the error estimate of spline polynomial in adjusted inequalities of Jackson and Bernstein type, we using this prove to obtain complete direct and converse theorems in approximation

of piecewise polynomial which is has a high degree of

The Besov space Bα_,σ is defined a set of functions f from L _,α 0,1 which have smoothness α. Where p < 1 we note that for μ< min 1, p and any sequence !f_"# , the following inequality holds (Chen, Wai-Kai 2009):

$∑ f"$ _,α ≤ '∑$f"$μ_,α( )

μ

. (1.1)

Let f ∈ L _,α 0,1 , α> 0 , 0 < p,σ< 1 , assume k = 1 + α , define the following quasi-norm of:

‖f‖./,ασ = '0 '

1

2α ω f; t ,α( σ₅₂

2 1

6 (

)

σ

(1.2) And

‖f‖./,ασ = sup

1

2α ω f; t ,α (1.3)

Where the quasi norm ‖f‖_./,α_σ is finite.

To begin proof the direct and inverse theorem for spline approximation. Denoted by Bα_,σ, the set of all functions f ∈ L _,α 0,1 such that the quasi norm ‖f‖_.

/,σ,9

α is finite.

Where

‖f‖._/,ασ,9= '0 '

1

2α ω f; t (

σ₅₂

2 1

6 (

)

σ

(1.4)

And α> 0 , 0 < p,σ< 1, k ≥ 1.

Now we can give some equivalent quasi-norms in Bα_,σ, [(Petrushev P. P. popov V. A., 1987)] Denote

‖f‖_.1_/,α_σ,9= !∑ ! 2 α<ω f; 2=< #

σ ∞

<>6 #

)

σ _(1.5)

‖f‖_./,

σ,9 α

? _{= @0 @}1

2α $∆2f . $_{C/,σ6,1= 2} D

σ₅₂

2 )

9

6 D

)

σ

(1.6)

The generalized steklove function for f ∈ L _,α is

f,E x = h = 0 … 0 I−f x + t₆E ₆E 1+ t?+ ⋯ + t L

+ 'k1(f@x + =1 t1+ t?+ ⋯ + t D + ⋯

+ −1 ' k_{k − 1( f 'x +}2)M⋯M29 _t

1+ t?+ ⋯LL

LLL+t #Ndt1… dt (1.7)

Theorem .1.1:

Let f ∈ L _,α 0,1 , α> 0 , 0 < p < 1 . Then

$f ,E− f$ _,α≤ω f; h ,α.

Proof:

The generalized Minkovski inequality in (1.7) implies

$f ,E− f$ _,α≤ P

1

h Q … Q R∆2)M⋯M29f x R dt1… dt

E

6 E

6 P _,α

≤ω f; h ,α ▄

Denoted by S k, n, a, b the set of all splines s ∈ S k, n restricted toa, b . a and b witch is the end points of the interval a, b also called knots of s. The degree of best approximation to a continuous function with respect to a polynomial spline on interval a, b ( Fadil, H., (2010), is given by

EW, f ∞= infI‖f − s‖∞ ; s ∈ S k, n X. (1.8)

The degree of best approximation of a function f ∈ L a, b with respect to a polynomial spline of degree ≤ n on a, b is given by

EW, f = infY‖f − s‖ ; s ∈ S k, n N. (1.9)

Define the degree of best weighted approximation to a given unbounded function f ∈ L _,α X with respect to polynomial spline of degree ≤n on X as

EW, f ,α= infY‖f − s‖ ,α ; s ∈ S k, n N. (1.10)

At the same time, when X₆= L _,α 0,1 , X₁= Bα_σ,α, , the K-functional is equipped to the kthL _,α-modulus of smoothness, ‖f‖_.

σ,α,9

α is a quasi-norm inB_σα_,α, . In this case,

K!f, t, L ,α, Bσα,α, # = inf_|E|]

δ^‖f − g‖C/,α+ t‖g

`_‖

.σα,α,9, s = 1,2, … a (1.11)

2. Auxiliary Results:

Let f ∈ L a, b , 0 < p < 1 and , 0 <δ<b=c , k ≥ 1 . Then we have

ω f;δ ≤ Cn 0 0₆δ _cb= 2e∆_Ef x e dxdt. (2.1)

Her we can prove the following lemma for the space L _,α.

Lemma 2.2:

For the unbounded function f ∈ L _,α a, b , α> 0 , 0 < p < 1 and,

0 <δ<1 , k ≥ 1, then we have

ω f;δ ,α≤ Cn 0 0 e∆Ee=αgf x e dxdt

b= 2

c .

δ

6 (2.2)

Proof:

ω f,δ _,α= sup

|E|]δ$∆Ef . $ ,α ,

α> 0 , 0 < p < 1

= sup

|E|]δ'0 e∆Ee

=αgf x e dx

b= 2

c (

) /

=ω e=αgf;δ

Using Lemma (2.1) to get

ω e=αgf;δ ≤ Cn 0 0₆δ _cb= 2e∆_Ee=αgf x e dxdt.

Hence

ω f;δ ,α≤ Cn 0 0 e∆Ee=αgf x e dxdt

b= 2

c .

δ

6 ▄

Theorem 2.3 (Saheb K. Al-Saidy, 2015):

For unbounded functionsf ∈ L _,α I α> 0 . 0 < p < 1, n ≥ 1 and I = 0,1. There exists a polynomial

P ∈ ∏W=1, where ∏W=1 is the space of all polynomials of degree less than or equal to zero, such that

‖f − P‖ ,α≤ CωW'f;

|k|

W( ,α. (2.3)

Where C = C k, p is a constant depending on k and p

Lemma 2.4:

Let f ∈ L _,α 0,1 , α> 0 , 0 < p < 1 and k ≥ 1 . Then

EW f ,α≤ Cω 'f;

1

W( _,α , n = 1,2, … WhereC = C k, p . (2.4)

Proof:

Let the open interval ∆_"= '"=1

W , " W(

By theorem (2.3), there exists a polynomial P_l of degree k − 1 such that

$f − P"$ _,α∆m ≤ Cω @f;e∆meD

,α

, C = C k, p

Using Lemma (2.2) gives

$f − P"$ _,α∆m ≤ C1n 0 0 e∆ne=αgf x e dxdu

m o= n mp)

o .

) 9o

6

Put ᴪ x for x ∈ ∆_" , j = 1,2, … , n then to get

‖f −ᴪ‖ _,α = '∑ $f − PW">1 "$ _,α(

) /

≤ C1n r0 r∑ 0 e∆We=αgf x e dx

m o= n mp)

o W

">1 s du

) 9o

6 s

) /

≤ C1n @0 0₆1= We∆We=αgf x e dxdu

) 9o

6 D

) /

▄

Lemma 2.5:

Let 0 <σ< < 1 and let the sequence Iu_WX_W>1∞ of nonnegative function u_W∈ L_∞ 0,1 such that ‖u_W‖_σ,α≤

λ_W ,α> 0 , λ_WM1≤βλ_W , n = 1,2, … where 0 <β< 1 and

‖u_W‖_t,α≤ Cδ_W

)

σ=

) u_λ

f = ∑∞W>1uW , f ∈ L ,α 0,1 We have ‖f‖ ,α≤ C @∑ δW /

σ=1_λ

W ∞

W>1 D

) /

. (2.5)

Proof:

Put v_W = u_Wσ then ∑∞W>1vW

/

σ≥ ∑∞_W>1u_W and

‖vW ‖1= 0 u₆1 Wσ x dx ≤λWσ

Therefore ‖v_W ‖_t≤δ_W

) u=1_λ

W , 0 < r < 1, Put

σ= s then we has

‖f‖ ,α≤ yz vW

∞

W>1

y

` `

≤ C σ, p,β ∑∞W>1δW`=1

= C σ, p,β ∑ δ_W

/

σ=1

∞

W>1 λW .

≤ C σ, p,β @∑ δ_W

/

σ=1_λ

W ∞

W>1 D

) /

. ▄

Lemma 2.6 (Petrushev P. P. popov V. A, 1987):

Let 0 <σ< < 1, { ≥ 0 . Then for every polynomial | ∈ P and every finite interval ∆ we have

'_|∆|1 0 || x |}_dx

∆ (

) ~_{≤ '}1

|∆|0 || x | dx∆ (

)

/_{≤ c '}1

|∆|0 || x |∆ }dx(

)

~ _(2.6)

Wherec = c q, k .

Note: the following inequality

!∑ |xl l| #

)

/_{≤ !∑ |xl l|}}_#)~ _(2.7)

can be used in following lemma

Lemma 2.7:

For u_l= eψ_?•m−ψ_?•mp)e, ψ_?•m ∈ E k, 2‚ and 0 <σ< v < 1. Then

‖ul‖t≤ 2‚m'

)

u=)σ(‖u_l‖_σ (2.8)

Proof:

From lemma (2.6) we have

‖ul‖t ∆< ≤ C|∆l|

) u=)σ$u_"$

Cσ,α∆ƒ , C = C k, r

Where

∆<= '<=1_?•m ,_?<•m ( , v = 1,2 … , 2‚m , then the view of inequality (2.7) we get

‖ul‖t= „z‖ul‖t ∆t …

?•m

<>1

†

1 t

≤ C2‚m'1σ=1t(„z‖u_l‖

σ∆_… t ?•m

<>1

†

1 t

≤ C2‚m')σ=

)

u('∑ ‖u_l‖_σ∆

… σ ?•m

<>1 (

) u

≤ C2‚m')σ=

)

u(‖u_l‖_σ. ▄

3. Main result:

In this section we shall prove direct and inverse theorems in approximation theory for spline approximation in the space L _,α a, b , α> 0 , 0 < p < 1

Theorem 3.1 direct theorem):

Let f ∈ L _,α 0,1 ,α> 0 , 0 < p < 1 ,σ= α+ p=1 =1 and k ≥ 1

SW f ,α ≤ CK 'f, 1

Wα , L ,α; Bσα,α, ( n = 1,2, … , C = C α, p, k (3.1)

Theorem 3.2 (inverse theorem):

For f ∈ L _,α 0,1 , 0 < p < 1 ,σ= α+ p=1 =1 and k ≥ 1

Then:

K 'f,_W1α , L ,α; Bσα,α, ( ≤ C n=α '∑

1

<!vαS< f ,α# λ W

<>1 (

)

λ

, n = 1,2, … (3.2)

Where λ= minIσ, 1X , C = C α, p, k .

Corollary 3.3:

For f ∈ L _,α 0,1 , 0 < p < 1 , k ≥ 1 and for ω be nondecreasing and nonnegative function on [0,1] such

that 2δω t ≥ω 2t for δ≥ 0, and

t ≥ 0 . Then

SW f ,α = ‰ 'm=βω n=1 ( , iff K 'f,

1

Wα , L ,α; Bασ,α, ( = ‰ 'm=βω n=1 (

Where 0 <δ+β<α, β≥ 0 .

Proof:

If K 'f,1

Wα , L ,α; Bσα,α, ( = ‰ 'm=βω n=1 (

Then by Theorem (3.1) (direct theorem) we have

SW f ,α ≤ CK 'f,_W1α , L ,α; Bσα,α, ( = ‰ 'm=βω n=1 (.

Let S_W f _,α= ‰ 'm=βω n=1 ( , 0 <δ+β<α

We need to prove that

ω mt ≤ 2m δ_ω t , t ≥ 0, m ≥ 1. _(3.3)

Since

ω 2t ≤ 2δ_ω t , _{then ω} 2`t ≤ 2`δ_ω t , t ≥ 0, s ≥ 0_.

Suppose that2`=1< Š < 2`. Then last inequalities give me

ω mt ≤ω 2`t ≤ 2`δ_ω t ≤ 2m δ_ω t _.

From inequality (3.3) we get

ω mt ≤ !2 λ+ 1 #δω t , λ≥ 0, t ≥ 0 . (3.4)

Now using Theorem (inverse theorem) (3.2) to get

K @f,_n1_α , L ,α; Bσα,α, D ≤ C n=α ‹z

1

v !vαS< f ,α#

λ W

<>1

Œ

1 λ

The fact that δ+β<α gives

K @f,_n1α , L ,α; Bσα,α, D ≤ C n=α ‹z

1

v 'vα=βω n=1 (

λ W

<>1

Œ

1 λ

Inequality (3.4) implies that

K 'f,_W1α , L ,α; Bσα,α, ( ≤ C1 n=α ‹∑ v=1rvα=β'

W <+ 1(

δ

ω n=1 s λ W

<>1 Œ

)

λ

≤ C_? n=α 'nδλ!_ω n=1 #λ∑W vα=β=δ λ=1

<>1 (

)

λ

≤ C_• n=α 'nα=β λ!S < f ,α#

λ

(

)

λ

= O 'm=β_ω n=1 ( _▄

Now we will offer some of the theorems that we need in the proof theorems above.

Theorem 3.4:

Let f ∈ B_σα_{, ,α,} ,α> 0 , 0 < p < 1 , k ≥ 1 . then

f ∈ L ,α 0,1 and

E =1 f ,α≤ C‖f‖._σα_,/,α,9 . (3.5)

Where E ₌₁ f denotes the best approximation to f in L _,α 0,1 by means of all polynomials of degree

Proof:

Let ψ_?•∈ E k, 2‚ such that

$f −ψ_?•$

σ,α= E?• f σ,α ,‚>1,? ,…. (3.6)

Define

E_?•m•) fσ,α ≤

1

•)γE?•m fσ,α ≤ E?•mp) f σ,α

(3.7)

For j = 1, 2 , … where γ= min 1,σ

Put u_"= •ψ_?•m−ψ_?•mp)• then by using (3.6) and ((3.7)) we get

$u"M1$_σγ_,α = ‘ψ_?•m•)−ψ_?•m‘ σ,α γ

$u_"M1$

σ,α γ

≥ ‘f −ψ_?•m‘

σ,α γ

− ‘f −ψ_?•m•)‘

σ,α γ

≥ E_?•m f σ,α

γ

− E_?•m•) f σ,α

γ

≥ 2E_?•m•) f σ,α

γ

(3.8) And

$u"M1$_σγ_,α ≤ E_?•m fσ,α γ

+ E_?•m•) f σ,α

γ

≤’_•E_?•m f σ,α

γ

(3.9) Hence

$u"M1$_σ,α ≤ '?_•( )

γ_$u

"$_σ,α , i = 1,2, …. (3.10) Assume λ_"= $u_"$_C

σ,α,δ"= 2

‚m_{, inequality (3.10) , (lemma 2.5) and lemma (2.7) gives}

$∑ u∞">1 "$ _,α≤ C k, p,σ @∑ 2‚m'/σ=1($u_"$

Cσ,α∆ƒ

∞

">1 D

) /

(3.11)

Inequalities (3.6), (3.7) and (3.11) give

2‚m'/σ=1($u_"$

Cσ,α∆ƒ ≤ C p,σ 2

‚m'/σ=1(!E

?•m f σ,α+ E_?•m₌₁ f σ,α#

≤ C p,σ 2‚m'/σ=1(!E

?•m=1 f σ,α#

Hence

$∑ u∞">1 "$ _,α≤ C k, p,σ '∑ 2‚m' /

σ=1(

∞

">6 E_?•m f σ,α( ) /

≤ C k, p,σ r∑ @2‚m'

)

σ=

) /(E

?•m f σ,αD ∞

">6 s

) /

(3.12)

By definition of ψ_?• and the fact that f ∈ L_σ,α 0,1 . Then we have the following series

∑ '∞">1 ψ_?•m−ψ_?•mp)( is a function f −ψ1 in Lσ,α 0,1

Inequality (3.12) implies that the same series converges in L _,α 0,1 , Hence in f ∈ L _,α 0,1 .

Also inequality (3.12) yields

E₌₁ f _,α ≤ $f −ψ₁$ _.

α≤ C k, p,α ‖f‖.σα,/,α,9 ▄

Theorem 3.5:

Forf ∈ B_σα_,α, ,α> 0 ,σ= α+ p=1 =1, 0 < p < 1. Then

f ∈ L ,α 0,1 and SW f ,α≤ C k, p,α n=α‖f‖._σα_,α,9, n = 1,2, … and k ≥ 1. (3.13)

Proof:

Since σ< , using the inequality (3), for f ∈ B_σα_,σ,α, then f ∈ B_σα_{, ,α,} . Then according to theorem (3.5) it follows that, for f ∈ B_σα_,α, , then

f ∈ L ,α 0,1 and

E =1 f ,α≤ C‖f‖._σα_,α,9 (3.14)

Suppose the function x = f u + v − u x , x ∈ 0,1 . Using inequality (3.14)we get

E =1 φ ,α≤ C‖φ‖._σα_,α,9 (3.15)

E =1 φ ,α6,1 = inf_“∈“

9p) '0 | f u + v − u x − P x e

=αg| dx 1

6 (

) /

= 1

<=n/)“∈“inf9p) !0 | f t − P t e =α2| dt <

n #

) /

= 1

∆)/“∈“inf9p) E =1 f ,α∆

Therefore we have

E =1 φ ,α6,1 =

1

∆)/“∈“inf9p) E =1 f ,α∆ (3.16)

On the other hand we obtain

$∆tφ . $_{C/,α”,)p9•} = '0₆1= 2e∆t e=αgf u + v − u x eσdx( )

σ

$∆_tφ . $_C

/,α”,)p9• =

1

|∆|)σ'0 e∆|∆|2 e

=αgf y eσdy

1= 2

6 (

)

σ_{, y = u + v − u x}

= 1

|∆|)σ$∆|∆|2f . $Cσ,α—,up9|∆|•

Therefore where h = |∆|t we have

‖φ‖_.σα_,α,9= @0 @t=α$∆₂φ . $

Cσ,α”,)p9•D

σ₅₂

2 )

9

6 D

)

σ

= „0 „

‘∆•9φ. ‘

˜_{σ,α! —,up9|∆|• #}

|∆|)σ2α †

σ 52

2 )

9

6 †

)

σ

= 1

|∆|)/r0 rh

=α$∆

Ef . $_C

σ,α! —,up9|∆|• #s

σ 5E

E |∆|

9

6 s

)

σ .

‖f‖. ∆ = @0 @t=α$∆2f . $_{Cσ,α—,up9•}D

σ₅₂

2 up—

9

6 D

)

σ

(3.17)

Therefore (3.15), (3.16) and (3.17) gives for each interval ∆_⊂ 0,1

E =1 f ,α∆ ≤ C‖f‖._σα_,α—,…. (3.18)

Let x₆, x₁, x_?, … , x_l=1 such that 0 = x₆< x₁< ⋯ < x_l=1< 1.

We define x_las

xl= sup ™y: xl=1< › ≤ 1, ‖f‖.σα,αœƒp),• ≤

1

n ‖f‖.σα,α”,) ž

For some m ≥ 1, x_‚≤ 1 and ∆_l= x_l=1, x_l . Then we have

∑ ‖f‖.σα,α∆ƒ

σ _{= ∑ 0 t}=ασ$∆

2f . $_C

σ,α!œƒp),œƒp9•#

σ ₅₂

2 e∆ƒe

9 6 ‚ ">1 ‚

">1

= 0 t)9 =ασ

6 ∑ $∆2f . $C_{σ,α!œƒp),œƒp9•#}

σ ₅₂

2

|∆_ƒ|Ÿ 2

≤ 0 t)9 =ασ

6 $∆2f . $Cσ,α”,)p9•

σ ₅₂

2

= ‖f‖_. σ,α α _6,1

σ

Therefore

∑ ‖f‖.σα,α∆ƒ

σ ≤

‚

">1 ‖f‖.σα,α6,1

σ _(3.19)

Hence

‖f‖.σα,α∆ƒ

σ ₌1

W‖f‖.σα,α6,1, i = 1,2, … , m − 1 (3.20)

Equality (3.19)and inequality (3.20)implies that there exists m ≤ n such that x_‚= 1. For definition of

x6, x1, x?, … , x‚ it gives that

‖f‖_.σα_,α∆

ƒ

σ ₌1

W‖f‖.σα,α6,1, i = 1,2, … , m. (3.21)

Now we Compensation the estimate inequality (3.18) to the function in ∆_l. In view of equality (3.21) we obtain

E =1 f ,α∆_ƒ ≤

C

n 1σ

‖f‖.σα,α,9n,< i = 1,2, … , m. m ≤ n

S_W f _,α ≤ '∑ E ₌₁ f _,α∆

ƒ ‚

">1 (

) /

≤ C k, p,α n=α‖f‖._σα_,α,9 ▄

Theorem (3.6):

For s ∈ S k, n, 0,1 , k, n ≥ 1, 0 < p < 1 ,α> 0 and = α+ p=1 =1 . Then

‖s‖.σα,α,9= C

‖ ‖/,α

opα , C = C α, p, k . (3.22)

Proof:

From equality ‖f‖_. σ,α,9

α = '0 !t=αω f; t _σ,α#σL52

2( 1

6 L

)

σ

Let s ∈ S k, n, 0,1 then we shall a partition of [0,1] such that

0 = x6< x1< ⋯ < x‚= 1 and polynomial P_"∈ P₌₁, j = 1,2, … , m such that s x = P_" x for x ∈ ∆_".

According to the above and ∆_Es x = 0 for x, x + kh ∈ ∆_". Therefore

ω s; t σσ,α,α= sup

6¡E¡10 e∆Ee

=αgs x eσdx

1= 2

6

ω s; t σσ,α,α≤ C ^∑e∆_me] 20 |e_∆m =αgs x |σdxL

L+∑ 0 |e=αgs x |σdx

∆_m∗

e∆me£¤¥ + 0 |e∆_m∗∗ =αgs x |σdxž

(3.23) Where

∆"∗= !x"=1, x"=1+ kt# , ∆"∗∗= !x"− kt, x"#

By lemma (2.6)[4]we have

‖s‖_Cσ,α'∆m∗₍≤ Ct

)

αe∆_"e=

) /_‖s‖

C/,α!∆m# then

ω s; t σσ,α≤ C ^∑ e∆"e

σα

‖s‖_Cσ_/,α!∆m#+

e∆me] 2 L ∑ e∆"e

=σ

/

e∆me£¤¥ ‖s‖C/,α!∆m#

σ

(3.24)

Now applying inequality (3.24) for ω s; t σσ,α to get

‖s‖.σα,α,9= 0 ! t

=α_ω s; t σ,α#

σ52 2 1

6

≤ C 0 t =ασM1 '∑ e∆_"eσα‖s‖_C

/,α!∆m#

σ

e∆me] 2 L L+∑ te∆"e

=σ α‖s‖

C/,α!∆m#

σ

e∆me£¤¥ D dt

1

6

≤ C ∑ !e∆‚_{">1 "}eσαL‖s‖_C

/,α!∆m#

σ _{0 t} =ασM1dt L+e∆

"e=

σ

/_‖s‖ C/,α!∆m#

σ _{0 t}•∆m•₉ =ασ

6 dtŒ

∞ •∆m•

9

≤ C ∑ ‖s‖_C

/,α!∆m#

σ ‚ ">1

≤ C '∑ ‖s‖_C/, α!∆m#

‚

">1 (

σ

/ _n 1=_/σ

= C α, p, k n ασ‖s‖ C/,α6,1

σ _▄

5.4 Proofs Of Main Results:

Proof of theorem (3.1)(direct theorem): Since

SW f ,α ≤ C k, p,α n=α‖f‖._σα_,α,9, n = 1,2, … and k ≥ 1.

Let f = f₆+ f₁ , f₆∈ L _,α , f₁∈ B_σα_,α,

Then by theorem (3.5) it follows that

SW f ,α ≤ 2

) /r‖ f₆‖_C

/,α+ sup_{`∈¦ ,W}‖f1− s‖C/,α s

≤ 2

) /_{'‖ f6}‖_C

/,α+ SW f1 ,α ( ≤ C '‖ f₆‖_C/,α+ n=α S

W f1 ._σα_,α,9 ( Hence

SW f ,α ≤ CK 'f,_W1α , L ,α; Bσα,α, ( ▄

f = f − s_?…+ s_?…− s_?…p) + ⋯ + s₁− s₆ + s₆ Therefore

$∆Ef$_C/,α≤ $∆E f − s?… $_C/,α+ z$∆E s?•− s?•p) $_C/,

α

<

‚>6

≤ 2 ‖f − s?…‖_C/,α+ ∑<_‚>6ω s?•− s_?•p); h _,α (4.1)

Where s_?…∈ S k, 2<

Let S_?< f is an existence set in L _,α Hence

‖s?…− s_?…p)‖_C/,α ≤ 2=

)

/'‖f − s_?…‖_C

/,α+ ‖f − s?…p)‖C/,α (

≤ 2 @2=1 S?…p) f C/,αD

≤ CS_?…p) f _C/,α. (4.2) From (4.1) and (4.2) to get

$∆Ef$_C/,α≤ 2 S?… f _,α+ 2h ∑<‚>62‚ S_?•p) f ,α (4.3)

Let |h| <1

W , n = 2< then (4.3) gives

$∆Ef$_C/,α≤ 2 S?… f ,α+

?

W9∑<‚>62‚ S?…p) f _,α

$∆Ef$_C/,α≤?

9

W9∑?<t>6 r + 1 =1Et f ,α (4.4)

Since

S?… f _,α≤

?…9

W9E?… f ,α

2‚ _S

?•p) f _,α≤ 2? =1∑? =1ℓ>?•p§ ℓ+ 1 =1Eℓ f ,α

Using (4.4), and(4.2), and( (3.22) to get For all m ≥ 0

K!f, 2=mα ,_L

p,α; Bσα,α,k# ≤ ‖f−s2m‖Lp,α+2

=mα_‖s

2m‖B_σα_,α,k

≤S₂km f _L

p,α+2

=mα_{™∑ $s}

2v−s2vp1$B_σ,α,k

σ

m

v>1 L L+‖s1‖B_σα_,α,k

σ _a

1

σ

Since‖s‖_B σ,α,k

α ≤C‖s‖p,α

npα , we get

K!f,2=mα ,Lp,α; Bσα,α,k# ≤S₂m k

+C2=mα™∑ @ 2vM1α$s2v−s2vp1$Lp,α

σ

D m

v>1 L L+‖s1‖Lp,α

σ a

1

σ

L≤C₁'S₂km f _L p,α

σ L +₂=mα_{^∑ '2}v=1α_S

2vp1 k

f _L p,α(

σ

m

v>1 L + L‖f−s1‖Lp,α

σ _{+ ‖f}‖

Lp,α

σ _a(

1

σ

≤C22=

mα_{^∑ ∑} 1 t't

α_S

t k

f Lp,α(

σ

2v t>2vp1 m

v>1 + ' S1 k

f Lp,α(

σ

+ ‖f‖Lp,α

σ a

1

σ

≤C32=

mα_{^∑ t}α2=1' _S t k

f Lp,α(

σ

2m

t>1 + ‖f‖Lp,α

σ a

1

σ Since K-functional is monotony

Hence (3.2) is satisfy ▄ REFERENCES

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Chen, Wai-Kai, 2009."Feedback, Nonlinear, and Distributed Circuits". CRC Press. pp. 9–20. ISBN 978-1-4200-5881-9.

Fadil, H., 2010. “On the constrained spline approximation”, M.sc., thesis, Education College, Babylon university.

Hawraa A. Fadhil, 2010 "On The Constrained Spline Approximation", M. Sc. Thesis , Babylon University , Mathematical Department College of Education.

Judd, Kenneth L., 1998. “Numerical Methods in Economics”. MIT Press. p. 225. ISBN 978-0-262-10071-7. Petrushev, P.P., V.A. popov, 1987. "Rational approximation real function" (Encyclopedia of mathematics and its applications; v. 28) Cambridge University Press.

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