ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Approximation of Functions in Lp,α(I) (0 < p < 1) Saheb K. Jassim Israa Z. Shamkhi Dept. of Mathematics/ College of Science / Al-Mustansiriya University Received in: 15 may 2012 , Accepted in: 3 march 2013 Abstract. In this paper we show that the function , ( )pf L Iα∈ ,0 ∈ ∈ we denote by Lp,α ,the space of all functions f such that , ( ) 1 , ,1 x pa p dx p b f f x e pα α −    ∫    = < ∞ ≤ < ∞ …(2.1) Definition (2.2): [1] If J is an interval then k-th order module of smoothness of f is defined by 1 , 0 0( , , ) : ( , , ) , p pk x k p h h t J f t J sup f x J e dxαα αω − ≤ ≤ >   = ∆    ∫ …(2.2) where kh∆ is the symmetric difference, such that 0 ( 1) ( ) ( , , ) : 2 2 0 k k i k ih k k k f x h ih if x h J f x J i otherwise − =    − − + ± ∈  ∆ =      ∑ Definition (2.3): [1] If , ( )pf L Iα∈ then the Ditizian – Totik modules of smoothness is defined by, 1 1 , 0 1 0 ,( , , ) : ( ) ( , , ) , p pk x k p h h t f t J x f x J e dxsupϕ αα αω ϕ − < ≤ − >   = ∆    ∫ …(2.3) where 0( ) ( 1) ( ( ) ( )) ( ) ( , , ) : 2 2 0 k k i k ih x k k k f x h x ih x if x h x I f x I i otherwise ϕ ϕ ϕ ϕ− =    − − + ± ∈  ∆ =      ∑ Monotone piecewise linear approximation Let 0 1 ... na bξ ξ ξ= < < < = such that adjacent 1[ , ] , 1,...,j j jI j nξ ξ−= = have comparable lengths, that is 1 0 j j I c I ± ≤ ...(3.1) with 0c an absolute constant. We denote to Y the class of all piecewise linear functions on [a,b] .Each function YS ∈ is completely determined by its left and right hand values 335 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ( ), 1,..., 1jS j nξ = − and the values 0( ), ( )nS Sξ ξ , if f is a function in , ( ) ,0 p 1pL Jα < ≤ ,J is an interval then a polynomial p of degree k is a near best , pL α approximation to f from among all polynomials of degree ≤k if ,, ( ) ( , )k ppf p J M E f J αα− ≤ ...(3.2) Where M is constant and ,( , )k pE f J α is the error of best approximation to f on J in the space ,pL α from among all polynomials of degree ≤k . Theorem (3.1): [3] For any interval [a,b] and 0 1 na bξ ξ ξ= < < < = there is a piecewise linear function S y∈ with the following properties : i ) S is non decreasing. ii) There is a constant M0 >0 depending only on p and the constant c0 such that for j=1,…,n, j satisfies (3.2) for an interval jI with 2 1 1 2:j j j j j j j jI I I I I I I I− − + +⊂ ⊂ =     (such that :kI φ= if 0k ≤ and nk > ). iii) 0 0 0 0( ) ( ) and ( ) ( ) S S S Sξ ξ ξ ξ+ + − −≥ ≤ Theorem (3.2): [3] Under the hypothesis of theorem (3.1) there is a nondecreasing piecewise linear function S y∗ ∈ satisfying (i) and (ii) of theorem (3.1) with jI replaced by 3 3 j j j vI I ∗ + −=  and the additional property that S ∗ is continuous. Definition (3.1): For , ( )pf L Iα∈ let us define, 1 1 , 0 ( , , ) : ( , , ) ...(3.3) t p pk x k p s J w f t J t f x J e dxdsαα − − = ∆    ∫∫ Lemma (3.1): , , ,For we have ( , , ) ( , , )p k p k pf L w f t J f t Jα α αω∈ ≅ Proof 1 , 0 1 1 0 0 = ( , , ) : ( , , ) , ( , , ) p pk x k p s s t J p pk x s s t J f t J sup f x J e dx sup t t f x J e dx α α α αω − < ≤ − − < ≤ >   = ∆      ∆    ∫ ∫ 1 1 , 0 , ( , , ) ( , , ) = ( , , ) t p pk x k p s J k p f t J c t f x J e dx ds cw f t J α α α ω − −   ≤ ∆    ∫∫ , ,( , , ) ( , , )k p k pf t J cw f t Jα αω ≤ …(3.4) 336 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 1 1 , 0 1 1 0 ( , , ) : ( , , ) = ( ( , , ) ) t p pk x k p s J t pppk x p s J w f t J t f x J e dxds t f x J e dx ds α α α − − − −   = ∆      ∆    ∫∫ ∫ ∫ 1 1 1 0 = (( ( , , ) ) ) t p pk x pp s J t f x J e dx dsα− −   ∆    ∫ ∫ Theorem (3.3): (Whitney's theorem in Lp,α) For any , ( )pf L Iα∈ on [0,1] and for any integer n≥1 , there is a polynomial p(x) of degree at most n-1 such that , ( ) , 1 ( ) ( ) 6 ( , ) 1 x n pf x p x e f n α αω −− < + Let p be the interpolation polynomial of = 1v v x n + If 1 , = , 0 , 1 h x vh t t h n = + ≤ < + Let ( ) 1 ( ) 0 ( 1) ( , ) ( , ) ( ) n v n x vy yn v f x f vh t f x vy e dy h αϕ ϕ − − −−= + = ∆ −∫ ( ) ( ) 1 1 1 00 1 1 , 0 1 1 , 0 1 1 , (( ( , , ) ) ) = ( ( , , ) ) = ( ( , , ) ) | = ( , , ) t p pk x pp s s t J t p p k p p t p k p p p k p t c sup f x J e dx ds t c f t J ds t cs f t J t t c f t J α α α α ω ω ω − − < ≤ − − −   ≤ ∆          ∫ ∫ ∫ , , , 1 , , 1 , (3.5) Then and From(3.4) and (3.5) we get , = ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) k p k p k p k p k p k p k c f t J w f t J c f t J c w f t J f t J c w f t J f t J α α α α α α ω ω ω ω − − ≤ ≤ ≤  , , This implies that and are equivalent ( , , ) p k p k k cw f t J w α α ω ≤ 337 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Thus for [ , ( 1) ],x vh v h∈ + ( ) , 1 1 ( , ) ( , ) 1n pnv f x f n α ϕ ω≤ + ...(3.6) , 0 Let ( ) , 0,1,..., n n v j j v x j x v n v j= ≠ − = = −∏ Before we prove Whitney theorem in Lp,α we need these results. Proposition (3.2): [6] Let Pn-1(f) be the interpolation polynomial for f at the points h,2h,…,nh , i.e. 1 1, 1 1 ( , ) ( ) ( 1), n n n j j x p f x f jh h− − −= = −∑  then 1 1 ,0 , , 0 00 ( ) (0) ( ) ( , ) ( , ) ( ) ( , ) ( ) tn n n n h n n n n j n n j j j x x x v f x p f f x f x h f jh v dv h h h ϕ ϕ ϕ−− = = − ′− = ∆ + − + +∑ ∑∫   Lemma (3.2): [6] Lemma (3.3): [6] ( ) { } ( ) 1 1 , , 0 0 1 1 : max ( ) ; 1 , 0,1,..., , 2 1 1 1 and 1 ... , 0 2 3 n n v v n v j n v n j v v n x v x v v v σ σ µ σ σ − + = + + − = ≤ ≤ + ≤ = = + + + + = ∑  Proposition (3.4) ( ) , 1, 1 , 1 0 For integrable on [0,1] , weget 6 7 min( , ) 1 ( ) ( ) ( 1) ( , ) , 1 1 1 1 For [ , ( 1) ] , , 0,1,..., , 1 ... , 0 1 2 3 p n v n v n j n pn j v v f L x f x f jh f h h x vh v h h v n n v α α σ σ ω σ σ − − − = ∈ + − − ≤ ∈ + = = = + + + + = + ∑  Proof Since the interpolation polynomial is , 1 1, 1 1 ( , ) ( ) ( 1), n n n j j x p f x f jh h− − −= = −∑  and knots are symmetric with respect to the middle of [0,1], we prove it for 1 x [0, ] 2 ∈ . For v = 0 from lemmas (3.2) and (3.3),using proposition (3.2)and using (3.6) we get, ( ) 1 , 0 : max ( ) ; 0 1 1 n n n j n v j v x x − =   = ≤ ≤ =    ∑  338 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ( ) ( ) ( ) ( ) , ,0 , ,0 1 0 1 0 0 0 , ,0 1 0 1 1 1 ( ) ( ) ( , ) max ( ) 1 max ( ) ( ) 1 1 ( , ) 3 1 max ( ) tn n x n p n n j n jn nt t j jv j n n p n jn t j v f x p x e f t t v dv h f t h α α α ω ω − ≤ ≤ ≤ ≤ = = ≤ ≤ =   ′ − ≤ + + +       ≤ + +    ∑ ∑ ∫ ∑     , 1 6 ( , )n pf h α ω≤ For 1 1, 2, 3,..., 2 n v − = , we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , ,1 0 0 0 , ,1 0 1 ( , ) 1 1 ( ) ( ) 2 max ( ) ( 1) 1 ( , ) 1 2 3 max ( ) tn nn p x n n v n j v n jn n nv u v j jv j j nn p n v n jn n v u v jv j f hf x p x e t v dv f h u α α α ω ω − ≤ ≤ + = = ≤ ≤ + =   ′ − ≤ + + +       ≤ +     ∑ ∑ ∫ ∑    ( ) ( ) ( ) 1 , , , 5 3( ) 1 ( , ) 6 7 1 ( , ) 6 7 min( , ) 1 ( , ) v v n pn v v n pn v v n v n pn v f h f h f h α α α σ σ ω σ ω σ σ ω + − + + ≤ + ≤ + ≤ Proof of theorem(3.3) By using proposition (3.4) we get immediately the proof of theorem (3.3) For v = 0 ( ) , 1 ( ) ( ) 6 ( , )x n pf x p x e f h α αω −− < and for v = 1,2,…, 1 2 n − we have ( ) ( ) , 6 7 min( , ) 1 ( ) ( ) ( , )x v n v n pn v f x p x e f h α α σ σ ω− − + − ≤ Since for every v = 0,1,2,…,n we have ( ) 6 7 min( , ) 6v n v n v σ σ −+ ≤ Then ( ) , 1 ( ) ( ) 6 ( , )x n pf x p x e f h α αω −− ≤ Auxiliary Results To construct monotone approximation, we shall need properties for the jξ .we begin recalling a construction introduced by [4] and used also in [5]. We approximate the truncated 339 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 functions ( ) : ( )j jx xϕ ξ += − by algebraic polynomial .If x=cos t we obtain algebraic polynomial ( ) : ( )j n jr x T t−= . Since , ( ) : ( ) 0,..., j j t t j j n n t t T t J J u du j nχ + − = ∗ = =∫ …(4.1) where , 0, ...,[ , ]: , := j j j j nnj t t jt πχ χ =−= and nJ be a Jackson kernel which defined by, 2 sin 2 ( ) , ( ) 1 2 r n n nt J t J t d t π π−    = =     ∫ …(4.2) jR is polynomial of degree ≤ nr is defined by,[3] [ ,1] 1 ( ) ( ) and : 1,..., j x j j jR x r u du j nξθ χ − = = =∫ …(4.3) Lemma (4.1): [3] With 2( ) 1x xϕ = − we have for any n ≥ 10 , i) 1 11 4 3 2 3 4( ) ( ) , [ , ], 4,..., 5j j j jc x n c x n x j nϕ ξ ξ ϕ ξ ξ − − + − − +≤ − ≤ ∈ = − ii) 1cos ( ), 0,..., 1j n j j jt c j nξ ξ ξ− +− ≤ − = − iii) 1 1 1 2 1( ) ( ), 1, 2,..., 1j j j j j jc c j nξ ξ ξ ξ ξ ξ+ − +− ≤ − ≤ − = − iv) 1 12 7 2 71 ( ) , 1 , 1 ( ) , 1nx c x n x and x c x n xϕ ξ ϕ ξ − − −+ ≤ − ≤ ≤ − ≤ ≤ ≤ v) 11 1 7( ) 1 , 1;n nc x n xϕ ξ ξ − − −≤ − ≤ ≤ ; 1 1 1 7 ( ) 1, 1c x n xϕ ξ ξ − ≤ + − ≤ ≤ where 1 2, and are constants dependent of and c c c n x . Lemma (4.2): [3] For 11,..., 1 let ( ) : 1 / ( )j j j jj n d x x ξ ξ ξ+= − = + − − then 1 ( ) ( ) ( ) , r j j jr x x c d xθ − +  − ≤   2 1( ) ( ) ( ) ( ) r j j j j jR x x c d xϕ ξ ξ − + +  − ≤ −   with the constant c Lemma (4.3): [3] If p is polynomial of degree≤ k, then for x∈ [-1,1] we have, 11 ( ) 1 max ( ) j j k j u j j x P x c P u ξ ξ ξ ξ ξ +≤ ≤+  −  ≤ +  −  Lemma (4.4): [3] If 02 and it is readily to seen that [ ] 2 1( ) ( ) i 1,..., -1i i i I d x dx c nξ ξ − +≤ − =∫ …(5.3) To prove that [ ] 21 1 2 11 1 ( ) 1 ii i i x d x dx dx ξ ξ ξ − − +− −  −  = + −  ∫ ∫ ( ) if > ( ) if < i i i i i x x x x x ξ ξ ξ ξ ξ − − =  − − If i ix xξ ξ− = − we take [ ] 2 21 1 1 2 1 1 11 1 1 ( ) 1 i i i ii i i i i x x d x dx dx dx ξ ξ ξ ξ ξ ξ ξ ξ − − − + + +− − −    − − + − = + =   − −    ∫ ∫ ∫ 11 1 1 1 1 2 = -( ) i ii i i i xξ ξ ξ ξ ξ ξ − + + + −  + − −  −  1 1 1 1 1 1 1 1 2 1 2 =-( ) i i i ii i i i i i ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ − − + + + + +     + − + −  − −    − −     343 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ( )( ) ( )( ) ( )( ) ( ) 1 1 1 1 1 1 1 1 1 1 2 1 2 =-( ) 1 2 1 2 =-( ) i i i i i i i i i i i i i i i i i i i ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + + + + + + + +  − − − − − + − −   + − − −  − − ( ) ( )( ) ( ) ( )( ) 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 2 =-( ) 1 2 1 2 i i i i i i i i i i i i i i i ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + + + + + + +  − − − − +   + − − −   − − −   + − − −  ( ) ( )( ) ( ) 2 1 1 1 1 2 = 1 2 1 2 i i i i i i i iC ξ ξ ξ ξ ξ ξ ξ ξ + + + + − + − − − ≤ − In the similar way we get the same result take ( ) .i ix xξ ξ− = − − If we applied this result we obtain, ( ) ( ) 1 1 1 1 1 , 1 1 1 1[ , ],1 1 ( ) max ( , ) ( , ) ...(5.4) ( , ) ( , ) i i j j n px n n i i j jxp i n px i i j jL pi j j L f P c p f x p f x e c c p f x p f x e α ξ ξα α ξ ξα ξ ξ ξ ξ ξ ξ + + − − + + < < = − − + + = + − ≤ − − ≤ − − − ∑ ∑ ( ) 1 1 1 1 1 1 [ , ],1 1 1 1 1 = ( , ) ( , ) = ( ( , ) ( , )) ( ( ) ( , )) j j j j j j n px i i L pi n px i i i px i i c p f x p f x e c p f x p f x e dx c f x p f x e α ξ ξα ξ α ξ ξ α ξ + + + − − − + = − − + = − = − − ≤ − ∑ ∑ ∫ ∫ n dx∑ , 1 ( , ) pk pc f n ϕ αω≤ Proof of theorem (5.2) (part 2) (monotone case) Here we use the continuous piecewise linear function S ∗ .For k=2 we can take ( )jp f in section 5 to be the polynomial of degree one and denote :=j j ja x b ∗ + of theorem (3.2) such that j ∗ is a near-best ,pL α approximant for kI ∗ . Definition (5.1): [3] 1 1 1 1 0 ( , ) ( , 1) [ ( ( ) ( )) n n j j j j L f x p f a x xϕ ϕ − ∗ + + = = − + −∑ …(5.5) 1 1 1 1 0 ( , ) ( , 1) [ ( ( ) ( )) n n j j j j P f x p f a R x R x − ∗ + + = = − + −∑ …(5.6) Now we note that since 1( ) ( )j jR x R x+− is increasing for j=1,…,n-1 and 0ja ≥ for j=1,…,n , the polynomial ( , )nP f x ∗ is nondecreasing in [-1,1]. 344 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Lemma (5.4): [3] ( , ) ( , ) for 1,..., -1n nL f x L f x j n ∗ = = Proof of theorem (5.2) 1 , 1 1 1 1 1 ( ) ( , ) ( , ) ( , ) = ( ) ( , ) ( , ) ( , ) ( ) ( , ) ( , ) ( , ) p p n n np p x n n n p x n n n f x P f x L f x P f x dx f x L f x L f x P f x e dx f x L f x e dx L f x P f x α α α ∗ ∗ ∗ − ∗ ∗ − − − ∗ ∗ − − = −  − + −   ≤ − + −  ∫ ∫ ∫ 1 1 p xe dxα− −   ∫ In view of theorem (5.1), we have only to estimate the second term, 1 1 1 1 1 1 1 ( , ) ( , ) ( )( ( ) ( )) ( )( ( ) ( )) ( ) ( ) n x x n n j j j j i n x j j j j i x j j j L f x P f x e a a x R x e a a x R x e a a e x α α α α ϕ ϕ ϕ − ∗ ∗ − − + = − − + = − +  − = − −  ≤ − − ≤ − ∑ ∑ ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( ) ( , ) ( , ) ( ) ( ) = ( , ) ( , ) ( ) n j i xn i i i i j j i i i n x i i i i i i i R x p f p f e x R x p f p f e α α ξ ξ ϕ ξ ξ ξ ξ ξ ξ − = −− + + + = + − − − + + + + = − − ≤ − − − − ∑ ∑ ∑ 1 ( ) ( )j jx R xϕ − Hence by lemma (4.2) and (4.3) we have ( ) ( ) 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 ( , ) ( , ) ( , ) ( , ) ( ) ( ) ( ) 1 max ( , ) ( , ) ( ) ( ) i i np p px x p n n i i i i i i i i i p n pi x p i i i i i i ix i i i L f x P f x e p f p f e x R x dx x p f p f e x α α α ξ ξ ξ ξ ξ ξ ϕ ξ ξ ξ ξ ξ ϕ ξ ξ + − ∗ ∗ − − − + + + + =− − − − − + + + + < < = +  − ≤ − − −   −  ≤ + − −  −  ∑∫ ∫ ∑ 1 1 ( ) p iR x dx − −∫ ( ) [ ] ( ) 1 1 1 1 ( 2) 1 1 1 1 1 1 11 ( 2)1 1 1 1 1 1 11 1 max ( , ) ( , ) ( ) ( ) ( ) max ( , ) ( , ) 1 i i i i p n p r pi x p p i i i i i i i i ix i i i r p n p ix i i i ix i i i x p f p f e d x dx x p f p f e dx α ξ ξ α ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ + + − − +− − + + + + + < < = +− − + − − + + + < < = +−  −  ≤ + − − −  −   −  ≤ − +  −  ∑∫ ∑∫ Then from the derivation of (5.4) and provided rp-2p>2 we obtain, 1 2 , 1 1 ( , ) ( , ) ( , ) p x p n n pL f x P f x e c f n α ϕ αω ∗ ∗ − −  − ≤ ∫ Then 345 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 2 ,, ,, 1 ( , ) ( , ) ( , ) 1 ( ) ( , ) ( , ) p p n n pp p p n k pp L f x P f x c f n f x P f x c f n ϕ αα ϕ αα ω ω ∗ ∗ ∗ − ≤ − ≤ References 1. Husain, L.A., (2010), Unbounded Function Approximation in some ,pL α Spaces, M.Sc Thesis, Al-Mustansirya University, Department of Mathematics, College of Science. 2. Hansen, B.M, (2010), Unbounded Functions, M.Sc Thesis, Al-Mustansirya University, Departmen Mathematics, College of Education. 3. Devore, R.A., Leviatan, D. and X. M. Yu, (1992), Polynomial Approximation in 0 1 ( )p pL < < , Springer-verlage, New York. 4. Devore, R.A. and X.M.Yu, (1985), Poitwise Estimates for Monotone Polynomial Approximation .Constr. Approx, pp.323-331. 5. Leviatan, D. and X.M.Yu, (1991), Shape Preserving Approximation by Polynomial in pL . 6. Sendov, B. and Popov, V.A, (1981), Average of Modulus of Smoothness of Continuity. 346 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Lp,α(I) ،(0 < p < 1)تقریب الدوال في الفضاء صاحب كحیط جاسم إسراء زاید شمخي الجامعة المستنصریة /كلیة العلوم /قسم الریاضیات 2013 أذار 3، قبل البحث في : 2012 أیار 15: استلم البحث في الخالصة pL,في ھذا البحث وضحنا ان الدالة الغیر مقیدة في فضاء ال�وزن α یمك�ن ان تقت�رب م�ن متع�ددة الح�دود الجبری�ة م�ع ,وجود خطأ ال یتجاوز مقاسات التعومة 1 ( , , )k pf t n ϕ αω للدوال الغیر مقیدة في الفضاء, ( )pL Iα. التقریب الرتیب ، متعددات الحدود، درجة التقریب. الكلمات المفتاحیة : 347 | Mathematics