Mathematics - 333 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Degree of Monotone Approximation in ,pL α Spaces S. K. Jassim and I. Z. Shamkhi Department of Mathematics, College of Science – Al-Mustansiriya University Received in: 18 February 2011 Accepted in: 18 March 2012 Abstract The aim of this paper is to study the best approximation of unbounded functions in the weighted spaces , 1, 0,p pL α α≥ > . Key Words: Weighted space, unbounded functions, monotone approximation Introduction With a great potential for applications to a wide variety of problems, approximation theory represents on old field of mathematical research. In the fifties a new breath over it has been brought by a systematic study of the linear methods of approximation which are given by sequences of linear operators .These methods became a firmly entrenched part of approximation theory. The problem of function connected with different polynomials was examined in many paper like[1] and [2]. In this paper we studied the degree of approximation of unbound functions by using piecewise monotone polynomials in the weighted spaces ,pL α . Definitions and notations Let f be any function such that ( ) [ ], 0, , ,xf x Me M R x a bα α≤ > ∈ ∈ we denote by ,pL α , the spaces of all functions such that ( ) 1 , d ,1α α −    ∫    = < ∞ ≤ < ∞x pa p p b f f x e x p …(2.1) See [7]. we approximated f by a piecewise polynomial of degree at most N . Definition 1 Let 1 2 1 0 1 1 2 1 1S (x ,x , ...., x ) = { s [a,b] ; s (x ,x ) , 1, 2,... 1}where ( , ) ( , )( , )...( , )i-1 ik n i i k k nC i k x x x x x x x xn n − − −∈ ∈Π = + Π = ,{x , x , ...., x }1 2 k is a space of spline with simple knots (x , x , ...., x )1 2 k , consider 0 1 1a = x x x x = bk k +< < .... < apartition on interval [a,b] .for k=1,2,…let 1 2(x ,x ,...,x ) k k k kS Sk n= ,for some k knot such that end points [bounded], 0 1a = x and b=x for each k k k k + . The mesh is denoted by ( )10,1,..., k k i ii k m Max x xk +== − denote , 1, 2,....,S S ny = the collection of all set Mathematics - 334 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 { }, 0 ... 1S SY y y= < < < , and ( ) (1) Y∆ be the collection of all functions [ ], 0,1,f f L p α′∈ which change monotonicity at the points of Sy . Set ( ) ( )1Six x yi=∏ =∏ − …(2.2) The differentiable function in [0,1],L p α is in ( ) (1) Y∆ If ( ) ( ) 0, 0,1 ,f x x x   ′ ∏ ≥ ∈ [4]. Definition 2: [7] The degree of monotone approximation by polynomials nP of degree not exceeding n will be denoted by ( ) (1) ( ) , (1) , , ,inf p Y Ln p E f Y f pnp pn α α α ∈∆ = −  …(2.3) For p = ∞ then ( ) ( ) [ ](1) , , , inf , 0,1n nE f Y f x p xα α ∞ ∞ = − ∈ Let ( ) ( )xxx −= 1ϕ , x∈[0,1] …(2.4) The spaces , ,p rL α ϕ ,r∈N are the spaces of all functions such that ( ) ( ) 1 1 0 ppr xx f x e dxα−     <∞∫ …(2.5) where ( )( )( )( ) x rrf f x e α−= and ( ) ( ) 1 2 lim 0r r x x f xϕ → = . Definition 3: [3] For 1k ≥ the Ditizian – Totik modules of smoothness is defined by ( )( ) ( ) ( ) ( ) ( ), , sup , 0r rr kk r kh h xp pf t x f x t ϕ ϕω ϕ= ∆ ≥ And ( )( ) ( ) ( ) ( ) ( ), , ,, sup , 0 r rr k k r kh h xp p f t x f x tϕ ϕα α ω ϕ= ∆ ≥ [7] …(2.6) Where k th symmetric difference is defined by ( ) ( ) ( ) ( ) ( ) 0 1 2 k kk jk h x j j k j h x g x g xϕ ϕ+ = −  ∆ = − −      ∑ is …(2.7) and the supermum is taken over all ( ) ( )1,0 2 , ∈xh k xx ϕ . Note that for [ ], 0,1pf L α∈ then ( ) ( ),0 , ,, ,k kp pf t f t ϕ ϕ α α ω ω= . Mathematics - 335 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Definition 4: [6] At the points 0 1, ,.., nx x x are defined by 0 1 0 0 ( ) [ , ,.., ; ] ( ) n j n j j i i i j f x x x x f x x= = ≠ = − ∑∏ Then [ ]gZZ ;, 10 stands for the first divided difference of function g at the knots 0Z and 1Z , and [ ]gZZZ ;,, 210 denotes the second divided difference at the knots 10 , ZZ and 2Z . The Main Result: In [5] Leviatan and Shevchuk proved that For every ( ) ( )1 rf Y Cϕ∈∆  then: ( ) ( )1 ( ), 1 , ,rn k rr c E f Y f n n ϕω  ≤     …(3.1) where c is a constant independent of n and f . Also they showed that Theorem 1: [5] If ( ) ( )rrf C Yϕ∈ ∆ with r > 2, then ( ) ( ) ( )(1) , , , 1 , , ,rn k rr c k r Y E f Y f n k r n n ϕω  ≤ ≥ +    …(3.2) Where ,rC r Nϕ ∈ is the space of functions ( ) ( ), 0,1r rf f C∈ for which ( ) ( ) 1 2 lim ( ) 0rr x x f xϕ → = and [ ]0 0,1C Cϕ = . Now we prove the following theorem when r ≤ 2 by c(s) denote the different constants which are constants depend only on s, while N(Y) the constants which depend on Y. Theorem 2: [7] If 1 (1), , ( )pf L Yα ϕ∈ ∆ then ( ) ( )1 2,1, , 1 , ,n p p c E f Y f n n ϕ α α ω  ′≤     , n ≥ N(Y) …(3.3) An immediate consequence of this theorem is that: Corollary 1: [7] If 2 (1), , ( )pf L Yα ϕ∈ ∆ , then ( ) ( )1 1,22, , 1 , ,n p p c E f Y f n n ϕ α α ω  ′≤     , n ≥ N(Y) …(3.4) Mathematics - 336 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Remark: It should be noted that in the case r=1, k=2 and r=2, k=1 the estimates of form (3.3) and (3.4) in the case 1rC (1) ( )Y∆ . Corollary 2: [7] For each ,sy Y∈ there is a function ( )( )Yf 1∆∈ with ( ) 2, 1 , 3n pE f nnα ≤ ≥ ( ) ( )1 2,n c E f Y n ≤ , n ≥ N(Y) …(3.5) where ( ) ,n pE f α is the degree of unconstrained approximation. Now, let [ ]1,0=I and put .,...,1,0,11 nj n j Cos n xj =    ∏−= We denote by nS the set of continuous piecewise polynomials in [ ], 0,1pL α . We put [ ]00 ,0 xI = and [ ]nnn xxI ,1−= Let ( ) ( ) ( )[ ]1 1 1 1 1, 2 ,L x f x x x x x f′ ′= + − be the linear polynomials which interpolates f ′ at 1x and 12x . We set ( ) ( ) ( ) ( )1 1 1 1 11 , ;n n n n n nL x f x x x x x x f− − − − −′ ′= + − − −   Lemma 1: [4, lemma 2] If (1)1 , ( )p Yf L ϕ∈ ∆ then there is a continuous piecewise polynomials n nS S∈  such that ( ) ( ) 2,1 1 1 1, 0 , , ( ) ( ) , ϕω               −       ′≤ ′∏ ≥ ∈ ′ − = ∈    n p p n n n n n c fn n x S x x x x S f S x L x x I …(3.6) Lemma 2 If (1)1 , , ( )p Yf L α ϕ∈ ∆ then there is a continuous piecewise polynomials nS S n ∈ such that Mathematics - 337 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 2,1 ,, 1 1 ( , ( ) 0 , [ 1 ) , ] ( ) ( ) , ϕ αα ω       −       ′≤ ′∏ ≥ ∈ ′ − = ∈    n pp n n n n n c fn x S x x S f S n x x L x x I …(3.7) Proof of Lemma 2 0 1 11 1 , 0 ppp p x xx np f S f S e dxn n fe S e dx α αα α   − −−      − = −∫   = −∫       = 1 0 1 ( ) ( ) p p g x G dxn x         −∫  = ( ) ( ) p g x Gn x−  such that ( ) ( ) xg x f x e α−= and ( ) xn nG S ex α−=  where ( ) 1 ,( ) pg x L Yα∈ and ( )Gn x is the continuous piecewise polynomial in S n . Then by lemma 1 2,1, 1, p p p cf S g G gn n n n ϕ α ω      ′− = − ≤  2,1 , 1, p c fn n ϕ α ω      ′= Therefore 2,1, , 1, p p cf S fn n n ϕ α α ω      ′− ≤ . Proof of Theorem (2) We first take n sufficiently large so that f monotone in 1I and nI . Then in view of lemma 2 at most what we have to correct the behavior of nS  on 1I and nI while keeping it close to the original function. A spline polynomial nS  satisfying. 2,1 1 1 ,, ( )1 ( ) ( ) , [ (0) 0 (1) 0 , ] (0) (1) 1( , )ϕ αα ω  −          = ∈ ≥ ≥ ∏ ∏ ′ ′ ′− ≤      n n n n n n n n pLp I I n S x S x x x S S x S S c f n …(4.1) Indeed Since 1 2, n cI I n ≤ then Mathematics - 338 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 1 , ( ) 111 [ ( ( ) ( ) ) ]x p pn n n n I II ILp I I nnn S S S t S t e dt dxα α −′ ′ ′ ′− ≤ −∫∫    = 1 11 [ ( ) ]p p I II I nn g G dt dx−∫∫  …(4.2) Then by holder's inequality we have 1 1 2 ( )1, ( ) ( )1 1 1 p q p n n Lp I I nLp I I I In n S S g G dx nα   ′ ′  − ≤ −∫           1 1 2 ( )1 1 q p p Lp I I n g G n +  ≤ −     1 1 2 , ( )1 1 q p p n n Lp I I n S S n α + =   ′ ′−      2 1 , ( )1α ≤ ′ ′−   n nn Lp I I n S S From (4.1) and (4.2) we get 2,1 2,1 ,2, , 1( , ) 1( , ) n n pp p cS S n f nn c f n n ϕ ϕ αα α ω ω= ′− ≤ ′  Which combined with (3.6) and (4.3) implies. ( ) 2,1 2,1 2,1 (1) , , , , , ( ) , [ , ]1 2 1 1 , , , , , 1 1( , ) 0 ( , ) 1( , ) n n nn n p p p n n n n n p Lp L L Lp x x n p p p f Y pE f S f S S S f S S S S S c cf f n n n n c f n n ϕ ϕ ϕ α α α α α α α α α α ω ω ω = − ≤ − ≤ − + − ≤ − + − + − ′ ′+ + ′=       References 1. Rempulska, L. and Skorupka, (2004), On Strong Approximation of Function by Certain Operators, Math .J. Okayama, Univ., 46, 153-161. 2. Miller, H. and Orhan, C., (2001), On AlMost Convergent Subsequences and Statistically Convergent Subsequences, Acta Math. Hungar, 93, 135-151. 3. Ditizian, Z. and Totik, V., (1987), Moduli of Smoothness, Springer Series in computational mathematics, Springer Verlage, New york. Mathematics - 339 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 4. Leviatan D. and Shevchuk, I.A., (1998), Nearly Comonotone Approximation, J Approx. Theory, 53-81. 5. Leviatan,D. and Shevchuk, I.A., (1999), Some positive Results and Couterexamples in Comonotone Approximation II, J .Approx. Theorey, 195-206. 6. R.k. Beston and D. leviatan (1993): "on comonotone approximation " . Canada. Math .Bull , 26,220-224. 7. Husain, L.A., (2010), Unbounded Function Approximation in some ,pL α Spaces, M.Sc Thesis, Mustansirya University, Department of Mathematics, College of Science. Mathematics - 340 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 pL,التقريب الرتيب في الفضاء α صاحب كحيط جاسم ، إسراء زايد شمخي الجامعة المستنصرية –كلية العلوم –قسم الرياضيات 2012اذار 18قبل البحث في: 2011شباط 18استلم البحث في: الخالصة أفضل تقريب للدوال الغير مقيدة في فضاء الوزن الغرض من هذا البحث هو دراسة درجة , (1 ) , 0,α α≤ ≤ ∞ >p pL. فضاء الوزن، الدوال غير المقيدة، التقريب الرتيب. : الكلمات المفتاحية Received in: 18 February 2011 Accepted in: 18 March 2012