I v r e s n a d di t r th o e r m e s o f r monoton e app o r i x m t a ion b a t 1. I r n o n a

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1 Iverse ad direct orems or mootoe aroxima tio ukeia Abdulla Al- B ermai B abylo Uiversity ciece College For Wome C omuter ciece Deartmet A bstract We rove that i s icreasig uctio o i or e ac h re a icreasig algebraic olyomial degree 8 uch that s c Where i s secod order Ditizia Totik modulus smooss w ere oitwe olyomial Also obtaied a estimates coverse These aroximatio results orem or same th comlemet tye or direct orem classical ucostraied I troductio a d Mai Results everal results a roximatio show that ucostraied aroximatio i some sese mootoe by algebraic olyomials erorms as well as F or examle oretz ad Zeller [7] have show that or each icreasig uctio i C ( I 7

2 sace all cotiuous uctios o I ) re a icreasig olyomial degree t hat saties c ( ) modulus cotiuity A geeral result or ( ) or ay k 0 re are icreasig t hat saties c k k () th result oretz [ 6 ] as geeral case was roved by DeVore [ 3 ] cases k 0 are much easier to rove tha geeral cases ice y ca be method i cotrast ro i [ 3 ] uses liear techiques It well kow that roved rar or usig ivolved liear o ucostraied a roximatio much imrovemet ca be made i estimates orm ( ) x ear ed oits I th mootoe t hes we are aroximatio iterested oly i I oitwe result th estimates tye k ow by Beatso [ ] He roved that t he estimate that or we x x x x c x x I holds or suitable icreasig olyomials wheever i s icreasig Devore ad Y have show that i 8

3 i s icreasig uctio o I or each re a icreasig olyomial degree s uch that x x x c i s secod order moduli smooss Amog or thigs we shall show that th ca be imroved to allow s ecod order 0 modulus smooss or saces T heorem I: I i s a icreasig uctio i I 0 t he or each t here a icreasig I o degree (8) s atyig olyomial i c( ) ( 3) Usig th orem we ca obtai our secod Iverse i equality: T heorem II: et b e a i creasig uctio i I 0 ce c m E m 9

4 A uxiliary emmas otios B eore we rove our orems we eed ollowig ad lemmas Our ro based o a two stage aroximatio We irst aroximate by a icreasig iecewe liear uctio We aroximate by a i creasig algebraic olyomial iecewe l iear uctio that iterolates at k k i we let s be s los s ( ) The ca x max x 0 be a s rereseted by usig uctio [4] () x s x s s x It clear that icreasig i i s also We that shall ow cotract aroximate a olyomial uctio x R as i [ 4 ] The costructio R begis with trigoometric olyomial T with 30

5 t 0 et K d eote Jackso kerel 8 t si K t a ( 3) t si a i s a costat deedig o c hose s o that t K dt Here ad throughout c c deote absolute costats deedig o ad o ccurrece D eie c c o same lie T ' s values may vary with each t t t K udu 0 [4] t t a d deie d t max dt t t t [4] ( 4) N ow let Ad or x d eie R r x T t x cost m x x r udu I articular R x x x ad R x 0 deied by R equati os I I w e deie [4] ( 5) oits are 3

6 sr s s R ( 6) with s deied by ( ) i icreasig s 0 a d sice we ca also write s R R Now rom deiitio olyomials T we have T T 0 icreasig W e ow estimate hece r r 0 ad reore R R it ollows that i s icreasig E x x x s s x R x ( 7) Now or x cost with 0 t we have 5 x R x c si t t t d t [ 0] ( 8) emma 9 c P ro: W e have x x s s x R x T he x x s s x R x x s s x R x Deiitio i mlies 3

7 x x s s x s T hus x c A d x c x s s R The ( ) imlies s T he c x c x x c x c x R ad by o c we have c 5 x c x c x t t d t c x c x 4 The by o llowig [4] emma 0 I g absolutely cotiuous ad g M a lmost every o I The or each ad each x I w e have x g x g x cm 33

8 W e ca rove emma I g absolutely cotiuous ad g M a lmost every o I or each x I w e have c g g ad each 3 ro t heorem I Firstly let us itroduce so called Ditzia Totik uctioal d eiitio as ~ K or I 0 i g g g W e have ~ K [] Give x I rom result above re a g s aties g c a d g c ( 3) g g g g T he by liearity ad bouded ess we obtai g g g g 34

9 emma ( ) imlies g g g c g g Usig ( 3) emma( ) ad liearity c c c c g we have B y virtue emma ( 9) w e have c ice c i s a icreasig olyomial degree 8 we h ave roved orem 4 ro F or ive by T heorem II i g m axi : x x x x we exad x b y x x x x 0 0 We recall that or m m c Em c c c E c E c i i c i a algebraic olyomial best mootoe aroximatio degree ot greater tha or equal i t mea 35

10 i E i (4) T he E c c i i i E c c T he by Berstei iequality we i i have i ce c A lyig iequality i i E i iv c v v m v N i m [5] W e g et c E c m Em m R eereces [] R Beatso( 008): o erators(rerit) [] E Bhaya ( 003): aroximatio Ph D O Thes Educatio b Al- H aitham hae costraied Baghdad reservig ad uiversity covolutio ucostraied College [3] R A DeVore ( 977) : Mootoe aroximatio by oly- o mials IAMJ Math Aal 8 ; [4] R A DeVore X M Yu ( 985) : P oitwe e stimates or mootoe olyomial aroximatio Costr Arox ;

11 [5] D Dryaov ( 989) : Oe sided aroximatio by trigoom- etric olyomials i - orm 0 Baach Cetre Pu- blicatios ; 99-0 [6] G oretz( 97): Mootoe aroximatio New York: Academic ress 0-5 [7] G oretz Z Zeller ( 968) : Degree aroximatio by mootoe olyomials II J Arox Theory ;