On Jackson's Theorem

Transcription

1 It. J. Cotm. Math. Scics, Vol. 7, 0, o. 4, O Jackso's Thom Ema Sami Bhaya Datmt o Mathmatics, Collg o Educatio Babylo Uivsity, Babil, Iaq mabhaya@yahoo.com Abstact W ov that o a uctio W [, ], 0 < < ad, i N( th st o atual umbs ), w hav / ω ( u) ( x) dx j ( x j ) c() ϕ, ω du j = 0 u wh < x < K < x < th oots o Lgd olyomial, ad ω m ϕ ( g, δ ), is th Ditzia-Totik mth modulus o smoothss o g i L. Kywods: Dict iquality, Jackso thom, Ditizia Totik modulus o smoothss. Itoductio Lt L, a,, such that [ 0 < < b th st o all uctios, which a masuabl o L / b :. a [ ] = ( x) dx < a, b

2 50 E. S. Bhaya W b th sac o uctios that ( ) L [ a, Ad lt [ a,, absolutly cotiuous i [ a,. ( δ ) ad ( ) is W bliv that o aoximatio i L, < th masu o smoothss ω ϕ, itoducd by Ditzia ad Totik [] is th aoiat tool. Rcall that wh Δ hϕ ϕ /, Δ hϕ x dx 0< h δ a b ( δ,[ a, ) = su ( )(, x, [ a, ) ω, ( x) ( x, [ a, ) : = k= 0,, 0 k h k ( ) x + kh, i x ± [ a, h o. w. ω Fo [, : = [, ] a o simlicity w wit ( δ ) : ω (, δ,[ a, ). ϕ, = ϕ = L [ ], a,b ad by Rcall that th at o bst th dg olyomial aoximatio is giv E ( ) : = i Π wh Π dot th st o all algbaic olyomials o dg ot xcdig. To ov ou thom w d th ollowig dict sult giv by: Thom..[] Fo, i N ad L [, ] ( ) c ϕ (, ) E ω () wh c is a costat ddig o ad (i <). Fo () was ovd by Ditzia ad Totik [] ad o 0<<, it has b ovd by DVo, Lviata ad Yu []. Now, cosid th Gaussia Quadatu ocss [3] ( x) dx ( x ) = I ( ) j= ω () j j :

3 O Jackso's thom 5 basd o th oots < x < K < x < o th th Lgd olyomial. Sic this xact olyomial o dg lss tha, w gt o th o ( ) = ( x) dx I ( ) i () by th diitio o th dg o bst aoximatio w hav wh ( ) E ( ) (3) (ot that ω 0 ad j j = j : = x su [, ] ( x) ω ). Th cud mthod o stimatig ( ) cosists o alyig Jackso stimat o th ight o (3) om () w gt th sha iquality ( ) c ϕ (, ) ω (4) which alady taks i to accout th ossibly lss smooth bhavio o at ±. Howv th sumum om i (5) is still too ough, ad th atual qustio is whth o smooth uctios o ca gt u bouds o ( ) usig ctai L, < quasi-om. R. A. DVo ad L. R. Scott [3] oud such stimats, thy ovd s ( s ( ) ( ) ) c s ( x)( x ) 5/ dx (5) ist o s= which obviously imlis wh ( ) E, L did by ( ) c E ( ) ϕ, (6) ϕ mas th bst wightd aoximatio with wight ( x) E ( ) ϕ : = i ϕ( ),. Π ϕ o i

4 5 E. S. Bhaya Thy th ocdd to stimat ( ) which ially yildd (5) o ay s. E,, usig high divativs o. Th mai sult I this sctio w itoduc ou mai sult. Usig (6) w obtai th ollowig thom W < < w hav Thom.. Fo [, ],0 / ω ( u) ( ) c( ) ϕ, 0 u du (7) O cous th covgc o th itgal o th ight imlis that is L quivalt o a locally absolutly cotiuous uctio. W us this quivalt stativ o i th quadatu omula ( Othwis, w do t hav v = o ) ( ) ( ) Poo. Lt Π b th bst aoximatig olyomial o i L [,], <. Th = + ( k+ k ) i [, ] L (i.. th xssio i th ight is th L quivalt o which w d ). Fom (6) ad Makov-Bsti ty iquality (s o xaml [4]) c E q ( ) ( ) q, c E ( ) q, ϕ c ϕ( ) q ϕ k + ( k k ) + c ϕ. q Th usig th act that ay two quasi oms a quivalt o th sac o algbaic olyomials o a ixd dg w hav I viw o () w gt k+ ( ) c( ) E ( ) < k k= 0.

5 O Jackso's thom 53 k k ( ) c( ) ϕ (, ) Now sic W [, ],0 < <, so that k k ( ) c( ) ω ϕ (, ) ω. c ( ) ( u) / ω ϕ, 0 u du. Povidd th last itgal covgc As a ial mak, w mtio that simila bouds holds o may oth systms o ods ad i (7) th ight had sid has th od x + x /, s o ay costuctd om aalytic uctios, x ± ad itatd logaithms o ths, which mas that (7) is th bst ossibl stimat o such uctios. Rcs [] Z. Ditzia ad V. Totik, Moduli o smoothss, Sig-Vlag, Bli, 987. [] R. A. DVo, D. Lviata ad X. M. Yu, Polyomial aoximatio L 0 < <, Cost.aox, 8(987), i ( ) [3] R. A. DVo, ad L. R. Scott, Eo bouds o Gaussia quadatu ad wightd L olyomial aoximatio. SIAM, J. Num. Aal, (987),

6 54 E. S. Bhaya [4] K. Kootu ad A. Shadi, O k-mooto aoximatio by kots slis. J. Math. Aalysis, 34(987), Rcivd: Novmb, 0