APPROXIMATION OF PERIODIC
SPL INES OF MIN IMAL DEFECT
A . A . L igun
FUNCTIONS BY
UDC 517.5
1. LetXbe anormedi inear space , and let xEX, FoX and Hc X*. We denote by E (F ,X ,x ) thebest
approx imat ion of the e lement x by the set F in the metr i c of the space X, i .e . ,
E (F, X , x) = in, ~ it x - - u [ix, (1)
uEF
and by E (F , H, X, x) the best approx imat ion with res t r i c t ion to H of the e lement x by the set F in the metr i c of
the space X, i .e . ,
inf i l x - -u l [x (F (x~=~) ,
E (F, H, X, x) = ,,~el*~ ' ' (2)
oo (F (x) = ~,
whe re
F (x) = F (14 x) = {u C F : V~, g (x) = g (u)}. (:3)
The fol lowing asser t ion is eas i ly obta ined f rom the dual i ty theorem for best approx imat ions (see, e .g . ,
[1, p. 281).
THEOREM 1. Let F be a convex set of the normed l inear space X, and let H be a subspaee of the space
X*. Then for any e lement x EX there is the equal i ty
E (Y,/4, X, x) = sup ff (x) - - sup f (u)). (4)
[EX* uEF
E(H,X*,f) ~< 1
In par t i cu la r , if F is a subspace of the space X, then
E(F, H, X, x) ---- sup f (x). (5)
I(:X*, Vu6Ff:U)=O
E(H,X*,f)gl
In a somewhat d i f ferent formulat ion the duality- re la t ions for p rob lems of best approx imat ion with res t r i c -
t ions were invest igated in [2, Chaps. 8 and 9]. We note that equa l i t ies (4) and (5) could be der ived also f rom
resu l ts p resented in [2]. In so doing it would be neeessary to repeat approx imate ly the same arguments as
those given below.
Proof of Theorem 1. We f i r s t note that
s up sup ff (x) - - g (x) - - sup (f (u) - - g (u))) = sup sup (~p (x) - - sup q) (u)) = sup ((p (x) - - sup q~ (u)) .
fEX*,I[fI[X,~I ,..EH u.EF gEH ~TgEX* uEF EX* uEF
[]~+gl/x* ~1 E(H,~*.cp)< 1
To prove Eq. (4) it thus suf f ices to es tab l i sh that
Z (F, H, X, x) = sup sup (f (x) - - g (x) - - sup (f (u) - - g (u))). (6)
fEX*.llfr[X . ~< 1 NEH uCF
if F (x) ---- ~ , then both s ides of Eq. (6) vanish at inf inity. Suppose that F (x) =/= 2~. By the convexity of
the set F the set F(x) is then a lso convex. Hence, the dual i ty theorem for best approx imat ions is app l icab le
[1, p. 28]. By v i r tue of th is theorem
E (F, H, X, x) = E (f (x), X, x) = sup (/(x) - - sup i (u)) - -
FEX*,IlfliX* <~I uEf(x)
--~ sup inf ( f (x ) - - f (u ) ) = sup sup inf (/(x) - - g (x ) - - ( f (u) - - g (u ) ) ) .
[EX*,IlfllX* ~ (t) dA (9)
�9 E(Sn,k__ 1 tl,r),g p,~l _JJ
g• Sn,r - -k
Since the set Sn , r _ k conta ins constants , i t fo l lows that g• const . There fore , there ex is ts an (r - k + 1) - th
per iod ic in tegra l f(t) of the funct ion g(t). Here we choose the add i t ive constant of in tegrat ion such that f(0) = 0.
r -k+ 1 Then the cond i t ion g E Lp , . g • Sn , r -k takes the fo rm f E Lp , ,n , and we can rewr i te (9) in the fo rm
II x (~ s~ (x)1i~ " (~ - - = sup ~ x (t) [~,-k+l) (t) dt = sup ~ x (r+l~ (t) f (t) dt =
te))~ 2a gEq.)~ ~ a
260
i i x v+l~ (t - - ~--- sup x (~+I) (t - - tl, r) q) (t) dt = sup t1,~) (q) (t) - - s~ ~ (% t)) dr,
{#E Lrp7 k+l,~,(tv,r) =0 (v=l,2,. . . ,2n) ~a E(Sn k--i @r--k4-1) P' ~< 1 '
(r--k~-l) -<1 E(Sn,k--t,q) p"
ff--k+l E (S~k--I (h.r), ff--k+I%, <~ 1 }. where ~ = {f : f~ ~,.~ ,
to the supremum on both sides of the last relat ion with respect to x EW~ +~ and recal l ing that by Pass ing
the duality theorem
def
sup S x (t) y (l) dt = inf II y - - )~ [Iv, = E, (y)q,, (10)
lixllq~l ~
x• - -~
we obtain the following assert ion.
THEOREM2. Letn , r= l , 2 . . . . . p, qE[1 ,oo]andk=l ,2 . . . . . r . Then
sup ]] x (k) - - s~]. (x) llp -~ sup E i (y - - s,~ (y))q..
q E(S~,k--I ,y(r--le-r'l))p,~
本文档为【周期样条函数】，请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑，
图片更改请在作品中右键图片并更换，文字修改请直接点击文字进行修改，也可以新增和删除文档中的内容。
该文档来自用户分享，如有侵权行为请发邮件ishare@vip.sina.com联系网站客服，我们会及时删除。
[版权声明] 本站所有资料为用户分享产生，若发现您的权利被侵害，请联系客服邮件isharekefu@iask.cn，我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权，请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传，仅限个人学习分享使用，禁止用于任何广告和商用目的。
浙公网安备 33021202002483