View of Best approximation by Chebyshevian splines and generalized Lipschitz spaces

B6 PAULINA PYCH

B ¡,4ATHEMATICA

_

REVUÐ D'ANALYSE NUIIIERIQUE

I4T DE THEORIE DE L'APPIì,OXIMATIOII

l"flNrtLYsrì

NUn'AÉR{QUE ET'

nrr T'ÍrÉoRr}ì FE

n,rLPPRoxrnfirtrnoN T'ome

5,

No 1e tr$?6, _nrp. gZ_Sõ

and for 12

|

( _2n2

l,o,er(r) l<{

_| ",,r"r

"'-t ^¡t+t

if 0( l<

^æ,

if

^f t

n.

Applying_ these inequalities

and

arguing as

in the

preceding tr,vo sections

s'e get the follos'ing

statements,

't'FrEoREl{

3.

Su.þþose that

f ^e I)*, r> 2.

^Tfu,en,

1s,,,,

(f ;x) _

_J.@),

. ;*å ^{*V,*,^i),}

nt

caclt

þoint x

^aJøerc lJ¿c cond.it'iott.

(2)

hold,s.

'r'IIlìoRElr '1. Giuctt, _{au'yt J^}

e _I)'t',

_{rsc ltot,a}

.[',-ïJ, ^if r:1, ,l/, " I _\

_{r¿ -i-lJn}

^{if r>r.}

BTiST APPROXIMATION BV CFIEBYST{EVIAN'

_SPLThTIìS

AND GENERAI,IZED I.IPSCIJITZ,SPACES

by

KARL SCIITìRER (Aachel)

RIi]IIlRENCES

first a direct

theorem

is

proved by JDRoMD

.l2l in

tlne

Cfa,bl

case to methods

_to in

_l-/1,

it is

sirorvä

tbat ;

moduli of continuity

_introduced ces.

fn thc third

section relations the ordinar)¡ _ones are establishecl.

yields

the

desired characterizãiion

1.

A

elilecú theoreila

(l) sp(Z,Â, *):{s - ^LZ(ø,b): LS(x):0, x u (x,_r,r,), t( i< ¡/,

r¡'here

A is a partition of the interval

_fa,

b),

i.e, ill Aljancic, ^S., Bojanic, R.,1'o,ric, \[.., O, ^{tltc degree} of co*uergence of tejér-

Lebesgul sums, L'Enseigfiernent tnathént,atique, tõ, 21-28 (1969).

[2] /{NBapüeliulsH¡u, ^A_, f., O cg.n.uupoaanuu cotrpnet{eHHotx pnôoc u ^pnôoe þgpoe

_.,[,an.cnga, ^Tpy¡sr Tóø,nnccr<oso Marev. I4uct., 22, 203-225 (19õfi).

l3l , I'lunrcepatt !,aumga u HeKomopbrc lonpocbr us aHarlLlaa, Tpy¡u Tó¡,'ruccxoro Mareu.

I4ttct., 26, 273-372 (1958).

[a] ^{S a k} s, S , ^'fheory of lhe integral, Nerv York, 1937.

iSl ^{1l a lr c r s k} i, Il., Sun¿mability of d,iJfercntiated inlcrþolaling þolynomials, Commentationes Mathernaticae !3, 197-213 (1970).

16l ^{V.¡r r rr u o} n ll. Jl., O rtpuíttuucenu.u q\gnx4uti, Cl6upcrorä Mareiv {ypxan, ó, 4lB- 437 (1e64).

ll7:, Z:'g rn 11 1r d A., Tri.gonometric scries, II, Carnbridge 1959, lÌeccivcd.31 I l97l

BB KÁ,RL SCHERER 2 3 ^BEST APPROXIMATION

89 (2)

L: ø:

^%o

_<

x,r

<.

^.

.{

xu

:

b,

4: ^min (*r-*,-r),

Â

:

max (x,

- ^x,-r),

L is a differential

operator

of the form

⁽⁰

<

^m

<

^n,

-

⁷⁾

(3) L:D,_t...

t)o,

rvh^ere

I¡,Í@) :d _ldxlf (x)lrøt(x)l witt strictly positive functions

_{u¿(x) =}

eC"-í

_[ø,

á]. Then there

exists

a

basis

of the null

space of

Z

of the fórm

uo@)

:

ryto¡*¡

of functions {"f*} in c"fa,bl which

converge

in Li(ø,b) to an

element

f ^{e Li@,b),} it is

seen

that (7) holds more

generally

for functions

in

Li

_@,

ó), 1< þ <

^æ.

The linear approximation method of

JERoMD

l2l ^(defined

^more

generally

for

nonsingular

lineai differeniiãl òperator u'ith suitable const

describèd as follows :

For _{_each}

( e points lr({),

. .

., t,(Z) be

consecutive points

of the

mesh

A< lrr- ^{El< 2A,;} lt, - ^{ql < þ,-}

^El

^{< ...}

and let

_{90, . .}

.,

_Ên

b" any (non-trivial)

solution

of the

system

\oui

^(to)

+

^.

^.. I þ,ui

^(t")

:

0

(4)

ur(x)

:

uo@) ur('cr)d\t

a

E E,

u, -r(x)

:

uo@)

a

91

u, t(€"-t)d\"-r...d4,

_þoui(to)

+ ... *

^g,,ui,þ,)

:0.

Setting

lo(E)

: _{, the

approximating

spline

S(ø)

:

S(f

;x) is

defined by The systerr

(uo,...,uu-r)

forms an extended complete Chebyshev (E.C.T)

system on la,bl ^(rcanr,rx

14,

p.2761), in particular the

Wronskian

W(uo,ur, . .

.,'u,-t) is strictly positive on _la,b],

i."., ^{(B) S(ø)}

: u(x) -f ^¡(x,l)Lf(l)

^d,(

uo@)

u,-

_t(*)

with

u:(x)

being as in (7)

and

y(x, €): ^ô(r,

^E)

-

^(1/p.)

(5)

> 0 ^{(x e} _lø,bl)

n

D"uo(x)

... D'u,-lx) I

_j:0

^p,6p,4¡

Denoting

by ui the

elements of

the

last column

of

W-|(uo,wr,

one sets ,1'(r-t

Obviously S(r)

belongs

to Sp(Z,A,z) by

^(6).

_:nnoup [2] ^has

^shown

that É ^{g¡îp,t,¡ has compact support} in _lto,t*l and that

_þ¡lþo< C

6@, e)

: ^0(*,4), 0, x<(, ^{x 2}

^E

forl(r<ø

ded.

By

theindependent of

the

choice

of

_{1,} provided

A/A

remains boun-

Taylor

expansion (see (6))

òle

has furtheimore o(x,y)

: (* - ^y)1,'l@ - ^\)! f

^0((ø

-

^y)"+),

rvhere 0(x,

1):D

^{uo(x)ui (E).}

^{Then by}

^definition

h:o

(6)

o',6@,

{)

¡:B

l-: ò7,r-r

^{(0 < _r}

^{< n} -

^t)

so that lor f ^e C"lø,bl the

generalized

Taylor

formula

so that one may

estimate

(e) tf @)

- ^s(/;ø)t

^<

x@,4)tLÍ(E)tdt

(7)

l)

l@) :

u(x)

| \6ø, zlr¡t|)aå)d.4

)

rvhere X@,1)

) ⁰

^{has support}

tnlx - ll<

^@

^+2)L

and satisfies _Z(ø,

{) (

( CoA,-r, the

constant

* _being

independent

of x

^and,

_f.

From this one easily

concludes

(with

constants Cr,Cr) holds, *1"1"_O(6)

: up(Q ur(E) ... a,_l|) and.u(x)

satisfies

La(l :

- 0 on fa,bl

and D¡u(a)

: _{Dif(a), 0<} /( ⁿ - L ^{ny taking} r

s"quénce

r

The coustants will be deuoted successively by Co, C,,

90 KARL scHERER

4

ll/ - ^{s(/)ll- <} c,Ã"¡¡ry¡¡, ^(.f ₌

L,L(q, b)),

lr - ^{s(/)ll, <} c,ñ'lvlll, (f ₌

L'i(a, b)).

By argunents of the theory of interpolatio' of

Banach spaces these inequalities

imply

(10) _lll -

^s(/)llp

^< cÄ'ltfllp,

(.f

=

^Lï(a,

ð),

¹ <

1<

^.o).

rndeed, by defi'itiou of ^peetre's K-functional o'c

has for

-f =

^{L'l(ø, b)}

K(t,f-S(f); L,(n,b), L,,

^(ø, ö))

< ^inf IlS(,f_s) _(í_s)ll,I¿lls(s)_sll*)

_<

t ^{u ¡,,!,,} k, t)

( rnax

(Cb

C2)L inÏ

_UIL(f

^_

g)ll, ^_l_ ¿ll¿Sll_) <

e. ^{r'L@, u)}

( max

(C1, Cr)L"I{(t,

Lf ;

^Lr(a, b¡,L"(a, b),\

Ø

But

since

I [l-r

ltt,K(t,-f ;Lr,

L.))d,tlt is a norln

equii,,alent

to _ll/lle

(see

tf, ^{p. fSSi), there}

fotlor,r,s (10).

rlrom this

one obtains

in a stairdard

lnanner

for the

best approxi-

mation E*""'(f ;2)

:.=.,,îllt^

,,,,11.f

-

^sll,,

^the

Jackson-t1,pe iuec¡rality E*'"')

(Í

_{; :þ)}

<

^c 4K(t,

f

^;

^L¡(a,

b), r|i,(r, b)).

IJsi'g an esli'rate in [3]

therefore establishes

the direct

theorcrrr THrior{D}r

l.

Und,er tJøe cond,,it,ion that tloe

ratio

^L,f

L,

rcruains l¡ouncl.cd.

th,ere holds

fctr

^cach

f = L¡(a,b), 1< < ^æ

^(or

/ ^{¿ C,i¿øj} iÍ þ: *)

(12) E*'"')(.f

^.,1) _< c,[Ã"ll _{.fltp 1_}

o,,(/;

À),,].

Here

co,(/;A),a

is the n th

rnodulus

of continuity

defincd bv

i

,

^{Following KARLTN.}

^{þ,- p.}

^S2Bl

^the

generalized

di'ided

differenc

e of f at x,,

. .

.,

x¡.1,

is

defined

by

5 BEST APPROXIMATION

2. An

inverse theorem

uoØi\

91

uu-t

G)

f(r¡\

(14) .f

(xt,

. .

.,

x¿+x)

:

^{uo(x¡+n) ... un_t (x¿+u\}Í(*¡+")

uo@¡) ttnþ¡)

uo@¡+À

..,

uu?¡1r) where _u,,(x)

:

zao@)

[;

^wr(4,,)

^.,. !2,-,

u,,(2,,)d,8,,

...

^dE,

The generalized divided difference has the property

(15) _Lf : o+f(r¡,

. .

.,

xi+^)

:0.

fn

generalization

of

(13) one defines (see [6])

with w,,(x):

¹

" lF,q{ f. l8'}f¡x'¡1t¡tø' 1(1<or

0 < Ul l < t. _{,,x +ilh} e (o,b)

I

(16) ar

_U.,

_f)p :

sqp

^ess.

sup l\'ìf(x)l,

0< _Jirf <1 x, x ^t, tùe la, bf

Here

L'i,f

(x) is the n

^Lh (foru,ard) difference of

f (x) with

increment /¿

" ^.sllp

^{ess..snp |}

8;/(z)1, þ:

^æ

u <1,/rl <l .t,t¡nhe(a,þ) r,r'ith

(17)

ài,f @)

:

h"

f

^{(x, t(}

+ ^h, ..., x f

^nto).

Since/(1,, ..., x¡r,) is.liue-ar.inl, thc

generalized. mod.ulus

of continuitv

r"or(l;

f)¡ ^is

srrblinear irr

/,

^i.c.,-

(lB)

_a,, (t _,

f, I _"f)p (

o¿ (l ;.fr)p

I ^o,r(t;

_"fr)p.

For the

follou'ing

the

relation

(re) sif";:-\

_{B(r, h)ã}

f ^{,, oi-'J@}

t

| ^rh)ß,

^{(x, h)}

together u'ith

(see

nrcHanns_lnvonn

_16])

(20) llB,(*,t)lB(x,l)ll,<C? (0<,{n;0<l<t), (21)

_llt3o@,

_{t)lB(x, t)ll.}

_>

CB

(0

< /<

t),

is

essential.

A

conseqrlence

of (lg),

(20) is

(22)

otr(t _;

f)o <

^Cnll.fli¡.

SU 0< l¿

p

¡<r l^iil@)tþ

,1(2<.o

'fi:Ø

(13) a^(f ;t)e:

^{[a, ¿¡ ]}

92 KARL SCHERER

'¿,N

I

*i, ì¡-rh

o

I

and

BEST APPROXIMATÎON

with the

help

of the

properties (1s),

(lB) and (22)

the foilowing converse theorem can be proved.

rHnonn¡r 2. Let

Eftra(.f _;

þ)

^{d,enote the best} øþþroxi uhere A*_

is the

equ,idistønt

þørtition of

Lo,

bl

into

(U

- ^{o)l\._ Then}

^oitc,

has-for'.f 7

Lo@,"t1,

\'< _e .**

þ : _æ, N

being odd. and."

N )

^S,

(23) _., (N-'

_{; "f)þ}

<

^{Cn[Etr,ot (,f} ;

p) + ^El!,]\(f

; þ)1.

Proof. _Since

it i _¡at

_{one of}

_the

corresponding theo_

rem in

, [7^]

for polyn _{a brief ,k"¿h} ir"ei""".

r,et s

denote

an gp-r"-oL;i;" "isiø,-A",

^{0) to}

f

^e

=

^Lp(o,

^b). Bv (18) i<

^r

rotlt; "f)p

< C,rlll -

^Sll¡

f

^{ar, (ú; S)e.}

By

(.15) and Qq) ^one

may

estimate

further

(0

<

^I

( Crrry-r with

suitable small constant Crr,

the

points

r¡,"

being

thJ kn;is-riäil

¡

lN-l or¿(l

i

^S)p

* _,tåt,l

Ð

< c,,{rr - ^{sl +;:p=,}

_{

But

the last term can be estimated

fV n(i,i\

(.f ;þ)

io

exactly the same manner as

in _[7]

using

property

(15).

3.

Connections between

moduli of

continuity The

following

theorem

is

proved:

THE9REM

3.

Tkere

exist

constøncs

Crt,

C,.! ønd.

ø ò,

0

< ^g< l,

suclt,

thatfor euervf = L¡(o,b)

ønd' 0

<r<

^s

ä,trii"ià-ïnl¿iitqiør,ities*

(24) atþ;.fp) 4 crnlt"

_llflb

+ ^a,

^(t;

_"f)pl,

(25) _{a,lt;.f)p <}

Crult"llfllp

+ ^6Llt;f)þ1.

. Pr3of.,lB1' relation (19) and the

estimates

(20), (21) there

folrorvs irnmediately

(26)

^, ^(t;.f)p " ,,f," _{lfll¡ +} .,(r;.f)p* b,

^{t, <ù,_,} 1t,

t)r]

t rn genetal the ter¡n

.!"llfllp ^cannot be dropped si¡ce ilris s,ould imply that the nul- space of Ð'is coutainea in thai"ãf Z,, or ãon".i".ry,

(27) a,

(t;

Í)¡ ( C;,

max

(t, ,)þ" _{IÍlb +} ar

(t;

f)e i'f ^{u^,,_,p,} ¡¡nf.

rt

remains

to

estimate

the

sum _{_in_.(26)}

or

(27)

in a

suitabre manner. By Marchaud's

inequality

(see

e.g. i3l)'

^oáe

has

^'

n-l â 1 -- I

(28) 'f _ur^-,

(t _;

.f)p( ^c,, 2 ^{*[WW -f} _iu--*,-,

^{øn (u,}

^{f)o d"l*}

t

" c,,[r" _Vll¡ -f rlu-*^,,'(u;f)ed.uf.

splitting up the

integration

from"r to ^g

and

from

^g

to l, and

observing

tt

"øn(tt _; ,f)¡"

<

^2" t2'"un (tr;

,f)p

^'(O

< ^tr(

^/.

( l),

one obtains

(29)

r"

rôr

f u-"a,(u;f)p du- l"Il u-,o-n(uifled.u*\-u-,-,.^(Í;u)odu]<

, '

{

^2"j fco,(r ;

¡¡, ^(;

'

- ^,,

^a-

^t,

^llfllei

^n(s-, I

,,0

* (

2'[co,

(t;f)p8 | nt,B-"llfllel.

Now

one takes

I :?..".-'.LCrÇ,t ma* (-1.,cr)l-'. _Then

_{(26) and the esti-}

mates (28), (29).estabtish tLe

iirit ir,"qoùitv

^'Q,+¡

"i ^tni

tìreorem. Further_

more,

inserting

(28), (29)

into

122)1viåds

*itì, lîir cttã""'or ¡

a,i(t;

f)p

" i.,(,

^;

^{Í)p -f C,,lt,}

^llfllþ

⁺

^az

^(t;.f)¡û,

whence

(25)

immediately _follows,

4.

Characterizationlof generalized

Lipschitz

spaces

The

generalized-r'ipschitz _spaces

in

question

are

defined

by

93

rlþ

l}s1, ¡x¡¡t

¿,

N-I ti,N rlþ

D

_{t:l t¡,¡¡-th}

!lsiftu)¡dx

Lip

^(<Þ, q, n I

þ) : e

Le@,b)

fO(r)-ro,, ^(t;

f)¡l ^ült|'to

^,

.?Tå ot,l-,^, _\t;.f)p,

{J l(l<oo

Ç:

^æ

94 ^{KARL SCHEREIT}

I I

^BEST ÀPPROXIMATION

95

for I

<1ó

ç.co

⁽ⁱⁿ ga.se _¿

:

co one assumes

-f = ^Cla, bl

instead of

L*

_@,b)),

where rÞ(i).is a positive non-decreasing function on (0,1] r,vith lim¿*o,O(l) ^1-

: 0.

Combining

the

previous results one arrives

at the

follor,ving châiac- terizaLion theorem

for

these

fipschitz

spaces

(in

case O

(l) : t""they

are

the familiar

Besov spaces

on the interval la, bl) in terms of the

best

approximatiç1

[ft,'n)

(f

^', _1>) ^:

tr{EoREM

4.

Let

Q

be as øboae alxd,

satisfyj*,r,-' t,-t dt { ^co.

^Tken

u'n'd'er tlre ^qboae notøt'ions thefollowing a.sscrtion,

)r,

equiaalentfor r

{

^q

(

oo:

1)

.f = I,ip

^{(O, q,}

n;

_þ),

I

li¡ {liop¡-lo¿ ^(r;f)t)o

^d.t¡t¡,ra

^ç.

^ee,

REFERENCES

!l

^{B u}

^tz,er

^p.

Ne¡v

^{of Oþcrøtors} and A.þþroxitttation, _Springer 12) J ^e r o m e, J.

-J. A

on.

_{, ui_ts+.}

.by -c_crtain gctrcralizetl sþli+te f.unct.ions.

[3'] Joo""ä#à,,289_g0a. _{ttle moduti} of s*toothness. r\{at. Vesnik g (pzr)

[a] ^{K a r I i} n, S., Totøt Positia_i.ty, yoi- I, Stanforcl Univ. ^press, Sta¡ford 1968.

[5] DeVore, tion. In ,,Splile Fulctio¡s Sy¡rposium, scr, ßasel R., Ricrr.ar<1 1973.Bclmontol í, r., 1gj2 soí"í"j¿oi'""¿,inuer.se aucl (a. Meiir: Aoproxi¡r.ti9" iir.ãiy,,l _ ¡. Sharnrá. áár.j _rheoreuts for isNlr proceedirrgs ^sþtine 21, aþþroxittta-Birt¡hüu-of the

[6] DeVore, R., Richards, (to appear). I¡., The rtegree of aþþroxim,atio, by Chebysheu,ian _s.fùiøes

[7] sc1rerer, K, sþtines, clt'araaterizali'on ,to _{appear oJ} generali.zed, Liþsch.itz classes lty best øþþroxirnatiott _taith

in srÀu J. ^{Numei. Anal.} li iínil.' ^'

Lehysttthl A _J.ür Møthemat.íh, Technische Hoahscltule Aacheti Received 2. IL tg?4.

ttt)

{Ë [o(N-r¡-r¿ *,ù ^(f

^;O¡1,

u,,-']"0. -

f n cøse g

:

æ end tn

:

0(O(l)) tlce foltowing øre equiualettt :

i)'f ₌ r,ip (o, æ,n,þ), ii)'

<o. (t;

"f)r:

^0(O(l)),

iii)'

Eø,.r

(f

;

þ) :0(1¡(N-1)).

If 7im¡*¡1inf O(Z¡-t¿':

ær. qsse.t.,tion

i)'

,imþlies that

f ^{is a}

þolynom,iøl of tlegree

n - l,

^wheleqs asserrtion

ä)' or äi)'

irnþty that

îJ:0,- I <

^y'

< *.'

Note that conditionf *U)-tr-t

d.t

< æ implies a smalle¡

decrease oT <Þ(l) 1.or

t *0 f

^than

^{åndition /'} :

0(o(l)), since from the boundedness

of ll-". integral it

^follows_

that t,:

O(O(t)). On

the other

^han<1, @(t)-1t

: : 0(l)

means

a

smaller decrease

than lirf,*¡* inf

O(l)-ttn

: Ø nìúi"t i,

g a pol¡.nomial oi

degree

n-l

or assertion

i)' - ^iii)'

^vanish identicalll- g theorerns yield the follorving chairr

.¿(N-' _{;,/)¡ <}

c,o¡.Ðç,',t

(f

;

Ð + ^E\ï,it t¡

^;

p)l

<

(

Czo [N-',||,/llp J-

.,,(N-. ;,f)¡]

<

( C,'[]/-'llflb + ar(N-' _f)pl,

from rvhich

the

equivalences stated above immediately follow. The satura-

tion

case is handled by the same arguments as

ir

nrinenns_DEvoRE _t6].