Construction of Continuous Functions with Prescribed Local Regularity

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Construction of Continuous Functions with Prescribed Local Regularity

Khalid Daoudi, Jacques Lévy Véhel, Yves Meyer

To cite this version:

Khalid Daoudi, Jacques Lévy Véhel, Yves Meyer. Construction of Continuous Functions with Pre-

scribed Local Regularity. [Research Report] RR-2763, INRIA. 1995. �inria-00073928�

ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Construction of Continuous Functions with Prescribed Local Regularity

Khalid DAOUDI, Jacques LE ´ VY VE´HEL and Yves MEYER

N˚ 2763

December 95

PROGRAMME 5

Unite´ de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)

Te´le´phone : (33 1) 39 63 55 11 – Te´le´copie : (33 1) 39 63 53 30

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¯ m

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Zãkš…2gjx…eŽ"ddH|~k‚€;gjbŒfkµk2x"dz"fŽæ|~dmtrŽKg#tçtrŽ"kŠ…—dxK|´thgjxth| m Œ y m y m gjxKy m gbŒk*€k|Œ|cteŽKgjx

Œ

i ,

Pékvx]…Ek~ATkk2xKy z"„)A*steŽ¤

“

_` ” ‘ bt–&_` b “ z ž

 w “‘ “ Š1 |

´xtrŽ"k?…2g|~k~A*Ž"k2bek o s;|µg‚…EdxHtrsx"dzK|d>z"xK…htrsdx[ATkŽKg ?kétrŽ"k¥>d€€d@Ašsx"fObek|´z"€ut5¤

£ ¡ )¢ )' ! 2 o " ~ "9" " " ;

wĨ

ž=<_ ;p ® k

<:ƒ<

wĨ

ž;)! 2 j %

o `qa*b x H

`^ab " a ¼ ;wƒ® ž=<À|

!!

• ($

j ( 2! %!

"€ #"

z w ! 2;l (

“o ` bt– o ` b “ ¯ “ – “E Š " `Vy

b¼ ;wD® ž=< )\;wD® ž=<À|

V"O

"

"

_t`qa*bdc

¶ = ?

©l¶

Š/ 

|~sx

` ¶ ab

K! 2;j %

 – y{si

/

6F p ‘ _Uc H `^abdc o `^ab |

abŒd{d-> ¤]|~k2kgj„"„‹k2xKy{sCF,

’j, ¬350T12s657%189350 6T8 0.1O2j1O4D6M057%189350 TT1X.

oµŽ$kÒteŽ"s9bry …—dxK|´tebezK…htes9dx d->…—dxHtrsxz"dzK|¬>z"xK…htrsdx\AšsuteŽ „"bek|Œ…EbŒs˜‹ky PWd€;y{kvb>z"xK…—tesdx

s;|Š˜Kg|´kyz"„‹dxQgfk2x"k2bhgj€sv}g#tes9dx}d->trŽ"kwx"dtesdxåd->D¬ˆ : ,oµŽ"s;|Š…EdxK|¬trbŒzK…—tesdx嘋kgjbh|O|´diwk

gjxKg€df-G)A*sutrŽåteŽ"ku]bh|¬tmdx"k6˜"z$tmŽ"k2bekATkæy{sbek…htr€CGiægjx"s„"z"€;g#trk?teŽ"k慗dxHtebhg…htrsdxåbrgjtesd|

d->mg ‚x"k >z"xK…htes9dxK|?sxK|´tekgy d]>¥ATdbXEsx"f[dx trŽ"k…Edk …—sk2xHt—|wd->teŽ"k}k=F{„KgxK|´sdx=sx teŽ"k

:

…eŽKgzKy{k2bé˜Kg|~s;|V,oqd˜‹k2fsxYA*sutrŽsAµkbŒk–…2gj€€c|~di kw˜Kg|~s;…¬>g…—th|gj˜‹dz$t´ˆ

: , @dbŒkwy{kEt—gjs€†|

…2gxҘ‹k~>dz"xKyæsx\;ž–“H‹‘Hž’ž#‘Hž#l‹%?@<+gjxKy dteŽ"k2bh|V,

!#"!

¥j¤ 22 ¡ !

5k2t ˜KkØg*…—di „Kg…htiwk2tebes;…|´„Kg…—kdA*Ž"dH|~ky{s;|´thgxK…—kçs;|y{k2x"dtekyé˜[G( `qa y4- b

>db a y4- ¼ \, 5kEt ˜‹kAteŽ"k‚|´kEtOd->Tgj€€+x"dxåk2i „{t G[…E€dH|~ky|~z"˜K|´kEt—|éd->£,qoTŽ"k2x s;|g

…—diw„Kg…ht&iwk2tebes;…O|´„Kg…—k¥AšsuteŽÒteŽ"k~PgjzK|ey{dbXB}i kEtrbŒs;…„;Ӟ–“ <[¤

‘ ` y !

bc iæg#F

6

|~z"„

/

= sxl>

3 = ( `qa y4- b yš|´z"„

/ = sxl>

3

=

( `qa y4- b 7

A*Ž"s†…eŽs;|&y{kžu]x"ky„A*Ž"kvx"kž?k2b gjxKy

!

gjbekm|´z"˜K|´k2th|&d->\,

5kEt,

g

¤ ’ >db ¼ 6

ž-yhyT|ž|ž|y.-Q7w˜Kk - …—dxHtrsxz"dzK|½>z"xK…—tesdx]|=,KoTŽ"k2x 6 y4,

g ¤

c ž]yhyž|ž|T|*y.-Q7Šs;|؅2g€€ky‚gxsutek2bhg#trky>z$xK…—tesdx‚| G"|¬trk2i ` ´ˆ

: b ,”Šk=u]x"kA ¤ ’ ˜[G

A ` bc

?

'

g#h

Œ , g ` b >db ¼ )|

¡

x[GÈ|~kEt ¼ |~zK…eŽÈtrŽ]gjt

A ` b'c

¨Pb%cedfeg

s;|Š…2gj€€k–ygjxÄg#tŒtrbrg…htedbšd->teŽ"k/´ˆ : 6 y4,

g ¤ c ž-yhyž|T|ž|*y -7±,

¡

x³´ˆ : gj€CA*gTG]|&gy{i sut—|

g#t€k–g|¬tdx"kg#tetebhg…htrdbT,M´xKy{k2ky.c|¬t—gjbŒt/A*sutrŽ_gx[G ¼ QqtrŽ"k2xãtrŽ"kȅ—€dH|~z"bŒk d->µteŽ"kÒ|´k2t

d->g€€6g…2…—z"iAz"€;gjtesdx‚„KdsxHt—|d->

6

A

` b 7 e

h Œ

A*steŽ;A

`

bc A ` A

Š © Œº

`

bb "s;|µgjx

g#tetebhg…htrdbTd->qtrŽ"k¬´ˆ : ,

€>~>db*|~di k o ¼ ;w yvž;{gjxKyg€€ ¼ 6

ž-yT|ž|ž|y.-Q7

( `, g

`qa*b y4,

g ` - bXb ¯ o ( `^a y4- b y " `^a y4- b½¼ )

trŽ"k2x}trŽ$k ´ˆ : s;|µtrk2bŒi kyŽ[G{„‹k2be˜Kd€s;…#,j´x}trŽ"s;|&…vg|´k A s;|*g慗dxHtebhg…htrsdxÈi‚g„"„"sx"f"{Ž"k2xK…—k

sutšgyi sut—|µgz$x"s†ƒz"kuOF „‹ds9xHt½AšŽ"s;…eŽs;|TteŽ"kz"x"s;ƒz"kg#tetebhg…htrdbµd->trŽ"k¬´ˆ

: ,

ZYŽ"k2xQtrŽ"kg#tetebhg…htrdb d->*gjx\´ˆ : s;|Oz$x"s†ƒz"k6sutAi‚g Gʋkæd˜{t—gjsx"k–y[g|¬>d€€d#Aé|;ž2œ¢<]¤Ï€kEt

„c ` Œ

yž|ž|T|*y ?b ˜‹kg?„"bed˜Kgj˜"s€st€G/?k…htrdb½A*sutrŽækg…eŽ g z w

gxKy g g c ž-, : thgjbŒtD>bŒdi

trŽ"kNulFㄋdsxHt a i d-> , Œ

gjxKy=y{k=u]x"kgå|´k–ƒz"k2xK…—k `qa

b ˜[G=…eŽ"ddH|~sx"få|~zK…2…—k|e|´sv?k2€CG a ¼ 6 , Œ

`^a

© Œ b yž|ž|ž|5y., ?`^a

© Œ b 7>©db) ¼ 6

ž]yhyr“lyž|ž|ž| 7AšŽ"k2bek‚„"bŒd˜Kg˜"s€sut G g

s;|Og#tethg…eŽ"ky

trdteŽ"k*kž?k2xHt a c , g

`^a

© Œ b ,HoTŽ"k2xjteŽ"k*dbŒ˜"st 6 a 7

=D?

s†|çy{k2xK|´k*sx ,HoµŽ$k g L|g€€d#A

trdOfk2x"k2bhg#trk*gŠz"x$s†ƒz"k؄"bed˜Kgj˜"s€9sut€GOi kg|~z"bŒkÒdx AšŽ"s;…eŽs†|ç|¬t—g#trsdxKgjbXG>db+teŽ"kšy{s;|e…—bekEtek

trsi k@ågjbEd#?‚„"bed$…—k|e|&y{k=u]x"kyÈg|D>d€€d#Aé|j¤

oTŽ"k„"bed˜Kgj˜"s€st€Gwd]>tebhgjxK| >k2b&d-> a³¼ tedæg aµdbŒk2€q|~z$˜K|~kEt

!

d-> s;|%¤

5`qa y ! bc

f g g

A

Š/  `! b

A*Ž"kvbŒk¤

A 3 ` !

bc R ž sC>G- ¼ !

w

sC>G-£°¼

!

ZãkO|~ŽKgj€€‹x"dt*y{kž?k2€d„"„‚Ž"k2bŒkŠtrŽ"s;|µg|´„‹k…—tTd->´ˆ

:

trŽ"k2dbXG{gjxKyWA*s9€€Kx"d#A>d$…—zK|dx teŽ"kzK|´k

d->'´ˆ : >db*…—dxK|´tebezK…htrsx"ffbhgj„"ŽK|Ød->…—dxHtes9xz"dzK|>z"xK…htrsdxK|¥;žvœ-<À,

sS?k2x g[|´kEtæd]>š„‹dsxHth|

6

`^a

g

y4-

g

bD¼ ;

wD®

ž=< )\;

® <Ày c w

yvž]yž||C|Cy.-7HdA*steŽ ` y lb)¼ :=<

¸ 

…—dxK|´s;y{k2bŠtrŽ$k´ˆ

:

fsv?k2xå˜[GtrŽ"k - …EdxHtebhg…htrsdxK| , g ` c ž-yž|C|C|Cy.- b y{k=u]x"kydx­;wƒ® ž=< )

; ® <©]˜[G*¤

, g

`qa

y4-

bdc ` g

`^ab

® g

`^a

y4- bb

A*Ž"kvbŒk g

s;|g&…—dxHtebhg…htrsdxštrŽKg#ti‚g„K|;wƒ® ž=<trd/;

a

g±© Œ ® a g

<gxKy

g

¤%;wƒ® ž=< );

®

<’ ; ® <

s;|µg>z"xK…htrsdx{…—dxHtrbrg…htesv?kA*steŽbek|´„‹k…htštrdtrŽ"k?|´k–…—dxKyN?gjbes;gj˜"€k{|´zK…rŽÒtrŽKg#tM¤

g

`qa

i

y4-

i

bc -

g±©

Œ ® g

`^a? y.- ?½bc - g | `’ b

oTŽ"kOg#tŒtrbrg…htedbd->teŽ"s;|´ˆ : s†|çteŽ"kfbhgj„"Ž‚d->gA…—dxHtesx{z$dzK|t>z"xK…htrsdx _ A*Ž"s;…rŽæsxHtrk2be„Kd€;g#trk|

trŽ"kO„KdsxHth| `^a

g

y4-

g b ;ž2œ¢<,

€>qtrŽ"k g L|šgjbekOg x"k g

`qabcp

g

a” ‘ g

]gxKy sC>~>dbµkg…eŽ ¼ 6

ž]yž|ž|ž|My.-7H

g ( `^a y.- b ¯ ( ` , g

`qa*b y4,

g `- bb

¯ o g ( `^a y4- b >db*gj€€ a y4- ¼ )y

§ §À¿