Constrained Hp Interpolation on Subsets of the Circle
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. . Constrained. interpolation on subsets of the circle Laurent Baratchart,. Juliette Leblond,. Jonathan R. Partington. N° 3763 Septembre 1999 THÈME 4. ISSN 0249-6399. apport de recherche.
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(145) . »Q¿ ». ^7o8 3 5 . C " N R/ P1 Q D N Rc/ P &1 Q D R @B C. . >. $°. = $ % ¡ P > Q # BC 2 O XBT 0" Lg +D oBC. ]³ . OC XBT d CLg. ³o D. = R v P " A A R Q J PTR^Q f* a " N $ V >oB P R^Q $ h = R h v >. ¯. + B0 G > + B > > a h . ï. R > . 6 8. G U ") $ h h UV ¥ [X Z [ = R/. . P" A A R Q J. Z,[Qht©lu[]_¡<nJh]p w _ w znJ^;r]ig_lufgnJpr}un7]io_^µ³o/fop }u_ w hz_tPlmn'lm[]_/<nJh]p w _ w _{Qlu}m_^jvi<r]}mnJ]ig_^ ³ ¢s´[]fkz[fgp"luh]}mpq}u_ w hz_5tlun;lu[_'¤_[1jv}mfr]}un7]ig_^³ qn7}u_n _}¢5f 1 [jJt3jvpfopClm_}mfon7} r<nJfgpCl¢JjJp w "s) N / N7P cQm¢>lu[]_plm[]_tyn7iohslmfon7pfkt3h]p]f
(146) Ch]_J¢ jJp w ¢sfoP "Ufkt,p]nvllm[]_alu}jJz_WnJ3jÇh]pzlufgnJpqfgp V ¢]lu[]_p v $ = R j]³ _J³3nJ4p &1¸³ B ¢Chsl/jtufg^cfgikjv}r]}mnQnv /p R ) _}tyfgnJpnv lu[]fkt,r]}mnJ]ig_^%´,jJt,tunJi _ w fop !gJ ( ¢QÇnJ},lm[]_zjJtu_ _{Qlu_p w t,lun;7_p]_}mjJi jvp w ´__5tljv]igfkty[lu[]_^cn7}u_a7_p]_}mjJi<}m_tuh]iolp]n@´³ = _hty_¡lm[]_'p]nJlmj@lmfon7pnJ r}un7]io_^]³gJ³ »Q¿ » ^7j C " N R )P1 Q xN R );P  Q # aC L<, R + B49 XB
(147) uX V. aXB CT $ h N. XY oB . . P") $ h A ) Q =. . P¢") $ A ) Q J I<C U $ h U$V X[Z|\]c[ = R DQXB C " S 3 ;%79 .
(148) " 1 e aS d) a XB C T $ XY 9 U $ UV X[ZY\]c[ "C"oB RG G B C C ih- $ h $ h = P v(' Q ' P v ' ""Q ryH ; P¢")dP v(' Q Q A 6#&8 BC
(149) " oB # - fhdB 4;9UG 6 8 D ' + YX 9 U $ h UV X[Z|\]c[ = Rc bh = $ h !#". y¯. . y¯. H`. lm[]_J_xnJtu_} j@lmfon7pÂfkt,lu[jvl¢]´}mfolufgp] . . ¯. ¢´_a[j _. 0 P ; P" )dP b h v Q Q A Q V X[ZY\]c[ = B. Çn7}jvigi fop"lm[]_almjJp]J_p7lturjJz_¡lunclu[]_'tur][]_}u_. . . V =
(150) b'N R ) PT &1 Q U b U V XZ|\ ]c[ = R J. ÄÇÆ
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(154) 7hf jvig_pClmiox7¢]´}ufolufgp] * = ¯ FP¢]´$_a[j _ ; P")dP b hsv Q = v * b h J 0J &1 A : lm[j@lfkt¢ v * b h fgt$lm[]_}u_5tlm}ufkzÒlmfon7pnJ
(155) jJp"*)¡ÇhpzÒlmfon7p ´folu[ ; P¢" & ] v * ryH & ZY\] v Py¯ * ryH Q & Z|\ ] Q =xA n7}OP * rH ""Q 0= ; P"()JPu¯ * rH Q Qu³q¨
(156) fop1jvigioxK´_´}mfolu_ ' = ¯ * + $ h = v P Q * ³. ¯J¢
(157) jvp w tunJi _cÇnJ}. óp fo_´SnJc6T';8,jvp w 6 #&8Ò¢´_jviktunc7_l,lu[1j@l $ h = Py¯ v(' Q ' h P¢" A A R Y ' Q A 6H&8 Çn7},lu[]_'tunJighslmfon7p h P" A A R ' Q3lunr]}mnJ]ig_^]³ j7tutunszfkj@lu_ w lmn."Ájvp w tuhz[lu[jvl U h UV X[Z|\]c[ = R ' ³©Z,[]fktfkt$lun;1_'znJ^;rjJ}u_ w lmn 6 8³ ¨nJ} d =?A ± ¢<fol¡ÇnJigion@´t/Ç}un7^²_
(158) Chj@lmfon7p7t 6ï 8¢ 6H 8lm[j@l $ h N ® * j7tWtunQnJpUj7t
(159) N ® * ¢ ´[fgz[ fkt"tyn7^c_´[j@lh]p]_{sr<_zlu_ w lu[]fktqzjvp <_Kr]hslqfop }m_Cjv} w lun«lu[]_ w _ptufolx ¬p]nJp w _ptuf lx r}un7r1_}ylmfo_5tnvPlm[]_n7}u_^;.t 's³ %jvp w ']³ `1³ _ w np]nJlWQp]n@´ f Plm[]fktWtylufgioi
(160) []n7i w tÇnJ}Wnvlu[_} jJioh]_5tnJ jJp w ´[j@lQfgp w nJ
(161) }m_ikj@lmfon7p"tu[]n7h]i w lu[_pq}u_^jvfgp9¢sf PjvpQx7¢s1_l´__p }~Ì| jvp w } ¦ | ³ *$Â
(162) _htu_ w []_}u_'tunJighslufgnJp1t$lun<nJh]p w _ w _{Qlu}m_^jJi#jvp w znJ^;r]ig_lufgnJp"r]}mnJ]ig_^t³ ;jJp w ]³g¡fgpqnJ} w _} lmntyn7i _WÇn7}jvp wQ igfg^cfolu_ ww fgtmz}m_lm_¡fgpClu_}ur<nJikj@lmfon7pfktutuh]_5t,fop¹ ® ³ []_p ± ¢Qfol$fkt$r1nCtutufg]io_/lunÇnJ}m^h]ikj@lm_WjJp w tunJi _/^;n7}u_WJ_p_}jvi_{Qlu}m_^jvi<r]}mnJ]ig_^tÌlu[jJp r}un7]io_^t] ³ = jvp w ]³gJ¢´[]_}m_´_}m_rigj7z_alu[_ty_l/0/Qxlu[_ty_l/0/ v nv
(163) Çh]pzÒlmfon7ptlm[j@l jJ}u_¡^;_}mnJ^;nJ}mr][]fkz¡fop"lm[]_ w fktuz¡´folu[Áj@l^;n7tyl r1n7io_5tjvp w jv}m_a1n7h]p w _ w fop F ¤ F ¤ ¯ ÇnJ} tunJ^;_ ¤ ¯7¢jvp w ´[]_}u_alm[]_'zn7ptlm}mjJfopClt > jvp w R jJ}u_¡p]nignJp]7_}zn7ptljvpCl³ _}m_Ç_},lm4n ! (ÇnJ} w _lmjJfoikt³ []_p ¢lm[]_j7tyxQ^;rslunJlufkz<_[j fgnJh]}nJ
(164) lm[]_znJptylu}jvfgpClmt > jvp w R jJt,lm[]_r1jv}jv^;_lm_} jJr]r]}mn7j7z[]_ t = a[? j7t,1__pq}m_z_pCluigxfop _tylufg7j@lm_ w fopJ! '-(#´[]_}u_tu[jv}mr_5tlmfo^jvlu_tjJ}u_¡_tylmjJ]igfgtu[]_ w ³ ' 5. . . . . .
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(166) lunalm[]_f w _pCluf21zjvlufgnJp;nvNigfop]_5jv}3lmfo^;_ fgp jJ}ufkjvpClPtyxstylu_^t z jvpa<_PÇnJh]p w fopO! `]¢J s¢-%J,( ³ ·Çh]}ulu[_}#jJr]r]igfgzj@lufgnJp¡lmn,lu[]_Ìtyluh w xWnJ /fg}ufkz[]ig_ly¥Q¤_h]^jvp]par]}mnJio_^;t fkt,Jf _pfgpJ!ä-(ï³ . È È îiÑCjæÒÑ.
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(179) lu[]_ zfo}zig_a´f lm[·¯° (¤ ± ³ XY =< "; 3& =gO T ¢#|` 3 `1Ò¥ 'v]¢17; #]³ !ä, ("HÌ|³ },jv}j@lz[jv}ul¢ ³ H9_]ignJp w ¢1¨³N¹jvp w }u_5j]¢]jJp w ~³ }¡³Nds%j +3³ª/n@´ zjJpq^c_}un7^;nJ}mr][]fkz¡jJr]r]}mn>{ fg^j@lmfon7p¹[]_igr¹luntyn7i _'tyn7^c_ % fgp _}mtu_ar]}un7]ig_^tÇn7}lu[]7_ H#jJr]igj7zfkjvp *|<, CLg ¢ , '6y- 83ì@@¥Q ]¢<J7]³ ! #("H̳ },jJ}mjvlmz[jJ}yl5¢ &³ H9_]ignJp w ¢sjJp w ³ ³Q|3jv}ulufgp]vlmnJp9³9ªWjv} w xcjJr]r]}mn>{sfo^j@lmfon7pclun R / Çh]pzlufgnJpt n7p¹tyh]1ty_lmt,nv lu[]_'zfo}zig_J³ XYT =<, "; 3& =gO T ¢#- %3 ` %v@¥C`77]¢]5J7]³ ! ("H̳ },jJ}mjvlmz[jJ}yl5¢ -³ H#_]ignJp w ¢@jvp w ³ ³5|3jJ}ylmfop]Jlun7p9³J|P}mnJ]ig_^t nv w jJ^¡~yjvpQB¥ }mn ¥ }u_fopWlxQr1_ n7p¬tyh]1ty_lmtanv$lu[_zfg}mzio_jvp w ^;fgp]fo^jJi©p]nJ}m^ _{Clm_ptufgnJpt³jvr]r<nJ}ul w _;}m_z[]_}mz[]7_ 5¤/ 5 '¢ ¤ J7& ']¢15J; #³
(180) Z njvr]r<_jJ}fop XY =< "; 3& =gO T ³ !g ( HÌ|³ },jv}j@lz[jv}ul¢ ³ H#_]ignJp w ¢ ³ ³<|3jv}ulufgp]vlmnJp9¢jJp w ¤'³<Z#n7}uQ[jJp]fï³n7]htylf w _pCluf 21zjvlufgnJp¹fgp lm[]_ w fktmzjvigJ_}mjÇ}un7^jJp wQ iofg^;f lm_ w¹w jvlmj³ * # "cXB T ¢` %E3g]- #>¥N& %&'s¢ 5JQQ³ !gJ ( HÌP³ }$jJ}mjvlmz[jJ}yl5¢ +³ H9_]ignJp w ¢jvp w ¤'³PZ#n7}uQ[jJp]fï³ }$_5tl;1n7h]p w _ w * )_{Qlm_ptufon7p«nJ/rjv}ulufkjvi Ç}m_
(181) Ch]_pzx w jvlmj^³ 5ódp '³ 5tyf w n7}ufï¢<dN³ }$f lulmjJp7lmfI¢ ~³9qnCtuzj]¢ ³ W_ H#hzj¢N¬³ ³ Wf }_p]_ w _lylmn¢ jJp w c5³ (W}ufgnJign¢v_ w folunJ}t.¢ K a 9 XB J KaC C D g1 ¢ nJigh]^;_ %s¢ r1jvJ_5ta7 >¥N7& 's¢CdQ_rslm_^1_}'7& 's³ !g- %( HÌY³ },jv}j@lz[jv}uljvp w ¨$³1dQ_xCÇ_}ul+³ p R ® jJpjvignJlunclu[_ . lu[]_nJ}mxN³ 5óp"r]}m_rjJ}mjvlufgnJp9¢N7J³ . . . . . . . .
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(200) ³ H©³ /h]}m_p#³ # B
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(217) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - B.P. 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Lorraine : Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - B.P. 101 - 54602 Villers lès Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot St Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - B.P. 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, B.P. 105 - 78153 Le Chesnay Cedex (France).
(218). . ISSN 0249-6399.
(219)
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