sol3 - Ñ g∈ Ð Î Ï Ï Í a Î Í ÌÈ Ë ÉÈ ÇÆ Å...

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Unformatted text preview: Ñ g∈ Ð Î Ï Ï Í a Î Í ÌÈ Ë ÉÈ ÇÆ Å Ä Ã Á ¤eƒ¦de‚qʦdʐ…dS¾¦¨À ½»º bU¹ sup (F ) (F ) s p |p m‚‚ r v t r s }  ‚ | p ‚ wduBv–‚–m‚¥s ≤ sup |Pn f − P f | ≤ max |(Pn − P )fi | f ∈F i x‰ © © ‘ p r | s } | s { s } |† y •®Svv‚rSvvˆ¤† ®·v6B‚vˆ† ¸ ¶ s tr s‚ | g }| v•v s„s 6Svt |p mS‚ ² … ± y x ” € ‡ x €— y … ‰ ˜rIrm®°S„d™ `t0¡ ≥1 2 $ $ & … ‰ ‡ € … ¯ € ‡ ™ ¨'(%t1®`#S„1–‚€ ” x  }| • v•v ­” r v t r s r p | p r } s s | €  | p wd6®u•–D6E‚Sv%“vmªs vwv ‰ žžS¡v…‚s ‚ªv6¨wA“v6S6ªv6%ˆ6S•–¦¨† } | v ’ ’ s Ž |† „ p ‚ s t r ‘ p r  y s „ s t r € r ‚ v Ÿ s y y† ˆvt € ~ } s „ p ‚ s ~ | v ‚ s ‚ v { y | v s }† ‚  ¬  v | p† y | s ’† šS6Ss S¡r‡•‚¡‚wd6z•e–Svlf•vm…v‚%–} † | y † } v „ ‘ p  v ~ s t vˆv•uq•¨•d0–ur  © q” ‰” s” † ‰ |† ’ s ~ p „ { s t r „ s }† y | p ‚ p r r | s †  ¢ s y† r† ˆ«‚ˆvm6 –ª–u'‚vˆ6v•‚¡6¨Swv vS¨ˆ¨“¡p ’ y † t { „ p ’ p y † v y † r † y £ `t” „ v s | †  y 6ˆvv…m%m6ˆ0¨'ˆ¤–lur ‰ •‚vˆ'†  } |v ’„ p | s tr y „sys„ v•§6mvBvuqSs ‚…‚…v{ Ž |† { { v ’ s t r r v t r œ ‚ s t ‚ p r € y v s y† vv–•ˆBvu¤dduE›‚–¦¥6E…•‚¤ˆqr rv tr t‚  dduk›v6y ‰ s  ¢† |  r y† Ÿ s –`ˆvv¡ ƒžs „ s t ‰ | p† r ‚ |  € | mˆuƒvvU‘ ˜•v • k” x 6‰ Svur }| v•v ‰ s„s € s ’ v “‚%•d| ‰ „ p œ ‚ s t ‚ p r € y v s y† r† | s mU‘ ‰ ›‚v›šu™•eDˆ˜ˆ—‚–t x  6Svt x ‘ p y† y v ~  v | p Ž p t r„ p | v } | Š | v ‚ •…•dG•dm•mvumŒ•vd‹•eXs ‰ rv t srp wdur 6•qo |† y „ p r ‚ ˆ‡…muƒ‚s € ~ }s |v {y y ˜‚v•v6zwv a i (f )φ i F φ1 , φ2 , · · · , φ r f∈F R r H ( , Q, F ) ≤ d log d r≤d L2 ||φi || = 1 (a1 (f ), a2 (f ), · · · , ar (f )) ∈ Rr f → (ai (f ))r=1 i 4R + . 4R + 4R+ ƒy td™ …†… `f‰ PH I¦ F C £ & C  7 9 7 & 53 GE(%DBA@8(64 %10('%#!  2 ") ¡ & $ " F = {f = 2 =1 θk ψk } k ( f 2 dQ)1/2 ≤ R g n x‰ g qf f¢ g g ‡ y ™ † ‡ € y x ” … ‰ ‡ … € x j h Iid1S#rm’f‚rlki’… t‰ …†… € € ™— ” y x ‡ ” y “‘ … ‰ ‡ † … ƒ € y x w P § edU˜–IU•t#b’fˆ`f„‚#r0%vS3 V h Y s T Y p i h cR g e c aR Y W V TR ¨rutr1q¨XU#`fdb`X#USQ   ©§ ¨¦ ¡£¡ ¥¤¢ d≥r F F ⊂ L 1 (P ) n 2¨$'$(&’tD`³Si–‚€ … ‰ ‡ € … ¯ € ‡ ™ ± y x ” —— “ Irm¨„t¢‰ € € ™— ” ™ ‡ ™ ‰ ‡ … € x j j “ P ¦ edU˜8G–t1¥‚Iltt¢®u63 (F ) n n i=1 ‰ x‰ g= αi fi ∈  • µ” |s Svt | p† r ‚ | ‘ € | v „ s }† y | mˆuƒvvU¥•¤‚vˆ6vmp ´ 1 g∈ |P n g − P g | = sup |Pn g − P g | ≤ sup |Pn f − P f | ≤ i=1 i=1 n n (F ) α i | (P n − P )f i | α i (P n − P )f i f ∈F f i ∈ F αi > 0 F⊆ ψk ∈ L2 (Q) (F ) x © –”   s džˆˆ•duƒ•v• v t € v  r ‚   • ¿” s‚‚v|ˆ6y ‰ •¾† †6„ur˜ˆ†™brˆwvd¢‚svˆ†‡s6y…„‚s s…„zvt † v y €† | s ’† v ‚ s t r y p „ { y† w¾‚v6%‚s 6v%ˆvt ½»º ¼¼U¹ |Pn g − P g | = sup |Pn f − P f |. f ∈F n i=1 Rr d αi = f= € … x ƒ † … ‰ ‡ … ¢fDt1U`fy v—— ™ † … y … f … † ƒidt`‚rx g n … ‰‡ ‡™ ‰ t1E–t1‡ f¨ x‰ … ” ™ j € …— e ™ † “ € ™ … h mdt‚U³1fedk!™ rx y ƒy td™ ‰ –m‡–m‚S6¥v6Amdp ” Sv6ˆ–•›‚¤ž¼“•dSvq6dBv6zSvt • s |p } |p‚sy s tr „p } s t y† ~ v r y s y† € r† v  ¢ s |† r y „ Š s t r | s ≥ = 2h2 (p1 , p2 ). √ √ ( p1 − p2 )2 dµ  x s d·s ‰ bˆˆ•d‚–q……dBvuA•dp vt €r† v  ¢ s |† r y „ Š s t r „ p g g V € … ‡— ™ “ h … y `U1Uidtuifi™x € g ||p1 − p2 ||1 = ‡ V x—— x‘ € ™ 7 T D RQ I UiIÊ%dB`db@PB 1 2 … † “ € ™ … h … ‡ y dfedkŒ1°#³¯ g V€ x—— x‘ € ™ 7T RQ I iiIÊ%gdBU!D S@PB … h x x ‡”… j€… ‰‡ kr‚€ 1‡ • l‚`† ° ƒy td™ ¦ "  ) ¤  & $ " ¢   ¢  ¨ ¦ ¤¢ 2(10('§%#¥! ©§¥£¡ p1 p2 σ− µ f ||p1 − p2 ||1 = 1 h2 (p1 , p2 ) ≤ ||p1 − p2 ||1 ≤ h(p1 , p2 )[2 − h2 (p1 − p2 )]1/2 2 √ √ √ √ |p1 − p2 | = |( p1 + p2 )( p1 − p2 )| √ √ ≥ ( p1 − p2 )2 , h (p 1 , p 2 ) = ≤ = = (2h2 (p1 , p2 )(4 − 2h2 (p1 , p2 )))1/2 = 2h(p1 , p2 )[2 − h2 (p1 , p2 )]1/2 , q |† r r „ iEs…wv x t ›‚ © q€t‚  (˜¦–•v ‰ r v tr r ‚ v‘ s tr ‘ wd6¤ƒ•iGvuf•p s •·…y  v ‚ s ~ }  p t s | †  t r „  p ‘ s t r } | € r †  v  ¢ (” v•‚‚r8ˆmvªvˆ(6…v•UB–u8v•v ¼“wd‚sx € ~ y } p t s |† } „† t r s €r† v  ¢ s | } | p ‚ s y s t r y s t y† ~ v r y s t ‚† bˆˆ•d‚vˆ† vmS‚6ˆ–u0Sv6ˆv•u…S™¦ˆvt ´ ˜¡–m–%vˆk6ˆvužvt y s† { ’† t ‚† ‚“ˆvžˆ¦ˆvt fy gi x—— ‘ … ‰ ‡ ‰ €— e ™ ‡ € iirx ‡f—eii³11ƒB7 ƒy td™ G D aB RB R W … ‰ ‡ … y ¯ … cYbQ `F YXf‡fq˜fƒ G F 7 E DB99 7 5 … ‰ ‡ ‡ ™ ‰ ‡ … y ¯ … H@ACA@86f¥–t1’f³fƒ € … ‡ € y … ƒ y … ± 3 P  `U°e#`f§ r°(4u63 √2 p2 ) dµ = 4 ||p1 − p2 ||1 = √ √ √√ | p1 − p2 || p1 + p2 |dµ |p1 − p2 |dµ √ √ ( p1 − p2 )2 dµ √ √ ( p1 − p2 )2 dµ |p1 − p2 |dµ |p1 − p2 |dµ. P1 µ f … …3 … ” y ™ ‡ €  ƒ † … ff y  — — … mtd1SitžryU„i`z x … † “ € ™ … h … ‡ y t‚dkX1°#³¯ ™ wg `1–f„™rfƒ v e ƒ … ‡ ™ y h x … † “ € ™ h v ‡— e ™ e x † j tev`… «`Ui„#1t%dt!x ¿f„‚#r0™¨u3 ‡ † … ƒ € y x w P t σ ” (X , A ) nn 1 2 dP1 − dµ dP2 dµ µ f € n … † “ € ™ … h f y ‡ ™ y h x ƒ … ‰ ‡ y x ƒ y … j … ƒ ‡ x y € … x fedk‡y„ ti™rtEtkr¿t`µfDdf0 ¢fƒ dP1 dµ (1/a) − dP2 dµ √ √ ( p1 + p2 )2 dµ (1/a) a 2 P2 µ = P 1 + P2 ) dµ 1/2 √ √ √ ( p1 − p2 )2 dµ + ( p1 + , 1/2 x  x‰ £ l”  vt ‘ ‘† y } p t €r† v  ¢ s |† p ~ v s t € ˆ¤vm–ˆbˆˆ•d‚vˆ¥s r•·v6r ‰ ˆ6„ v s ” s d¥s xwe ´ x y | p† r ‚ | ‘ | v t r „ s t r v „ y „ s ~ ’  | y –m6SvvU8•duA‚v6wu­6SrDv`­wv v•v ‰ we…u(•e¥s 6Svt }| rvs„r |v‚ s„s p p0 √ √ √ √ ( p − p0 )2 ≤ 16( p − p0 )2 , ¯ p ¯ p = p0 p = p0 x‰ fe€ ƒy td™ ” s „  y v s ’ Ž |† r v |† ’ p } s t r | p } | s { s } r p | y s p 6v…•e%Dvˆ6wd%mv¥–u(mzv‚rSv­•vqS`v} • µ” s ‚ | v ry† } ‚–•›v¨„ s s „  y v s ’ Ž |† r v |† ’ p } s t r y 6v…•‚ž¡vˆ6wd%m–¥–uA•v v…v(•vm6–uDm­v‚6…Gžwu•y Ž |† y  t Ž  p „ t r p Ž y { s r y s ’ v •‚mŽ | †  s ¸  s t r y  B” –ˆ ‚q¥vu¤–`t r‚vŸs s ƒ•–ƒDvt y † r † „ s ’ € } p œ† oq | p } v ¶ r  p ~ v r  y s „ } „ v } | v r y v y s y  €r† v  ¢ s r y v s ‚s uwv 6Sv} `•®bim–•®B mr•Bˆv…‚…š6•d–•›ˆ¥‚6–zbˆˆ•dSG•¾Dvt = dP1 d(λ + µ) µ (1/a) st –ur ‰ rv tr t‚  wvu¦–6y  | p † r ‚ |  © ¥ † r v Ž s | | p | v ‘ p s ‚ | s ” r y¥Ÿ s s t r y s †  { ’ ’ s „ p s x t ’ € } p œ † o q | p } v m6Sv–U­” s 6w˜Svvm–A•ˆSv” ‚6…ˆ–x¦%vuG‚“ˆv%† ‚…m‚–ur `–`•ˆ®bimv•q¶• ‘ ”† s‚| Sv† s„ yvs ’ s t 6v…•e%0–ur r „ wumSuˆzu•r·6ˆ…„ u¤(•‚6wu6…0vt y v „ Ž s r |† t r p ~ s r† p r y† € Ž s r v „ r y s x ( )” rvt wd6r  &$ '% "" ) #20!"   p ty  0 v…¨ˆ s }| v•v    & ¤ ¢  ¨ $ !C§!$ 10'& ©§¥) £ X¢ ¡&  ¨ ¦¤ Ñ P1 dP1 dλ λ = = (1/a) P2 − dP1 dµ dµ d(λ + µ) dP1 dµ dP1 dµ p λ dP2 dλ (1/a) (1/a) (1/a) a p0 g f € … ‡ € y … y … ± P 1 `i1Uet`tƒ `r°(3 u'3 p = (p + p0 )/2 ¯ √ √ ( p − p0 )2 = h2 (p, p0 ) ≤ 16h2 (¯, p0 ), p p − p0 ¯ ≤ 16 √ √ p + p0 ¯ √ √ = 16( p − p0 )2 ¯ √ √ √ √ 2( p + p0 ) ≥ p + p0 ¯ − dP2 dµ dP2 dµ (1/a) ” s ‚ |† y y } p t € r† v  ¢ s |† s t r s „ s Sv…­–m–ˆ¼“wd‚vˆ¥vu¥6Svt x − dλ = = 16 = 16 − dP2 d(λ + µ) (1/a) a (1/a) a p − p0 √ √ p + p0 dµ dP2 dµ d(λ + µ) p − p0 ¯ √ √ 2( p + p0 ) (p − p0 )/2 √ √ 2( p + p0 ) dµ dP1 dµ (1/a) a 2 (1/a) dµ d(λ + µ) d(λ + µ) λ+µ d(λ + µ) 2 (1/a) a 2 2 rddur vt n x p ty pr tŽ p | v6¥u(mv•vl¬ …†… t‰ g − dP2 dµ d(λ + µ) (1/a) a µ λ+µ dµ/d(λ + µ) dµ € 1v © ...
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