B. Annexe, u r le s corps d e H ardy Nous rappelons que lq ues déments conc er na nt les corps de Hard y, po ur le conte nu dêt aillr HO ll S nous référon s à [ta, pp.5-72

@. Tous-les-l-cor-ps-de-hardy, L signifie loga,.ithmÎco-expon elltit'f j, un L-co rp s éta nt un co rps de germes de fonctions obtenues par R(I ) en répét ant les adjonctio ns de fonctions alg ébriques à valeu r réelle, de logar ithnws de fonctions positives. et d 'exponent iels df' fo nctio ns (e.g., le co r ps A(I

I. Si and . Es-t-un-et-qu-e-[-un-Éléme-nt-non-nu-l-de-fi-', ced implique que f( t) of 0 si l E A es t suffisame nt grand. P uisq ue f' E 1\' . / est dériva ble pou r l E A suffisamenr g ra nd, pa r conséquent / cet co nt inu et soit to ujou rs positif o u soit toujours négat if po ur t suffisa me nt gran d . Donc cha que / E f{ es t asy rnptoriquement zér o , ou touj ours positi f, ou toujours nég ati r, c'est -à-dire que le signe de f est as ym ptotiquement invar ia ble, Ce fait est égalem e nt vrai pou r f' E I{

E. Chaque-f, E. «. Es-t-asymptotiq-uement-monot-one-f, and _. Lim, /{t) ex iste com me un é lém ent de R U {-oo, + oo}. Ce so nt les pro priétés tes plus fondamentales des corp s de Hard y. Nou s rem a rq uons que la ca ractérisation d u corps de Ha rd y l'l' dut les comporte men ts oscil latoir es il -ë cc

/. Si and E. T{, alors à pa r tir d ' un t suffisement gra nd nous avo ns f ?: 9 o u Y ::= / O n dit que [ ",t 9 sont com pa ra bles si et seul em ent s'i l existe des e-ntiers N,' 14 > 0 et. des const a ntes réelle s n , fi > 0 tels qu e, po ur 1 suffisa rnon t gra nd

. Si, 0, rempl açon s-les par ± l/fou ± I/g. On d it que la clas se de com para bilité de f es t plus grande q ue ccii,' de 9. si, J. 9 croissan t infinime nt. f > y N qu el q ue soi t N . Le te rme cie "com pa ra bilité" nt:' s 'ap pliq1le qu 'a ux é lé me nts non nuls de !Î dont les limit es sont 0 o u ± 1Xllo rsq ue t ..... 00 . C'e st facile de vérifi er que la com parabilité est une rela t ion d'équiva lence . LI:'rang d'un cor ps de Hard y II' est défin i pa r le nombre de la clesse de comp ar abi lité de f{ Pa r exemple, co rps de Ha rd y fi est de ra ng 0, si ct seu lement si K CH

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