Department of Mathematics and Computer Science, University of Calabria, Cosenza, Italy

Transcription

1 Advaces Pure Mahemacs, 5, 5, 48-5 Publshed Ole Jue 5 ScRes. h:// h://d.do.org/.436/am Aalyc heory of Fe Asymoc Easos he Real Doma. Par II-B: Soluos of Dffereal Ieuales ad Asymoc Admssbly of Sadard Dervaves Aoo Graaa Dearme of Mahemacs ad Comuer Scece, Uversy of Calabra, Coseza, Ialy Emal: aoo.graaa@ucal. Receved 3 Arl 5; acceed 7 Jue 5; ublshed 3 Jue 5 Coyrgh 5 by auhor ad Scefc Research Publshg Ic. hs wor s lcesed uder he Creave Commos Arbuo Ieraoal Lcese (CC BY). h://creavecommos.org/lceses/by/4./ Absrac Par II-B of our wor coues he facorzaoal heory of asymoc easos of ye (*) f ( ) = a( ) a( ) o( ( ) ),, 3 where he asymoc scale ( ) ( ) ( ),, s assumed o be a eeded comlee Chebyshev sysem o a oe-sded eghborhood of. he ma resul saes ha o each scale of hs ye remas assocaed a mora class of fucos (amely ha of geeralzed cove fucos) ejoyg he roery ha he easo (*), f vald, s auomacally formally dffereable mes he wo secal seses characerzed Par II-A. A secod resul shows ha formal alcaos of ordary dervaves o a asymoc easo are rarely admssble ad ha hey may also yeld sew resuls eve for scales of owers. Keywords Asymoc Easos, Formal Dffereao of Asymoc Easos, Facorzaos of Ordary Dffereal Oeraors, Chebyshev Asymoc Scales 7. A Bref Iroduco hs s a couao of a revous aer [], abou he facorzaoal heory of asymoc easos he How o ce hs aer: Graaa, A. (5) Aalyc heory of Fe Asymoc Easos he Real Doma. Par II-B: Soluos of Dffereal Ieuales ad Asymoc Admssbly of Sadard Dervaves. Advaces Pure Mahemacs, 5, h://d.do.org/.436/am

2 A. Graaa real doma. 8 coas he ma resul he aer: o each Chebyshev asymoc scale ( ( ), ( ),, ( ) ) remas assocaed a mora class of fucos ejoyg he roery ha a asymoc easo accordg o hs scale, f vald, s auomacally formally dffereable mes he wo secal seses characerzed 4,5 Par II-A. Uder he regulary assumos of he facorzaoal heory, hs class s characerzed by a h-order dffereal eualy whereas he ye-o-be-develoed geomerc heory wll be he class of geeralzed cove fucos as he secal case of olyomal easos ([], 4). I 9, dscussg formal alcao of sadard dervaves o a asymoc easo, we characerze he esece of cera olyomal easos a a edo where dervaves may fal o es ad such ha he growh-order esmaes of he remaders of he dffereaed easos follow ueeced algebrac rules. coas he roofs ad coas a few remars abou our heory. Whereas he resuls Par II-A show ha formal dffereao of asymoc easos s usually admssble oly f suable oeraors led o he gve scale are used, he resuls hs Par II-B shed furher lgh o hs classcal roblem by ehbg a meagful ad o oo secal case where suable formal dffereaos are auomacally admssble ad by showg ha sadard dervaves are admssble very secal cases oly ad ha hey may yeld formulas algebracally sew from a classcal vewo. We coue he umberg of secos ad formulas [], bu we ado a deede umberg of he refereces he bblograhy. I order o agree wh some classcal ermology abou he maer hs aer, s covee o secfy he sgs of cera Wrosas, so we ls he fudameal roeres of he scale we shall use ae from ([], Def.. ad Pro..3): he oeraor AC,, ; (7.) [ [, ; (7.) ( ) [ [ W,, > o,, ; (7.3) ( ) [ [ W,,, o,, ; (7.4) [ [ o,, amely sg =,. (7.5) L : (,,, ) (,, ),,, u = W u W s he uue lear ordary dffereal oeraor of ye (.),, acg o he sace AC [, [ ha er L sa (,, ).,, = Easos we are sudyg are of ye f ( ) = a( ) a( ) o( ( ) ),, 3, (7.6) ad such (7.7) ad we are suosg 3 as he wo-erm heory has bee horoughly suded [3]. Oeraors L ad M are defed formulas (3.) o (3.4) Par II-A; roeres of he L s are reored he frs few lemmas 4 ad roeres of he M s are o be foud Prooso 3. wh he sgs secfed by (3.9), due o our rese assumo (7.3). We recall he acroym C.F. for caocal facorzao ([], Pro..). 8. Absolue Covergece ad Soluos of Dffereal Ieuales he heory develoed Par II-A becomes arcularly smle whe he volved mroer egrals are absoluely coverge ad sll more eressve for a fuco f sasfyg he h-order dffereal eualy [ [ L a.e.o,.,, f (8.) Uder he assumos (7.) ad (7.3) hs s a subclass of he so-called geeralzed cove fucos wh re- 48

3 sec o he sysem (,, ) of a asymoc easo (7.7) he hs easo s auomacally dffereable A. Graaa. he ce resul saed he e heorem clams ha: f such a fuco adms mes he seses of boh relaos (4.3) ad (5.6). heorem 8. (Comlee asymoc easos). If f AC [, [ sasfes (8.) he he followg are euvale roeres: ) here es ( ) real umbers a,, a such ha ( ) f = a a O,. (8.) ) here es real umbers a,, a such ha ( ) f = a a a o,. (8.3) 3) he followg se of asymoc easos holds rue: L f = al al o,, ; see (4.3). cosa 4) he followg se of asymoc easos holds rue: = M f a M am o M, ; see (5.5)-(5.6). 5) he followg egral codo s sasfed:,, f ( ), (8.4) (8.5) L d < ; see (4.3). (8.6) 6) he followg egral codo s sasfed: L d, see (5.9) ad (5.9)., f, < (8.7) o hs ls we may obvously add he oher roeres heorem 5. ad f hs s he case he remader R of he easo (8.3) adms of boh rereseaos: whece follows ha f ( ) f ( ) L,, R ( ) = d L,, = d, [, [, R ( ) [ [ (8.8),. (8.9), he above euvalece ) ) smly meas ha, uder codo (8.), a relao f ( ) = O ( ), mles he esece of a fe f ( ) ( ) a. lm I addo o he euvalece 3) 4) here s aoher remarable crcumsace where he wo yes of formal dffereaos are smulaeously admssble amely whe he covergece of he ere mroer egrals s absolue. f AC, he followg hree egral codos are euvale: heorem 8.. For [ [,, f ( ) L d < ; (8.) 483

4 A. Graaa P L,, d, f < d d d where P ( ) : = ; 3 (,, ) (,, ) L f W d L,, f d. < W,, (8.) (8.) Hece each of hese hree codos mles boh ses of asymoc easos (4.3) ad (5.5)-(5.6) (here he sgs of he Wrosas are mmaeral). A drec bref roof of he euvalece (8.) (8.) ca be based o heorem 8., bu also follows from he followg remarable relao vald for ay sgs of he Wrosas (7.3), (7.4): ( ) (,, ) W,, P,. W (8.3) Usg heorems 4.4 ad 5., we ca also ge he aalogues of heorems for comlee asymoc easos ad here s a cocse saeme, all asymoc relaos referrg o of course. heorem 8.3 (Icomlee asymoc easos). Le f AC [, [ sasfy (8.) ad le {,, } be fed. he he followg are euvale roeres: = ( ) f a a O (8.4) ; = ( ) f a a a o (8.5) ; L f = al al o L, ; L h f ( ) = al h ( ) ahl h h( ) o( ), h, (whch las relaos are wre (4.8) a eaded form); (8.6) M f ( ) = a M ( ) am ( ) o( M ( ) ), ; L,, f M f ( ) O = d, ; (8.7) ( ) L,, f M f ( ) O = d ; ( ),, f ( ) L d < ; (8.8) P, L,, f d, < d d where P, : = f ; ( ),, f ( ) (8.9) d d L d <. (8.) o he foregog ls we may obvously add roery ) or roery 5) heorem 4.4 ad roeres )-3) 484

5 heorem 5.. For reduces o relao (5.6). = relao (8.4) reads f ( ) O( ( ) ) A. Graaa = ad he frs grou of easos (8.7) Noce ha relao (8.3) ad he defo of P ( ) (8.) mly ha lm ( ) ( ) =, hece (8.8) does o geeral mly he covergece, as, of ay of he er egrals aearg (8.); s he sroger codo (8.) whch mles he covergece af all he egrals (8.) (remember ha, by (.38) ad (.45), ad are o subjeced o ay egrably cosra). Moreover each of he O -esmaes (8.7) s meagful wheever he volved egral dverges as.e. wheever he asymoc easo (8.5) cao be mroved by addg more meagful erms of he form a j j( ). As soo as oe of hese egrals coverges o a real umber as he we may aly he heorem wh a greaer value of. Ad he case of dvergece, uder he rese assumo of oe-sgedess, s ossble o fer from codo (8.) sharer esmaes o deedg o L f heorem 8.4 (Sharer esmaes for M f ( ),,., ). Uder he assumos heorem 8.3 le 3, ad suose ha all he egrals aearg he O -esmaes (8.7) dverge as,.e. he he esmaes (8.7) for M f ( ) f ( ) d L Q : = d, ;. (8.),, ca be relaced by: M f ( ) o =, ; (8.) d d d M f ( ) = o, ;. (8.3) I he rese coe he above esmaes are by o meas obvous or aural : hey have bee obaed by adag he sadard calculaos he roof of he Abel-Drchle s es for covergece of weghed mroer egrals (Lemma. below). As a smle chec of her valdy we reoba classcal esmaes for he dervaves of h-order cove fucos, ad o be cosse wh he meag of he rese seres of aers, amely = dmeso of he Chebyshev sysem (,, ), we sae he resul for cove fucos of order accordg o a sadard ermology. Corollary 8.5 (Raes of crease of dervaves of hgher-order cove fucos). Assume ha: f AC [, [,, ad ( ) [ [ he he followg asymoc relaos hold rue as f a.e. o, ; f ( ) = O( ( ) ),, for some {,,, }. :,. f = a o, for a suable cosa a, ( ) f ( ) = ad ( ) o ( ) Here he asymoc scale s:, case = assers ha f a ( ) h-order cove fuco f o [, [ he case f ( ) : = ( ) (8.4) (8.5) ; M d d ; ad. he secal s bouded a he as : ( ) ( ) f = a o for some cosa a; f = o,. (8.6) shows ha he esmaes heorem 8.4 are he bes ossble, geerally seag. he esmaes Corollary 8.5 also follow from old resuls by Ladau, Hardy ad Llewood abou dffere- 485

6 A. Graaa ao of asymoc relaos volvg real owers, uder assumos of mooocy o he dervaves, resuls ha were dscussed [4] ad he eeded [5] o asymoc easos real owers. he secal case = has also bee obaed deedely by Poovcu ([6],. 8). Secalzao of heorems o he scale,, yelds aalogous esmaes a ([], h. 4. ad Remar,. 8), f use s made of a echcal resul ([], Pro. 5.,. 83). A mora remar. I heorem 8. he wo yes of formal dfferebly,,, mes are euvale facs whereas s o so for a geerc f such ha L f ( ),, chages sg o each deleed lef egh- borhood of. hs has bee roved for olyomal easos [] ad for real-ower easos [5] a drec way by eressg he wo ses of dffereaed easos as suable ses of easos volvg he sadard oeraors d d ; he ew ses of easos made evde ha wha we called wea formal dffereably, led o he C.F. of ye (I), s deed a weaer roery ha wha we called srog formal dffereably, led o a C.F. of ye (II). hs wll be also roved rue Par II-C, 5, for a secal class of easos cludg he real-ower case. he same crcumsace occurs for a geeral wo-erm easo ([3]; Remars,. 6) bu s o a self-evde fac. I each of hese hree cases drec roofs could be also rovded worg o he corresodg egral codos. Hece hese cases he locuos of wea or srog formal dffereao are legmae. Bu he geeral heory for 3 we face a orval suao ad sae Oe roblem. For 3 cosder he wo yes of formal dffereably characerzed heorems 4.5 ad 5.. Ivesgae wheher or o he roery heorem 5. always mles he oe heorem 4.5, he wo roeres beg euvale he case of absolue covergece descrbed heorem 8.. We shall o dwell o hs margal asec of he heory hough leaves usolved wheher or o we may use rereseao formula (4.38), alerave o (5.)-(5.3), uder codo (5.). 9. Asymoc Admssbly of Sadard Dervaves 9.. Asymocally-Admssble Oeraors Before vesgag cases where sadard dervaves d d are formally alcable o a asymoc easo s good o gve a rgorous defo of he volved coce, cursorly reaed ([], 3) ad ([7], 3), wh a few eamles. Defo 9. (Asymocally-admssble oeraors). Le be a lear oeraor acg bewee wo lear saces of real- or comle-valued fucos of oe real varable, :, ad le (,, ) be fucos formg a asymoc scale a, ossbly or :,, (9.) whou ay furher regulary assumos. (I) (A defo vald secal cases bu hghlghg he coce). s sad o be asymocally admssble wh resec o a gve asymoc easo ( ) f = a a o,, (9.) f s formal alcao o boh sdes of (9.) yelds a ew asymoc easo f = a a o,, (9.3) hs mlcly mles ha f ad ha he oeraor chages he asymoc scale (9.) o a ew asymoc scale,. (9.4) Pu hese erms he defo s well-osed f oe of he fucos ( ) f s he zero eleme of whch meas he fuco decally zero o some eghborhood of C. I geeral, o avod cosseces, he defo mus be modfed as follows. (II) (A geeral defo). Frs, f ( ) =, ha s f er, he he coce ueso s o defed. If hs s o he case he we u 486

7 { { } } ad say ha s asymocally admssble wh resec o (9.) f A. Graaa m: = ma,, : (9.5) m,, (9.6) f = a am m o m,, (9.7) afer suresso of all he zero erms. A alerave locuo for a asymocally-admssble s s formally alcable o he asymoc easo (9.) ; ad he valdy of (9.6) may be eressed by sayg ha reserves he asymoc herarchy (9.). A frs grou of eamles clarfes he ecessy of secfyg afer suresso of all he zero erms. I each of he followg hree eamles he sadard oeraor of dffereao dd s asymocally admssble accordg o Defo 9. oly f all he decally-zero erms have bee suressed. f : = log e, >, f ( ) = log o( ),, f ( ) = o( ),. f : = 3, >, f = 3 o,, f = 3 o,. f3 : = log, >, f3 = log o,, f 3 ( ) = o(, ). (9.8) (9.9) (9.) ha he sadard oeraor of dffereao does o reserve asymoc herarches s ue elemeary bu a secod grou of eamles shows ha may o reserve asymoc herarches eve whe acg o a -ule formg a Chebyshev asymoc scale (sgs aar) o a eghborhooh of whch, he eamles below, s ae as. ) Elemeary eamles showg ha f he ay asymoc cogecy may occur for he ar (, ) : α β ( ) : = ; ( ) : = ( α > β) ; (9.), ;,. : = ; : = ;, ;,. : = ; : = ;, ;,. (9.) (9.3) ( ) : = s cos ; ( ) : = e ;, ; oe of he wo lms : lm ( ) ( ), lm ( ) ( ), does es because of : ( ) = ( cos ). (9.4) 487

8 A. Graaa ) Eamles of Chebyshev asymoc scales ( ) such ha suable ermuaos of ( ) form asymoc scales:,,, 3, 3 : = log ; : = ; 3 : = ; 4 : = ;, ;, ) Eamle of a Chebyshev asymoc scale (,, ) such ha o ermuao of (,, ) asymoc scale: 3 3,, (9.5) forms a ( ) : = s cos ; ( ) : = e ; 3( ) : = e ; (9.6) 3, ; see ( 3.4 ). he above eamles are varaos o he eamles Bourba ([8]; Par V, 4,. V.-V.3). 4) Eamle of a Chebyshev asymoc scale (,, ) such ha (,, ) s a asymoc scale as well bu here ess a fuco f such ha f ( ) = a( ) o( ( ) ),, = f ( ) has o asymoc easo of ye a ( ) ; = (9.7) ad here ess a fuco g such ha Jus ae wh dffere values of α >. g( ) = b( ) o( ( ) ),, = g ( ) has a comlee asymoc easo, say g ( ) = b ( ) o( ( ) ),, wh <. =, ; f = g : = s α (9.8) (9.9) 9.. Asymoc Admssbly of Sadard Dervaves Le us as he ueso: Wha are he aural scales grag he asymoc admssbly of he sadard oeraors d d? Some of he above eamles show ha hese oeraors do o auomacally ur a gve Cheby- shev asymoc scale o a asymoc scale ad, so, a rash aswer o our ueso mgh sugges a scale ( ) (,,, ) or a scale ( ),, such ha (,, ) s aga a asymoc scale bu hs s glargly dsroved eve for he famlar scale a, f : = s α, α ; here we have ha f, bu o f, adms of a asymoc easo wh resec o he meoed scale. Of course a secal case occurs whe oe of he ses of oeraors eher L or M,, cocde wh d d ; he our oeraor, whch we deoe by he secal symbol D, adms of he facorzao for some fuco > ad of class, by a fuco such as ( ) u u Du,, (9.) AC o some erval; ad he erel of ( ) ( ) ( ) D s saed by,,,. (9.) 488

9 A. Graaa For or = we have resecvely he assocaed Chebyshev asymoc scales:, ; (9.),. (9.3) Here he oeraors d d are o be aled o o he fuco f whose easo s gve bu o he rao f. From formulas (.7), (.8) Par II-A, we ge he C.F. s of D. Lemma 9.. (I) If he (9.) s a C.F. of ye (II) a whereas he C.F. of ye (I) a s ( ) u D u ( ) ( ). (9.4) ( ) ( ) ( ) (II) If = he (9.) s he C.F. of ye (I) a whereas a C.F. of ye (II) a, assocaed o (9.3), s ( ) u Du. (9.5) ( ) he reader mus o h ha we are ow fllg a few ages wh rvales abou aylor s formula; as a maer of fac f we aly our heory o he oeraor D he case = we oba he resuls abou asymoc arabolas for he fuco f whose heory s horoughly suded []. Bu for he frs facorzaoal aroach characerzes a se of asymoc easos where (ue surrsgly) he esmaes of he remaders he dffereaed easos may follow algebrac rules dffere from hose vald boh he case = ad he case of he sadard aylor s formula; ad he secod facorzaoal aroach gves aylor s formula as a lm of aylor s formulas whch s a classcal elemeary resul o be commeed o our coe. I hs las case oe mus ay aeo o he fac ha formal alcao of he sadard dervave s geeral ermssble oly a umber of mes relaed o he growh-order of he remader he gve asymoc easo. See also eamles ad a fal comme a he ed of hs seco. heorem 9.. Le: ; f, AC [, [ ; > o [, [. he h-order weghed dervave assocaed o facorzao (9.4) s u L u: = ( ) ( ( ) D),, ; (9.6) ( ) ( ) whereas he oe assocaed o facorzao (9.) s he sadard dervave ( ) Cosder ow a geerc olyomal of order of ye Mu: = u,, ; (9.7) j j (9.8) j= = ( ) P : a. (I) (he couy roery of aylor s formula). he followg are euvale roeres for a fed {,, } : ) he se of asymoc easos as : ( ), ( ) f = P o ( ) ( f ( ) = P ) ( ) o ( ),. (9.9) 489

10 A. Graaa As cocers he boud whe comarg wh heorems oce ha, he rese seg, he =. ) he mroer egral (volvg eraed egraos) s of he geeral heory are gve by whch for = mus be read as ( ) ( ) ) O accou of he hyohess f AC [, [ whch s euvale o codo d d f d coverges, (9.3) f d coverges. (9.3), codo (9.3) or (9.3) s euvale o codo ( f ) ( ) lm, (9.3) ( j) f ( ) j { } e f C [ ] lm R,,,..,. (9.33) Hece relaos (9.9) are ohg bu aylor s formula of order of f a ogeher wh he sadard dffereaed relaos u o order obaed as he lm of he aylor s formula a he o ξ as ξ. (II) (A olyomal easo a a edo where dervaves may fal o es). he followg are euvale roeres for a fed {,, } : 3) he se of aymoc easos as : ( ), (( ) ) f ( ) = P o ( ) ( ) o,, L f ( ) = L P( ) ( ) o ma ( ; ). 4) he se of aymoc easos as : ( ), (( ) ) (( ) ) f = P o o, ( ) ( f ( ) = P ) ( ) o,, where he wo esmaes cocde for =, amely ( ) ( ). (9.34) (9.35) f = P o (9.36) he odffereaed easo s wre dfferely (9.34) ha (9.35) o correcly aly he oeraors L as defed (9.6). Noce ha for = he remader he secod relao (9.34) s o for, ad he remader he secod relao (9.35) s o( ( ) ) for. 5) he eraed mroer egral (volvg egraos) ( ) ( ) d d f d coverges. (9.37) We mus comme o he above clams. Par (I) s a classcal elemeary roery whch may be raced bac o Waler ad Ford ([9], Lemma II,. 35), 9, ad a roof s reored Auma ad Hau ([], Ch. 8, 8.9.., ) vald uder weaer regulary assumos volvg oly he esece of he hghesorder lef dervave of he gve fuco ad s lm as. Pu geomerc erms assers ha: If he osculag arabola of a cera order a a geerc o ξ adms of a lm oso as ξ he hs las s he lef osculag arabola of order a. hs fac hsorcally s he dea uderlyg he geomerc heory of lm arabolas a, see ([], ) where he wo ma resuls characerze easos volvg remader-esmaes a eher of he form ) ( f ( ) = o( ),, or f = o,. From a al- 49

11 A. Graaa gebrac vewo he frs sroger form follows he same formal rule as aylor s formula (9.9) ad relaos (9.35) for =, whereas he secod weaer form has o couerar for ad. he euvalece 4) 5) s o rval fac ad le us have a closer loo a he se of relaos (9.35) whch may seem srage ad eve correc a a frs sgh. For smlcy we u =. Frs, he owers aearg he o-erms decrease wh whch amous o say ha we have worse esmaes for hgher dervaves; ad hs s a aural heomeo. Secod, f for wo fucos f, f we have ( ) f = o, ; f = o, ; ( ) f fo each { } he he esmae for f s sharer ha ha for,,. hs s easly checed for = whereas for ad for < he wo esmaes cocde. hs fac smly says ha he esmaes (9.35) for dffere values of are cosse. Wha may seem uaural, o he corary, s he ga of wo us bewee he eoes sde he las o-erm (9.35) corresodg o cosecuve values of. Eamles. he followg smle eamles volvg oscllaory fucos wll reassure he doubful readers (ad he auhor hmself was he umber). For = ad = we have he euvalece a eamle beg rovded by { ( ) } s f o, f o, s ds f d coverges, = = (9.38) f ( ) f =, : = α s ( ) f >, ( < α < ), (9.39) whch s dffereable a = bu f has o lm as ad s ubouded. For = 3 ad =, we have he followg cogeces, assumg f AC o a deleed eghborhood of = : { f ( ) = o( ), f ( ) = o(, ) f ( ) = o( ), } d f d coverges; { f ( ) = o( ), f ( ) = o(, ) f ( ) = o( ), } ( ) d d f d coverges. I s obvous ha boh cases f ca be eeded so as o be of class ad a eamle for boh cogeces s rovded by he fuco: for whch α (9.4) (9.4) C o a comlee eghborhood of = f : = s, < α < 3, (9.4) = o = ; ; = ;. o f o f f o (9.43) A fal comme. he dscusso hs seco shows ha formal alcaos of ordary dervaves o a asymoc easo s o admssble geerally seag, ad eve for he very secal asymoc scale (9.) he frs (bu o he secod) facorzaoal aroach ca gve seemgly-uaural resuls. I s rcle rue ha each of he wo ses of easos characerzed Par II-A, 4,5, ca rovde asymoc formao (o always meagful ad o ecessarly easos) for he ordary dervaves; however hs s easly acheved for he frs-order dervave bu s raccally umaageable for hgher-order dervaves ad yelds o heorecal resul. I s also rue ha for easos arbrary real owers as wo lemmas of a algebrac characer erm o rasform each se of easos volvg he ere oeraors L or M o a meagful se of easos volvg he ordary dervaves ad here aga he frs facorzaoal aroach 49

12 A. Graaa yelds ucommo resuls, ([5], Lemma 7.3,. 96, ad Lemma 7.4,. ); bu as a ossble aalogue of heorem 9.-(II) for arbrary owers s comlcaed by he fac ha s ecessary o searae may a case for he eoes. O he corary, he use of weghed dervaves defed by caocal facorzaos yelds a cohere ad alcable heory.. Proofs Proof of heorem 8.. he oly hg o be roved s he ferece ) 5) 6), he oher roeres beg cluded heorems 4.5 ad 5.. We use a rocedure already used ([],. 93) ad ([5],. 3). From rereseao (4.5) we ge (usg he smlfed oao L L ): f L f d, [, [. (.),, c o = By he assumo (8.) he lef-had sde has a fe lm as, ad for he rgh-had sde we have: ( ) L f ( ) lm d ( ) ( ) d by.4 ad.43 L f lm = b P ( ) P( ) (.) L f ( ) ( ) d lm = b = = lm, b L afer alyg L Hosal s rule mes (whch s legmae as all he deomaors dverge o ). By he osvy of he egrad hs las lm ess ad cocdes wh he lm of he lef-had sde (.) hece mus be a real umber ad (4.5) ca ae he form: L f f = a c c d,,, (.3) [ [ wh suable cosas c,, c. From hs we ge: f ( ) a ( ) ( ) L f ( ) c o = d, [, [. (.4) Here aga he lef-had sde has a fe lm as whereas he lm of he rgh-had sde, by (4.), euals: 3 lm L, d L f lm b P ( ) P( ) L f ( ) ( ) d = lm b = = b 49

13 afer alyg L Hosal s rule ( ) ad (.3) ca be rewre as: A. Graaa mes. Hece hs las lm, whch ess, mus be a real umber = f a a c c L f d, [, [, (.5) wh suable cosas c,, 3 c. I s ow clear how hs rocedure wors ad by duco oe ca rove he valdy of rereseao: wh a suable cosa ad (.6) ur mles: = f a a c 3 L f d, [, [, c. As a las se we observe ha (8.) mles: (.6) f a a = O,, (.7) ( ) L f ( ) ( ) L f ( ) ( ) by 4. d d = O,. By he osvy of he egrad hs las relao mles (8.6) ad he frs rereseao (8.8) for (.8) R. o rove (8.7) we aly he same deas sarg from rereseao (5.) ad dvdg by ; recallg ha = we ge whch mles f ( ) L f d,,. (.9) [ [ c o = ad (5.) ca be rewre as L f d < ; (.) L f f a c c d, (.) = wh suable cosas c,, c. From hs we ge = f a L f (.) c o d. Evaluag he lm of he rgh-had sde by L Hosal s rule ad usg formula (.3), ge L f d H lm = lm L f ( ) ( ) d, =, we 493

14 A. Graaa ad hs las lm, whch ess, mus be a real umber. hs meas ha 3 L f d <, (.3) ad (.) ca be rewre as = f a a c c L f d, (.4) wh suable cosas c 3,, c. For he clary s sae we mae elc he ses hs secod ar of our roof. Assume by duco ha he followg wo codos hold rue: L f d < ; (.5) = L f ( ) ( ) ( ) f a a c c d, (.6) for some,, ad suable cosas c,, c. Dvdg boh sdes of (.6) by ad ag accou of (8.) we fer ha he lm of he uay L f d ( ) (.34) L f ( ) d ess. Alyg L Hosal s rule mes o evaluae hs lm we ge he ew lm lm L f d (.7) whch, by he osvy of he egrad, ess hece mus be a real umber. We fer ha codo (.5) holds rue wh relaced by ad hs mles rereseao (.6) wh relaced by ad suable cosas c,, c. By hs ducve rocedure we arrve a rereseao: wh some cosa = L f ( ) ( ) ( ) f a a c c. Dvdg by d,,, [ [ ad usg (.) we may ow coclude ha L f ( ) (.8) d = O(, ) (.9) ad f we ry o evaluae he lm of he rao o he lef alyg L Hosal s rule lm L f ( ) ( ) d mes we ge he, whch ess ad mus be a fe umber. hs s codo (8.7) whch al- lows he secod rereseao (8.8) for he remader (8.3). he roof s over. Proof of heorem 8.. he euvalece bewee (8.) ad (8.) easly follows from Fub s heorem by 494

15 A. Graaa erchagg he order of egraos (8.) whereas he euvalece bewee (8.) ad (8.) s by o meas a obvous fac. A cocse roof based o heorem 8. s as follows; ug we have F( ) : = L f ( ) d, [, [,,, (.) [ [ [ [, ; a.e.o, ; (.) F AC L,, F = L,, f,, ad heorem 8. mles he euvalece bewee (8.) ad (8.). Now we rove relao (8.3) recallg ha all he volved fucos ad Wrosas are srcly oe-sged o he erval. he symbol of asymoc euvalece s referred o of course, ad, wheever used, s graed by he herarches of he Wrosas (.4) ad he dvergece of he volved egrals. We reor a classcal dffereao formula used Prooso.4: hece F sasfes L F( ) a.e. o [, [ W( g( ),, g ( ), g ( ) ) W g,, g W g,, g =,, W( g ( ),, g ( ), g ( ) ) W g,, g ( ) vald for ay ordered ( ) -ule of fucos (,,, ) es. Now for =,3 we have: (.) g g g a ay o where he reured dervaves d d P ( ) ( ) = (, ) ( ) W( ) =, ; W (.3) d d P 3( ) = ; 3 W( 3, ) W( 3,, ) ; 3( ) ( ) = ( ) ( 3) ( ) 3( ) ; W( 3) W( 3,, ) d = W ( 3, ) W (, 3, ) W(, ) W(, ) d d, W ( ) (, 3) (, 3), W W 3 where he las bu oe assage we have aled formula (.) o he ordered rle (, 3, ) W ( 3, ) W(, ) W(, ) P3 ( ) =. W(,, ) W(, ) W(,, ) Hece (.4) (.5) Now, for a fed 3, we use he rocedure (.4) o rove by duco o ha (,,, ) (,,, ) W,. W (.6) As (.4) we ca rove ha (.6) s rue for = ; assumg o be rue wh relaced by we have: 495

16 A. Graaa d d W (,,, ) (,,, ) W(,,,, ) W (,,, ) (,,,, ) W,,, W,,, W,,, W W ( ) W( ),,,, W(,,,, ) d,,,, W(,,, ) W ha s (.6). Usg hs relao for = ad he eresso (.43) for we, fally, ge: P ( ) W W ( ) ( ) d ( ) ( ),,, W,,,,, W,,, ha s (8.3). Proof of heorem 8.3. he oly hg o rove s he O-esmaes (8.7). From rereseao (5.) L f M f( ) c M ( ) cm ( ) = d for some cosa c, whece he esmae for M f ( ), L f = = by 3.3 ad 3.4 d c o d (.7) follows. Ad so o for he oher esmaes. Proof of heorem 8.4 s coaed he followg lemma oly vald uder he saed oe-sgedess resrcos. Lemma. (Growh-order esmaes for eraed mroer egrals wh oegave egrads). Assumos: [ [, loc,, ;, ; h g L m (.8) [ [ [ [ h > a.e. o, ; g a.e. o, ; (.9) h d <, m ; (.3) m m H( ) : = h( ) d ( ) d m( m) d m d, ; h h g m Hm ( ) : = g; hess. he followg esmaes hold rue: (.3) H coverge ad H dverge as, m. (.3) H( ) o h =, ; H( ) = o h h h,, m. (.33) 496

17 Proof. All fucos For H we have he smle esmae H are oegave ad odecreasg ad sasfy A. Graaa H = h H, m. (.34) H h d h H d = by he covergece of H = o, whch mles he frs relao (.33). o esmae H 3 we egrae by ars as follows 3 = h H d H d h H h h h H d. Now he egral o he lef s coverge by hyohess ad he frs egral o he rgh s coverge by (.34), hece he secod egral o he rgh coverges as well ad we ge he eualy whece, aga by (.34), he odecreasgess of H 3 mles = 3 h H d H h h h H d, (.35) = H d. 3 h h o (.36) (.37) H h h d H h h d = o, 3 3 whch yelds he esmae for H 3. Relaos (.36)-(.37) gve he ey o roceed by duco. Pug we wa o rove H H( ) o (.36), whch reads H ( ) H ( ) o whece = H : h h h, (.38) =. For he sae of smlcy, we wre ou he calculaos for H 4 usg 3 3 =, ad egrag by ars he egral aearg (.36): d d Ad ow he odecreasgess of H 4 mles H h h H H = H H H h H d = H H H h H d, = H h H d o. (.39) H ( ) 4 H h H d H h H d = o. Relaos (.39)-(.4) allow erao of he rocedure o esmae H (.4) 5 ad so o. Proof of heorem 9.. Par (I) s ohg bu heorems 5. ad 5. aled o he rese case where f s relaced by f ad ( ). he euvalece bewee (9.3) ad (9.33) s a elemeary fac uder he ac assumo f ( ) ( ) : = a for =. I ar (II) relaos (9.34) are hose gve heorems secalzed o our case ad he oly hg o be roved s he euvalece 3) 4) coaed he e lemma where we smlfy all formulas by assumg = ad. Lemma.. Le f be of class AC o some deleed eghborhood of = ad le 497

18 A. Graaa For a fed {,,, } Lu : = ( u ) wh dervaves,,. (.4) he followg se of asymoc relaos as :,, o f ( ) = o( ); L f ( ) = o( ), ma ( ; ), (.4) s euvale o he se of asymoc relaos, as, volvg oly sadard dervaves: ( ), o f = o ; f ( ) = o,, (.43) where he wo esmaes cocde for =, amely ( ) f = o. (.44) For L f = o for all, ; ad (.43) we have ( ) f ( ) = o( ), for all,. Proof. I s easly roved by duco ha he eaded eressos of he L s are = (.4) we have Lu u u; Lu u c, u c, u; ; j ( j) Lu c, j u, c, =, j= (.45) wh suable cosas c, j whose elc values are o eeded our calculaos. Le us ow rove (.4) (.43). Frs se. From he frs eualy (.45) we ge 3 f ( ) = L f ( ) ( ) f ( ) whece o o = o f, = o o( ) = o f =, 3 o f, f ( ) = o f =. Secod se. From he secod eualy (.45) we fer = f L f c, f c, f o o o = o, 3, = o o o = o, =, o o( ) o( ) = o( ), =, havg used (.47), ad from hese las esmaes we ge 5 o, 3, f ( ) = o,. (.46) (.47) (.48) (.49) 498

19 A. Graaa Suose ow ha he relaos (.43) for he dervaves have bee roved rue for ( ) <, ad le us rove he corresodg relaos for f. From (.45) we ge ( ) f, f,, f, ( ) 3 j =, j j= ( j ) L f f c f, (.5) whece 3 ( ) =, j j j= ( j ) f L f c f. (.5) Now for ( ) we use he frs esmae (.4) for L f ( ) ( j (.43) for he dervaves f ) ( ) so geg from (.5) 3 ( ) j j f o o o j= ad he frs esmaes = = f. (.5) For he remag values of we mus use he secod esmae (.4) for L f ( ) ( j esmaes (.43) for f ) ( ) so geg from (.5) 3 ( ), j j j= ( j ) ad he suable f = o c f = (.53) ( j) Now sde he sum each f mus be relaced by oe of he wo esmaes (.43), assumed o be rue, ad we have wo ossble cogeces ( j ) j f j j j j j o = o f j, = o = o f j. (.54) I he frs case he resrco mles, ad he secod case he resrco j mles j as well. I each case he whole sum (.53) s o( ) as ad (.53) gves ( ) = f. (.55) 3 f o Fally from (.5) we ge he sough-for esmaes for f ( ) ( ) f ( ) ( ) ( ) o f, = o f. : (.56) he roof of (.4) (.43) s over. he coverse mlcao s checed a oce relacg he esmaes (.43) o he sum (.45) eressg L f ( ) : j= j j o o( ) = j o( ) o j= j = o. (.57). Some Remars o Facorzaoal heory.. O he Use of No-Caocal Facorzaos We show by wo eamles ha he use of o-c.f. s s urelable o cosruc a geeral heory. Le us refer, e.g., o he characerzaos of a asymoc easo for a geeralzed cove fuco. 499

20 A. Graaa Frs eamle. For f AC [, ), we ow ha f = a b c o, < f f d, (.) ad ha he ferece ca be easly roved whe a C.F. of he oeraor Lu : = u of ye (I) a s used, see ([], roof of h.4.). Suose ow o use he followg facorzao (whch s o C.F. a ) ad he relaed rereseao 3 u ( u ) u AC, ), (.) 3 = d d d. (.3) f c c c f 3 Assumg he valdy of he easo (.), we ry o fd a ecessary egral codo volvg f. From (.3), we ge ad by reasos of cosa sg, we fer a much weaer codo ha by ( 5.4) 3 lm = lm d d d f c f H. 3 s = c lm s ds f d, 3 s s f ( ) d d s <, (.4) f d <. Hece, hs eamle he used facorzao does o allow o characerze he easo a had. Secod eamle. Le us cosder he oeraor Lu : = u u ad he followg hree facorzaos: We have er L sa ( e ; ;) a global facorzao o, Lu e ( e u ) : ad he C.F. of ye I a ; a facorzao vald o, Lu ( ) ( ) ( ) e ( e u ) : ad a C.F. of ye ( II) a ; a facorzao vald o, Lu e e ( u) : bu o C.F. a. ] ) = ad he followg characerzao for a f AC [, ) f ( ) = ae b c o(, ) f ( ) f ( ) f ( ) f ( ) ] ) (.5) (.6) (.7) d <, (.8) obaed from he resuls 8 based o he use of C.F. s. However, hs case he euvalece (.8) ca be also obaed usg he o C.F. (.7) f oe sars from he corresodg egral rereseao for f ad reales he same rocedure used he roof of heorem O he Use of Iegral Rereseaos Iferred from Facorzaos We sar ocg ha he covergece of a eraed egral d g( s) d, s ( ) (.9) 5

21 A. Graaa where >, =, <, he case of codoal covergece, may deed a uredcable way o he choce of as he followg wo elemeary eamles: α cos sdscoverges ( = π, ) f < α < ; (.) ( = = ± ) d s s s ds coverges π, Z, or. (.) Now a egral rereseao led o a C.F. of ye (I) ad more geeral ha (4.5) Par II-A s f ( ) = c ( ) c ( ) L f ( ) d, [, [, (.),, where he c s are suable cosas ad he fed edos are such ha. From hs rereseao, oe fers a oce ha,,, <, such ha he eraed egral If here ess a -ule L f ( ) d coverges, (.3),, ( ) he, recallg ha ( ) = b ( ), we ge ( ) f = a a o,, (.4) wh a arorae rereseao of he remader. Bu, by he al remar, such a egral codo s almos useless for geeral resuls as well as for raccal alcaos f he s are fed a ror ad dsc from ; may be well fulflled by some very secal f wh a ocllaory Lf, bu cao be sasfed by ay f such ha Lf > o maer how small s order of growh a : fac f he mroer egral L d f,, rereses a osve umber, he he eraed egral (.3) dverges. Moral. Worg wh he C.F. of ye (I) a, he sole egral codos whch ca be used for suffcely geeral resuls are hose aearg heorem 4.4 Par II-A. he suao s echcally dffere whe worg wh a C.F. of ye (II). Referrg o a egral rereseao of ye f ( ) = c ( ) c ( ) L f ( ) d, [, [, (.5),, more geeral ha (5.) Par II-A, we see ha as soo as we may choose = for some,.e. f some of he ermos mroer egrals coverge, he auomacally he remag ouer egrals coverge as well. Moreover, a codo le L f ( ) d coverge (.6),, ( ) does o whmscally deed o,, f hey are dsc from : f oe of hem s allowed o cocde wh, hs smly meas ha codo (.5) may be relaced by a sroger codo yeldg addoal asymoc formao. Refereces [] Graaa, A. (5) Aalyc heory of Fe Asymoc Easos he Real Doma. Par II-A: he Facorzaoal heory for Chebyshev Asymoc Scales. Advaces Pure Mahemacs, 5, [] Graaa, A. (7) Polyomal Asymoc Easos he Real Doma: he Geomerc, he Facorzaoal, ad he Sablzao Aroaches. Aalyss Mahemaca, 33, h://d.do.org/.7/s [3] Graaa, A. () Aalyc heory of Fe Asymoc Easos he Real Doma. Par I: wo-erm Ea- 5

22 A. Graaa sos of Dffereable Fucos. Aalyss Mahemaca, 37, h://d.do.org/.7/s [4] Graaa, A. () he Problem of Dffereag a Asymoc Easo Real Powers. Par I: Usasfacory or Paral Resuls by Classcal Aroaches. Aalyss Mahemaca, 36, 85-. h://d.do.org/.7/s [5] Graaa, A. () he Problem of Dffereag a Asymoc Easo Real Powers. Par II: Facorzaoal heory. Aalyss Mahemaca, 36, h://d.do.org/.7/s [6] Poovcu. (944) Les Focos Covees. Herma & C e Édeurs, Pars. [7] Graaa, A. (5) he Facorzaoal heory of Fe Asymoc Easos he Real Doma: A Survey of he Ma Resuls. Advaces Pure Mahemacs, 5, -. h://d.do.org/.436/am.5.5 [8] Bourba, N. (976) Focos d ue Varable Réelle héore élémeare. Herma, Pars. [9] Waler, M. ad Ford, B. (9) Codos Suffsaes our u ue Foco Admee u Déveloeme Asymoue. Bulle de la Socéé Mahémaue de Frace, 39, [] Auma, G. ad Hau, O. (974) Eführug de reelle Aalyss. I: Fuoe eer reelle Veräderlche. Waler de Gruyer, Berl. h://d.do.org/.55/