Hardy Spaces, Hyperfunctions, Pseudo-Differential Operators and Wavelets
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1 Hrdy Sps, Hyprfuntions, Psudo-Diffrntil Oprtors nd Wvlts Colltions from lrtur Th Hrdy Sp nd Hilbrt Sls Lt : s /, t thn th Rimnn Hypothsis is th sttmnt tht / ( s is nlyti on th hlf-pln Th ppropr Hilbrt sp frmwork is th Hrdy sp H ( of ll nlyti funtions F on suh tht F sup / F( dt Any F H ( hs lmost vrywhr on th ril lin non-tngntil boundry vlu funtion F ( t : lim F( L ( R (dfind lmost vrywhr suh tht / F F ( dt Thus th Hrdy sp ( H my b idntifid vi th isomtri mbdding F F wh losd subsp of th L -sp of th ril lin wh rspt to th Lbsgu msur sld by th ftor / W not tht th oprtor H of th prvious stion pplid to ompl-vlud funtion produs s onjugt ompl funtion On dfins th Fourir-Mllin trnsform : L (, H ( M by: M ( f ( s : f ( d s, f L (,, s, whrby M is n isomtry Th rltd Fourir-Hilbrt sl thory is built on th Rimnn mpping thorm This ssrts tht ny opn rgion in th ompl pln, boundd by simpl losd loop, n b mppd holomorphilly to th intrior of th un irl th boundry bing lso mppd ordingly D :,
2 Du to rsult from Hrdy th mn funtion i ( r : r d, is inrsing, i s hr divrgnt or is boundd, s r for u ( bing rgulr, nlytil funtion to th intrior of th un irl, i on th opn disk D : Thn th Hrdy sp H ( onsists of thos funtions, whos mn squr vlu on th irl of D rdius rmins boundd s r Lt H ( b th Hrdy sp of D L funtions on th un yl wh n nlytil ontinution insid th un disk D Th innr produt is dfind s follows: u, v : t v( t dt For point D lt (t st of funtions dfind by ( t : Applying th Cuhy intgrl formul thn th funtions (t dfin linr ontinuous mpping C : H ( H ( D of funtions on to n nlytil funtion in D dfind by: whrby C( u ( : u u ˆ( :, u, t ( dt t i dt i y dy y This mpping C is n isomtry of th Hilbrt sps H ( nd H (, whr th innr produt on H ( is dfind by D u, v : lim r r v( r dt Th mpping C ould b invrtd by n oprtor C H ( D H (, t lim uˆ( r : r
3 Th rproduing oprtor : C C on H ( is lld th Sgö singulr intgrl oprtor S P Considrd on H ( th Sgö oprtor S P is n orthogonl projtion on s losd sub sp H ( For H L ( nd s losd vtor subsp H H L ( ( H, th following hrtrion holds tru : u H ( if nd only if u for Supposing tht u ~ H (, i tht u ~ hs Fourir offiints wh u ~ for, thn th lmnt u of th Hrdy sp ssod to u ~ is th holomorphi funtion u ( u, Th proprtis of th Hilbrt trnsform lds to u ( H ( u, Rmrk: For ompl vlud funtion onjugtd funtion n b rprsntd by Lt ;, b n priodi funtion f ( iv( f ( lim f ( f ( ot d f ( ot d i i, b th Fourir offiints of f Thn n n n onjugt nd holds, s ; b, r th Fourir offiints of s n f ( d f ( d rsp f ( d n bn Rmrk: In th on-dimnsionl s hyprfuntions r th distributions of th dul sp C of th rl-nlytil funtions of rl vribl C, dfind on som onntd sgmnt R In th on-dimnsionl s ny ompl-nlytil funtion, s ny distribution f on R, n b rlid s th jump ross th rl is of th orrsponding in C R holomorphi Cuhy intgrl funtion F( : i f ( t dt, t givn by ( y f, lim F( iy F( iy ( d 3
4 Empl: Th prinipl vlu P v(/ of th not lolly intgrbl funtion is th distribution g dfind by ([BP] d ( g, : lim ( log ( d for h C Th rltion of this spifi prinipl vlu to th Fourir trnsform is givn by P v( i sgn( t nd P v( P v( Rmrk: Th Dir distribution funtion n b intrprtd s th jump ross th rl is of orrsponding holomorphi Cuhy intgrl funtion in C R : Lmm: If C nd thn y ( d ( d y rg( iy In th on-dimnsionl s hyprfuntions r th distributions of th dul sp th rl-nlytil funtions of rl vribl R Any rl-nlytil funtion is holds C of C, dfind on som onntd sgmnt C, but not vry funtion C is nlytil, g ( : C but ( C ( n From ( for ll n for th Tylor sris follows n! n, wht s diffrnt to ( pt t, i ( C is not n nlytil funtion Th sution is diffrnt in s of ompl-nlytil funtions, whih r holomophi nd nlytil t th sm tim This mns tht th dul (distribution sp C of th sp of th rl-nlytil funtions C hrtris th so-lld hyprfuntions A hyprfuntion of on vribl f ( on n opn st R is forml prssion of th form F ( i F ( i, whr F ( is funtion holomorphi on th uppr, U U Im(, for ompl nighborhood rsptivly lowr, hlf-nighborhood U stisfyingu R Th prssion f ( is idntifid wh if nd only if F ( grs on s holomorphi funtion 4
5 If th lims ist in distribution sns, th formul givs th nturl imbdding of th sp of distributions into tht of hyprfuntions Hyprfuntions n b dfind on rl-nlyti mnifolds Fourir sris r typil mpls of hyprfuntions on mnifold: i ( onvrgs s hyprfuntion if nd only if O( for ll Z Som mpls of gnrlid funtions intrprtd s hyprfuntions r i Dir s dlt funtion k ( lim os kdk i i i, i ii Hvisid s funtion Y( log( i log( i log( i Th Hvisid funtion n b hrtrid ([BP] B E Ptrsn, 6 by lim log( iy log iyˆ for y nd Yˆ ( Y ( iii ( isin for Z m m ( ln( for m Z i iv th Fynmnn propgtor (Grn s funtion is th solution ( S S i of th distribution wv qution ( t m S( t, ( t ( wh ( m S ( t, ik dkd ( k i ( k i ( m S ( t, ik dkd ( k i ( k i 5
6 Psudo-Diffrntil Oprtors Th lss of distributions, whih is dfind by divrgnt intgrls, is th lss of osilltory intgrls lding to th onpt of Psudo-Diffrntil oprtor Thy r in th form A( (, d i(,, whr th phs funtion (, is subl rl vlud funtion suh tht th intgrnd osillts rpidly for lrg nd th mplud funtion (, bing llowd to hv polynomil growth in It would b too rstriv to rquir th intgrl to dfin funtion Thrfor s intrprtd in th distribution sns Thus on is tully b onrnd wh intgrls of th typ A, v (, v( dd i (, Th study of th Hilbrt trnsform nd th study of oprtionl lulus for non-ommuting oprtors in quntum mhnis ontin som bsi ingrdints of th thory of psudodiffrntil oprtorsth Hilbrt trnsform is lssil psudo-diffrntil oprtor wh symbol isign(s Its slint fturs nbld th introdution of th lgbr of singulr intgrl oprtors Singulr distributions n b gnrtd by Hdmrd s fin prt of divrgnt intgrl; thniqu for trting fin prt from divrgnt prt, building psudo funtions ppying Cuh s prinipl vlu onpt, whr turns out tht this fin prt dfins singulr distribution W not (if( th fin prt rprsnttion ( t ( t Fp dt lim( dt ( log t t Holomorphi funtions in th distribution sns r dfind in th following wy: Dfinion : Lt g b funtion dfind on opn subst U C wh vlus in th distribution sp Thn g is lld holomorphi funtion in U C (or g( : g is lld holomorphi in U C in th distribution sns, if for h C th funtion ( g s, is holomorphi in U C in th usul sns Th phs funtion (, of osilltory intgrls is subl rl vlud funtion suh tht th intgrnd osillts rpidly for lrg nd th mplud funtion (, bing llowd to hv polynomil growth in It would b too rstriv to rquir th intgrl to dfin funtion 6
7 Wvlt A wvlt is funtion ( L ( R wh Fourir trnsform whih fulfills ˆ( : d Clssil Hilbrt sps in ompl nlysis r mpls of wvlts, lik Hrdy sp of L funtions on th un irl wh nlytil ontinution insid th un disk Th wvlt trnsform of funtion f ( L ( R wh th wvlt ( L ( R is th funtion W f t b (, b : f ( t b, ( t dt f ( t ( dt For wvlt ( L ( R s Fourir trnsform is ontinuous nd fulfills, R, b R ˆ( Th wvlt trnsform to th wvlt ( L ( R is isomtri nd for th djoint oprtor W g (, b : W W (t d ddb : L ( R L ( R,, ddb : L ( R, L ( R t b g( t ( g(, b ddb holds W W Id nd W W P rng ( W Th ontinuous wvlt trnsform is known in pur mthmtis s Cldrón s rproduing n formul, i for L ( R rl nd rdil wh vnishing mn, i ( ˆ( d For ( : ( n holds Cldrón s formul f d f 7
8 Rimnn-Stiltjs intgrl dnsis nd Hyprfuntions W brifly skth th link btwn Rimnn-Stiltjs intgrl dnsis nd hypr funtions nd distributions in ordr to motivt th svrl following dfinion: Lt ( : in (, b boundd vrion sptrl funtion, whih builds ording to th Grn funtion G( d ( th two holomorph Cuhy-Rimnn rprsnttion in R( s, R( s by G ( iy G( iy d ( ( iy ( iy Thn th Stiltjs invrs formul is vlid for ontinuous points ndb, i b ( b ( lim G iy G iy d y i ( ( If thr ist sptrl dnsy funtions (, holds E ( lim G( i G( i i In th on-dimnsionl s ny ompl-nlytil funtion, s ny distribution f on R, n b rlid s th jump ross th rl is of th orrsponding in C R holomorphi Cuhy intgrl funtion F( : i f ( t dt, t givn by y ( f, lim F( iy F( iy ( d 8
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