On the Frame Properties of System of Exponents with Piecewise Continuous Phase

Transcription

1 Alid Mahmaics h://dxdoiorg/436/am3456 Publishd Oli May 3 (h://wwwscirorg/joural/am) O h Fram Proris of Sysm of Exos wih Picwis Coiuous Phas Sad Mohammadali Farahai Tofig Isa ajafov Isiu of Mahmaics ad Mchaics of ASA Bau Azrbaija ahchiva Sa Uivrsiy ahchiva Azrbaija sadzfarahai@gmailcom ofi-cfov@mailru Rcivd Jauary 4 3; rvisd Aril 3 3; accd Aril 3 Coyrigh 3 Sad Mohammadali Farahai Tofig Isa ajafov This is a o accss aricl disribud udr h Craiv Commos Aribuio Lics which rmis ursricd us disribuio ad rroducio i ay mdium rovidd h origial wor is rorly cid ABSTRACT A doubl sysm of xos wih icwis coiuous comlx-valud cofficis ar cosidrd Udr dfii codiios o h cofficis h fram rory of his sysm i Lbsgu sacs of fucios is ivsigad Such sysms aris i h scral roblms for discoiuous diffrial oraors Kywords: Sysm of Exos; Fram Prory; Prurbaio Iroducio Cosidr h followig sysm of xos i () Z C is a suc of comlx umbrs Z ar igrs Sysms () ar modl os whil sudyig scral roris of diffrial oraors Udr suiabl choic of h boudd variaio fucio o h sgm aa hy ar igfucios of firs du ordr diffrial oraor Du wih a igral d codiio of h form u d a a For his raso may mahmaicias aald o sudy of basis roris of h sysms form () i diffr sacs of fucios If h oraor D is cosidrd i h Lbsgu sac L a a h is aural domai of dfiiio is h Sobolv sac W a a i h sac cosisig of absoluly coiuous o a a fucios whos drivaivs blog o L a a ad h rlaio du Du u () d holds a o all h sgm aa Aarly h firs rsuls for basis roris of h sysms of h form () i h sacs L L C a a blog o h famous mahmaicias Paly P- Wir [] ad Lviso [] I sul his dircio was dvlod i h ivsigaios of may mahmaicias For mor daild iformaio s h moograhs of R Youg [3] A M Sdlsii [4] Ch Hil [5] O Chriss [6] (ad also h ars [7-9]) ad hir rfrcs Thr is also h survy ar [] May roblms of mchaics ad mahmaical hysics rduc o discoiuous diffrial oraors i o h cas wh h domai of dfiiio of a diffrial oraor is o cocd I should b od ha h sysms of h form i (3) Z as has h rrsaio sig (4) aris as ig fucios of aroria diffrial oraors whil solvig may roblms of mchaics ad mahmaical hysics by h mhod of saraio of variabls Th followig sysm is a rivial xaml of h cas udr cosidraio si s cos L J J I is obvious ha s ar h ig fucios of h followig scral roblm Coyrigh 3 SciRs

2 S M FARAHAI T I AJAFOV 849 wih a scrum i boudary codiios u u J J uu u u u u Cocrig hs issus s also h ars [-4] Aohr rmarabl xaml is cosidrd i V A Ili s ar [5] Hr h cosidrs a mixd roblm wih cojugaio codiios a h ir oi x l wih rsc o h wav uaio u axu x x x l T wih codiios xx ux x ul a u x u ux u x x u u x ux x x a x x ax a x x l a a (wav vlociy i mdium) ad (mdium dsiy) ar osiiv cosas a ar Youg moduls wih addiioal codiio of ualiy of assag im of wav h sgms x ad x l : x l x a a Th comlss i L of h sysm of igfucios of a ordiary diffrial oraor ha corrsods o his roblm is sablishd i h ar [6] Th clos class of roblms was arlir cosidrd i h ar [7] Ths xamls vry clarly dmosra xdicy of sudy of fram roris of h sysms form (3) Th rs ar is dvod o ivsigaio of fram rory of sysm (3) i L L Prviously som rsuls of his ar wr aoucd wihou roof i [8] This wor is srucurd as follows I Scio w rs dful iformaio ad facs from h horis of bass ad clos bass ha will b usd o obai our mai rsuls This scio also coais h mai assumios abou h fucios ad which aar i formula (4) I Scio 3 w sa mai rsuls o h basiciy of h rurbd sysm of xos (3) i Lbsgu sacs L cssary Iformaio ad Mai Assumios I sul w will d h followig oio ad facs from h hory of bass ad frams W will us h sadard oaio will b h s of all osiiv igrs; will ma hr xis(s) ; will ma i follows ; will ma if ad oly if ;! will ma hr xiss uiu ; K R or K C will sad for h s of ral or comlx umbrs rscivly; is Krocrs symbol Th Baach sac will b calld a B-sac X is a sac cojuga o sac X By LM w do h liar sa of h s M X ad M will sad for h closur of M Dfiiio Sysm x X is said o b a ba- sis for X if x X! Dfiiio Sysm x l i X if K : x x X is said o b com- L x X I is calld miimal i X if x L x Dfiiio 3 Sysm x X is calld -li- arly idd i B -sac X if from ax imlis a I holds h followig Lmma L X b a B-sac wih h basis x ad F : X X b a Frdholm oraor Th h followig roris of h sysm y Fx i X ar uival: is coml; ) y ) y 3) y 4) y is miimal; is -liarly idd; a basis isomorhic o x W will d h followig oios Dfiiio 4 Th sysms x ad y i a B-sac X wih h orm ar said o b -clos if x y Dfiiio 5 Th miimal sysm x X i a B-sac X wih cojugad x X is said o b a x X : xx l l is a -sysm if for ordiary sac of sucs a of scalars wih h orm a a l I h cas of basiciy such a sysm will b calld a -basis Th followig lmma is also valid Lmma L X b a B-sac wih -basis x ad h sysm y X b -clos o i: Coyrigh 3 SciRs

3 85 S M FARAHAI T I AJAFOV Th h xrssio Fx x xy g- ras a Frdholm oraor i X x X is a sysm cojugad o x O ca s hs or ohr facs i h moograhs [39] ad also i h a rs [7-] W will d h followig Kri-Milma-Ruma s Thorm [] Thorm KMR X b a B-sac wih h orm ad wih h ormd basis x x X b a sysm biorhogoal o i If h sysm y X saisfis h codiio x y su x h i for rhic o ms a basis isomo x for X Whil obaiig h basic rsul w will us h fol- lowig asily rovabl lmma Lmma 3 L X b a B-sac wih h basis x ad x X b a sysm biorhogoal o x Th sysm y X diffr from x by a fiily may lms i y x Th if d x y h sysm y is o miimal i X Proof So X b a B-sac wih h basis x ad y X diffr from x by fiily ma y l ms i y x Exad y by his basis y y a x y (5) a x L A firs assum ha Th i is o bvious ha a As a rsul i follows from xrssio (5) ha y blogs o h c losur of h liar sa x ad so h sys m y is o miimal Cosidr h cas i y a x a x y y a x a x y (6) aa aa I is obvious ha if a a for or h h sysm y is o miimal Ohrwis xcludig x i (6) w hav: a y a y x a y a y a y ay a y ay a y ay I dircly follows from hs rlaios ha y y blogs o h closur of liar sa of h rmaiig lms y i is o mii- y Coyrigh 3 SciRs y mal i X Cosuly for h sysm y dos form a basis This rasoig is a o a arbirary v ry asily ڤ Bfor rocdig h mai rsuls w acc h followig basic assumios cocrig h fucios of ad ) is a icwis-holdr fucio o r : r r ar is discoiuiy ois of firs id; Do h jums of h fucio a h ois r r by : L h codiio ) Z r b fulfilld 3) Th fucios hav h followig asymoic rlaios O (7) 3 Basic Rsuls A firs w cosidr h sysm of xos i Z (8) sig Z For h basiciy of sysm (8) i L h rsuls of h ar [3] will b usd Rrs sysm (8) i h form i i i i ; Z (9) ( Z ar o-gaiv igrs) L h codiio ) b r fulfilld Fidig i Z from h followig i- ualiis : assum i i i i r () r () Basd o Thorm of h ar [3] w ca dircly coclud h followig Sam L h codiios ) ) b fu lfilld for h fucio Suos ha Th sysm (9) forms a basis for L (for = a Risz ba- sis) if ad oly if i holds h iualiy W will us h followig sam obaiig from h rsuls of h ar [4] Sam If sysm (9) forms a basis for L

4 S M FARAHAI T I AJAFOV 85 h i is isomorhic o h classic sysm of i xos Z So l sysm (8) form a basis for L Do by L a sysm biorhogoal o i L Z f L a d b is biorhogoal cofficis by f Z sysm (8) i f f d Z is comlx cojugaio Th followig horm ca b dircly coclud d from Sam Thorm L sysm (8) forms a basis for L Th hr hold: ) L ad f L Th f l ad Z f m f Z l is fulfilld m is a cosa id d of f is a ordiary orm i L ) L ad h suc of umbrs a Z blog o l Th f L such ha f a Z morovr f M f M is a cosa Z l idd of f Z ow sudy h basiciy of sysm (3) i L W hav i i i M! c! c is a cosa idd of Th las iualiy follows from (7) Cosidr h diffr cass ) L W hav i i c Assum ha all h codiios of Sam ar fulfilld Th sysm (8) forms a basis for L Thus by Sam i forms a -basis for L i his cas L b a sysm biorhogoal o i Cosidr h Z oraor F : L L : i () Ff f f L f f d Z By Lmma oraor () is Frdholm i L I is asy o s ha i i F Z Th h sam of Lmma is valid fo r sysm (3) ) L I is clar ha Cosuly for f L i is valid f c f c dd s o ly o Assum ha all h codi- ios of Sam ar fulfilld Cosuly sysm (8) forms a basis for L I is clar ha f L ad Th from Thorm w obai ha f l Z f ar h orhogoal coffiz cis of f by sysm (8) From h sam ho rm w obai: f m f M f f L Z l h cosa M is idd of f Thus sysm (8) forms a -basis i L I is asy o s ha sysms (3) ad (8) -clos i L Cosidr oraor () Furhr w bhav similarly o cas I Hc h validiy of h followig horm is rovd Thorm L asymoic Formula (4) hold h fucio saisfy h codiios ) ) ad for h fucio h rlaios (7) b valid Assum ha i holds max ; mi is dfid from xrssios () () Th h followig roris for sysm (3) i L ar uival: ) Coml; ) Miimal; 3) -liarly idd; i 4) Forms a basis isomorhic o I sul w will cosidr a cas wh I his cas i is obvious ha i holds Z i i L all h codiios of Thorm b fulfilld Th h i sysm forms a basis for L Do by Z Z L a sysm biorhogoal o i Assum su I is clar ha : i i Cosidr h fucios Thus i holds i i Th as i follows from Thorm KMR h sysm i Z forms a basis isomorhic o i Z for Coyrigh 3 SciRs

5 85 S M FARAHAI T I AJAFOV L Sysm (3) ad h basis i diffr by a fiily may lms By do a biorhogoz al sysm o his basis Cosidr Z i i i a a (3) : I is obvious ha i i a d Z Do by h followig drmia I is clar ha if i lms d a ij i j (4) i h xasio (3) h may b rlacd by h l- i ms i Th h sysm Z forms a basis for L sic f L has h xasio f f i Hc i dircly follows ha if h f L has a xasio by s ysm (3) i i is com l i L Cosidr h oraor Ff f i W hav i i Ff f f i i f f I T f i I : L L is a idiy oraor ad T is a oraor grad by h scod summad Frdholm rory F i L follows from fii-dimsioaliy of h o raor T I is clar ha i i Z F Th from Lmma w obai h basiciy of sysm (3) i L Covrsly if sysm (3) forms a basis for L h as i follo ws from Lmma 3 Thus w sablishd ha udr accd codiios sysm (3) forms a basis for L if h drmia drmid by xrssio (4) is o zro Thus w rovd h followig Thorm 3 L all h codiios of Thorm b fulfilld Th drmia is drmid by xrssio (4) Sysm (3) forms a basis for L if ad oly if ow cosidr h cas wh L for xaml I his cas as i follows from Thorm of h ar [3] h sysm forms a basis for i i Z (5) L Cosidr h sysm i (6) Z L is a fucio L h codiios ) ) b fulfilld for sysm (3) ad max ; Th i is asy o s ha sysm (6) ad basis (5) ar -clos i L is drmid by h formula Cosuly sysm (3) is o coml i L Th rmaiig cass wh ar rovd i h similar way Cosidr a cas wh for xaml I h is cas agai as i follows from Thorm of h ar [3] h sysm i (7) forms a basis for L If h codiios ) ) ar fulfilld i h basis (7) ad h sysm ar -clos i L Cosuly sysm (3) is o miimal i L Th rmaiig cass wh ar rovd similarly Thrfor w obai h followig fial rsul for h basiciy of sysm (3) i L Thorm 4 L asymoic formula (4) hold h fucios ad saisfy h codiios ) ) 3) Th variabl b drmid from rlaios () () ad l max ; Th for sysm (3) is o miimal i L ; for i is o coml i L For ris of sysm (3) i ) Coml; ) Miimal; 3) -liarly idd; ba orhic o 5) h followig ro- L ar uival: i 4) Forms a sis isom ; Z is drmid by xrssio Coyrigh 3 SciRs

6 S M FARAHAI T I AJAFOV 853 (4) Idd uiv alc of roris )-4) follows dircly from Lmma Euivalc of codiios 4) ad 5) is rovd 4 Coclusios Taig io a ccou h obaid rsuls w ca summariz his wor as follows Prurbd sysm of xos h has of which may has diffr asymoic bhavior i diffr ars of h basic irval is sudid i his wor I should b od ha i s robably h firs im h roblm of basiciy is co sidrd for such a sysm Udr crai codiios o h fucios dfiig h has w rov ha his sysm may hav a fii dfc i L Morovr i ihr forms a basis for L or i is o coml ad o miimal i L 5 Acowldgms Th auhors xrss hir ds graiud o Profssor B T Bilalov for his aio ad valuabl guidac o his aricl Aalysis Iogi aui i Thii Srmaya Mamaia i Prilozhiya Tmaichsi Obzory Vol [] L H Lars Iral Wavs Icid uo a Kif Edg Barrir D Sa Rsarch Vol 6 o [] S A Gabov ad P A Kruisii O Lars s osaioary Problm Zhural Vychislil oi Mamaii i Mamaichsoi Fizii Vol 7 o [3] P A Kruisii Small o-saioary Vibraios of Vrical Plas i a Chal wih a Sraifid Fluid USSR Comuaioal Mahmaics ad Mahmaical Physics Vol 8 o [4] E I Moisv ad Abbasi Basis Prory of Eigfucios of h Gralizd Gasdyamic Problm of Fral wih a olocal Oddss Codiio ad wih h Discoiuiy of h Gradi of Soluio Diffrial Euaios Vol 45 o [5] V A Ili Mixd Problm Dscribig h Damig Procss of a Bar Cosisig of Two Scios of Diffr Dsiy ad Elasiciy Providd ha h Tim of Wav s Passag i Each of Ths Scios Coicid Trudi Isiua Mamaii i Mhaii Uro RA Vol [6] I S Lomov o-smooh Eigfucios i Problms of Mahmaical Physics Diffrial Euaios Vol 47 REFERECES o [] R Paly ad Wir F ourir Trasforms i h Com- [7] L M Lujia Rgulariy of Scral Problms wih Adlx Domai Amrica Mahmaical Sociy Provi- diioal Codiios a h Ir Pois Mamaichsdc 934 i Zami Vol 49 o Ga ad Dsiy Thorms Amrica [8] B T Bilalov ad S M Farahai O Prurbd Bass of [] Lviso Mahmaical Sociy Providc 94 Exoial Fucios wih Comlx Cofficis Tras- [3] [4] R M Youg A Iroducio o o-harmoic Fourir Sris Srigr Brli A M Sdlsii Classs of Aalyic Fourir Trasforacios of AS of Azrbaija Vol 56 o [9] I Sigr Bass i Baach Sacs I Srigr Brli maios ad Exoial Aroximaios Fizmali doi:7/ Moscow 5 [] I T Hochbrg ad A S Marus O Sabiliy of Bass [5] Ch Hil A Basis Thory Primr Srigr Brli of Baach ad Hilbr Sacs Izvsiya Aadmii au 534 doi:7/ Moldavsoj SSR o [6] O Chriss A Iroducio o Frams ad Risz [] B T Bilalov ad T R Muradov O Euival Bass bass Srigr Brli 3 44 i Baach Sacs Uraiia Mahmaical Joural Vol [7] D L Russll O Exoial Bass for h Sobolv 59 o Sacs Ovr a Irval Joural of Mahmaical Aadoi:7/s lysis ad Alica ios Vol 87 o [] B T Bilalov Bass from Exos Cosis ad Sis doi:6/-47x(8)94- Big Eig Fucios of Diffrial Oraors Diffrial Euaios Vol 39 o [8] X H ad H Volmr Risz Bass of Soluios of Surm-Liovill Euaios Joural of Fourir Aalysis ad Alicaios Vol 7 o doi:7/bf585 [9] H Milos Ivrs Scral Problms ad Closd Exoial Sysms Aals of Mahmaics Vol 6 o doi:47/aals56885 [] A M Sdlsii oharmoic Aalysis Fucioal [3] B T Bilalov Basiciy of Som Sysms of Exos Cosis ad Sis Diffrial Euaios Vol o 99-6 [4] B T Bilalov O Isomorhism of Two Bass Fudamalaya i Priladaya Mamaia Vol o Coyrigh 3 SciRs