Representation of linear operators by Gabor multipliers

Transcription

1 Rprsttio of lir oprtors Rprsttio of lir oprtors by Gbor multiplirs Ptr C Gibso, Michl P Lmourux, Gry F Mrgrv ABSTRACT W cosidr cotiuous vrsio of Gbor multiplirs: oprtors cosistig of short-tim Fourir trsform, followd by multiplictio by distributio o phs spc (clld th Gbor symbol), followd by ivrs short-tim Fourir trsform, llowig diffrt loclizig widows for th forwrd d ivrs trsforms For giv pir of forwrd d ivrs widows, which lir oprtors c b rprstd s Gbor multiplir, d wht is th rltioship btw th (o-clssicl) Koh-Nirbrg symbol of such oprtor d th corrspodig Gbor symbol? Ths qustios r swrd compltly for spcil clss of comptibl widow pirs I dditio, cocrt xmpls r giv of widows tht, with rspct to th rprsttio of lir oprtors, r mor grl th stdrd Gussi widows Th rsults i th ppr hlp to justify tchiqus dvlopd for sismic imgig tht us Gbor multiplirs to rprst osttiory filtrs d wvfild xtrpoltors INTRODUCTION This ppr is cocrd with th rprsttio of lir oprtors by Gbor multiplirs For us Gbor multiplir is oprtor of th form: short-tim Fourir trsform, followd by multiplictio by distributio o phs spc clld th Gbor symbol followd by ivrs short-tim Fourir trsform W xplicitly llow th lysis widow, upo which th forwrd short-tim Fourir trsform is bsd, to b diffrt from th sythsis widow, which srvs s th bsis for th ivrs short-tim Fourir trsform Mor prcisly, w study th rprsttio of grl (i, o-clssicl) Koh- Nirbrg psudodiffrtil oprtors s for xmpl [4] by ms of Gbor multiplirs As is wll-kow, th clss of grl Koh-Nirbrg oprtors is simply ll lir oprtors (sstilly this is vrsio of th Schwrtz krl thorm s (5) i Sctio ) Howvr, our im is to lyz th corrspodc btw th Koh- Nirbrg symbol, which i th grl sttig is rbitrry distributio, d th Gbor symbol, of giv oprtor, providd th lttr xists Th Gbor symbol of oprtor dpds o choic of lysis d sythsis widows; idd for fixd widow pir, thr my b limitd clss of lir oprtors tht my b rprstd s Gbor multiplir As fr s w kow th problm of rprstig Koh-Nirbrg oprtors xctly by (our vrsio of) Gbor multiplirs is w, d i this ppr w cofi ourslvs to th followig two fudmtl qustios CREWES Rsrch Rport Volum 5 (003)

2 Gibso, Lmourux, d Mrgrv For giv pir of lysis d sythsis widows, which Koh-Nirbrg oprtors my b rprstd s Gbor multiplir? Ar thr xplicit formuls rltig th Gbor d Koh-Nirbrg symbols? Much of th ppr rsts o ky tchicl otio of comptibl widow pir For such widows w hv complt swr to both of th bov qustios A commo widow choic for th short-tim Fourir trsform is Gussi, both s lysis d sythsis widows, d this srvs s th prototyp of comptibl widow pir O cocrt rsult tht mrgs from our lysis of comptibl widows is w typ of widow, which w cll xtrm vlu widow lytic, rpidly dcrsig widow, with lytic, rpidly dcrsig Fourir trsform It is mor grl th Gussi i th ss tht th clss of oprtors tht c b rprstd s Gbor multiplir bsd o Gussi widows is propr subst of th clss tht c b rprstd usig xtrm vlu widows Th bsic qustios tht w ddrss hr r motivtd by prticulr pplictio, sismic imgi whr psudodiffrtil oprtors ply importt rol [6,7] d Gbor multiplirs hv b succssfully implmtd s fficit d flxibl mthod of sismic dcovolutio [8] Ths implmttios rst o th fct tht thr xist fficit umricl schms, dscribd i [5], by which to vlut discrtizd vrsios of Gbor multiplirs But somthig tht hs b lckig so fr is thorticl lysis of th rltioship btw th origil oprtors i xtrpoltors or filtrs which r sily xprssd s Koh-Nirbrg oprtors, d th corrspodig cotiuous Gbor multiplirs tht hv b usd to std i for thm Th rsults prstd hr srv to fill i th gp Thr is importt issu cocrig th vlutio of lir oprtors i prcticl pplictios tht w do ot ddrss i th prst ppr, mly, to qutify th dgr of pproximtio to cotiuous Gbor multiplir tht is ttid by discrtizd vrsios of it For rsults i this dirctio, s, for xmpl, Fichtigr d Nowk (003) d [9] Th ppr is orgizd s follows Th mthmticl frmwork i which w formult th problms dscribd bov is md prcis i Sctio, whr w fix th bsic ottiol covtios Thorm of Sctio rfsc-schwrtzkrl givs strightforwrd wy to xprss th Schwrtz krl of lir oprtor i trms of its Gbor symbol, which i tur yilds xprssio for th Koh-Nirbrg symbol i trms of th Gbor symbol Th covrs problm of xprssig th Gbor symbol i trms of th Koh-Nirbrg symbol sms to b much mor difficult, d tks up th bulk of th ppr Sctio 3 provids th bsic motivtio for our ky tchicl dfiitio of comptibl widows i Sctio 3 A ltrt dfiitio of comptibility, provd i Thorm of Sctio 3 to b quivlt to th origil, lds to squc of tchicl rsults i Sctio 3 tht ly th groudwork for our mi Thorms Ths r Thorm 3 d Thorm 4 of Sctio 33 I Sctio 34 w pply Thorm 4 to show tht xtrm vlu widows srv to rprst widr clss of lir oprtors s Gbor multiplirs th is possibl with Gussi widows W discuss brifly i Sctio 35 turl xtsio of th otio of comptibility to sigulr widows Lstly, Sctio 4 provids short summry of th ppr CREWES Rsrch Rport Volum 5 (003)

3 Rprsttio of lir oprtors PRELIMINARIES Nottio d covtios Our ottio is mostly stdrd, with th xcptios tht: (i) w us th vrsio of th Fourir trsform tht hs fctor of π i th xpot, d (ii) w tk tmprd distributios to b cotiuous, cojugt lir, rthr th lir, fuctiols o th spc of Schwrtz clss fuctios Grlly spki w dl with fuctios d distributios o R, whr th vlu of is fixd withi giv cotxt, d R is lwys th domi of itgrtio, which w omit W idict poit i R by pir (x,y) of poits xy, R, d itgrtio ovr R is idictd by pir of itgrl sigs For covit rfrc, w hv compild i Tbl list of som of th fuctio spcs d oprtors tht w us rptdly W work withi th bsic frmwork of L, th spc of cotiuous, lir oprtors L ' : S S, mkig frqut us of th corrspodc btw L d S ' [0] Mor prcisly, ' giv lir oprtor L L, thr is uiqu distributio K = K( L) S, its Schwrtz krl, tht stisfis th qutio K, θ ϕ = Lϕ, θ ϕ, θ S () ' Ad covrsly, giv K S, th qutio () vidtly dtrmis uiqu L L (Hr dots th tsor product: f gxy (, ) = f( xgy ) ( )) Th djoit of lir * oprtor L L is th lir oprtor L L dfid by th qutio * Lϕ, θ = Lθ, ϕ ϕ, θ S () I ordr for th oprtor T ψ, listd i Tbl, to b wll-bhvd, som rstrictios hv to b plcd o ψ ; i this rgrd w itroduc th otio of tmprd chg of vribls, s follows Dfiitio W sy tht smooth, ivrtibl mp m m ψ : R R, * * is tmprd chg of vribls if ch of th oprtors T, T, T, T mps S m ito S m ψ ψ ψ ψ Thr is o tmprd chg of vribls o R whos corrspodig oprtor w ssig spcil ottio: CREWES Rsrch Rport Volum 5 (003) 3

4 Gibso, Lmourux, d Mrgrv T_ = T, whr ψ _( x, y) = ( x, y x) (3) ψ _ Giv rbitrry tmprd distributio oprtor dfid by th formul σ( x, ξ) S, w writ σ ( X, D) for th ' ix ( X D) ( X D) ( x) π ξ σ, : S S ; σ, ϕ = σ( x, ξ) F ϕ( ξ) dξ (4) (Hr X : R R dots th idtity mp, so tht, for xmpl, X ( x) = x D stds for th diffrtil oprtor D = πi ) W rfr to σ ( X, D ) s Koh- Nirbrg psudodiffrtil oprtor; th distributio σ ( x, ξ ) is its Koh-Nirbrg symbol I th cotxt of clssicl psudodiffrtil oprtors, whr th symbol σ ( x, ξ ) is rquird to b smooth d of boudd growth, th itgrl o th right-hd sid of (4) is ihrtly wll-dfid A simpl wy to giv th itgrl umbiguous itrprttio i th prst much mor grl sttig is to dfi σ ( X, D) i trms of its Schwrtz krl: K( σ ( X, D)) = T_F σ (5) ' Sic ch of th oprtors T_ d F crris S bijctivly oto itslf, it is vidt from th rprsttio (5) tht th clss of Koh-Nirbrg psudodiffrtil oprtors o R is idticl with L itslf Amog th vrious wys to rprst lir oprtor, howvr, th Koh-Nirbrg symbol d ccompyig forml rprsttio (4) r of prticulr itrst sic, from th physicl poit of viw, thy r turl both for prtil diffrtil oprtors d for osttiory filtrs [4] I othr words, i pplictios o is somtims giv th Koh-Nirbrg symbol of lir oprtor dirctly Not tht th Koh-Nirbrg symbol σ L d th Schwrtz krl K(L) of lir ' ' oprtor L : S S both blog to S But thr is ss i which ths r distict vrsios of ' S, i tht σ L is distributio o phs spc, R R, whil K(L) is distributio o th cross product R R of th udrlyig spc with itslf Idd, sic it is somtims usful to mk this distictio btw spc d frqucy, w rsrv ξ d η for frqucy vribls, usig othr lttrs (such s x,y,τ,t,v,w) for spcil vribls 4 CREWES Rsrch Rport Volum 5 (003)

5 Rprsttio of lir oprtors Tbl Nottio Fuctio d distributio spcs S Schwrtz clss fuctios ϕ : R ' R S (cojugt lir) tmprd distributios u : S C P C fuctios ϕ : R R such tht ϕ is boudd by polyomil P for vry multi-idx Oprtors Dscriptio Symbol Actio o fuctios Adjoit Compositio with chg of vribls m m ψ : R R Fourir trsform Prtil Fourir trsform Prtil Fourir trsform Modultio Trsltio Multiplictio by ' S m T ϕ ϕ ψ * ψ T F F F S S : ' ' ( S S) S S : ' ' ( S S) S S : M T ' ' ( S S) S S ξ : x ' ' ( S S) S S : ' ' ( S S) N : m ' m ϕ( x) ϕξ ( ) πξ i x = ϕ( x) dx ϕ( x, y) F ϕξ (, y) = dt J T ψ ψ ψ F = F F = F πξ i x = ϕ( x, y) dx ϕ( x, y) Fϕ( x, η) F = F πη i y = ϕ( x, y) dy ϕ M = M πξ ( ) i x x ϕ( x) ξ ξ ϕ() t ϕ( t x) P S ϕ ϕ T = T x x CREWES Rsrch Rport Volum 5 (003) 5

6 Gibso, Lmourux, d Mrgrv Th Schwrtz krl of grl Gbor multiplir If g : R C is msurbl d boudd by polyomil, th th formul V ϕ( x, ξ) = M T ϕ g dfis short-tim Fourir trsform V g : S P with lysis widow g Th bsic thory of short-tim Fourir trsforms is dscribd i [4] For S, th rg of V lis i S ; th djoit V of V is th mp ξ x V : S S ; V u, ϕ = u, V ϕ If th distributio u S hpps to b L fuctio, th V u is giv by th formul Vut () = ux (, ξ ) MT() tdxdξ ξ x Dfiitio Lt g : R C b msurbl d boudd by polyomil, d lt S If i dditio 0, th w sy tht ch of ( ) d (, g) is widow pir o R It is bsic fct bout th short-tim Fourir trsform tht for y widow pir ( ), th mp VV g : S S (6) is th idtity Rcll from Tbl tht w us th symbol by N to dot multiplictio Dfiitio 3 Giv widow pir ( ) o M V N V g R d distributio 6 CREWES Rsrch Rport Volum 5 (003) = Gbor multiplir; w rfr to th distributio s its Gbor symbol S, w cll (Not tht Fichtigr d Nowk [] us th trm short-tim Fourir trsform multiplir for Gbor multiplir bsd o idticl widows ( gg, ), whil i Fichtigr d Nowk (003) Gbor multiplir rfrs to mor grl objct th w hv dfid) A Gbor multiplir, i th ss of Dfiitio 3, is lir oprtor blogig to L Its djoit is lso Gbor multiplir, d th prcis coctio btw th two works out s follows

7 Rprsttio of lir oprtors Propositio 4 For y widow pir ( ) d y distributio S, th djoit of th Gbor multiplir M is, g = M M Roughly spki th dfiig structur of Gbor multiplir ms tht it crris implicit digoliztio o phs spc From th thorticl poit of viw, this fct mks it dsirbl to xprss, if possibl, giv lir oprtor L L s Gbor multiplir, th structur of th oprtor th big codd i its Gbor symbol It is lso dsirbl to xprss oprtor s Gbor multiplir from th poit of viw of pplictios, sic thr xist fst computtiol mthods to vlut discrtizd Gbor multiplirs [5] Th mi problm tht w r cocrd with i th prst ppr is to xprss giv Koh-Nirbrg psudodiffrtil oprtor s Gbor multiplir Bfor cosidrig th issu i dtil, w dl brifly with th covrs problm, of xprssig giv Gbor multiplir s psudodiffrtil oprtor I light of th xprssio (5) for th Schwrtz krl of psudodiffrtil oprtor, th lttr problm is quivlt to computig th Schwrtz krl of Gbor multiplir This turs out to b rltivly strightforwrd, d c b crrid out i full grlity I sttig th bsic rsult w mk us of th followig ottio Lt S : S4 S dot th mp dfid by Th corrspodig djoit is th mp Sρ( x, ξ) = ρ( x, x, ξ, ξ) S : S S ; S u, ϕ = u, Sϕ 4 Thorm A rbitrry Gbor multiplir M hs Schwrtz krl K( M ) = V gs (7) Proof Not tht M ϕ, θ = V, gϕv θ d π ( ) ( ) it ξ πiτ ξ Vg V x, = t g( tx) dt ( ) ( x) d ϕ θ ξ ϕ θ τ τ τ πit ξ πiτ ξ = ϕ () t gt ( x) θτ ( ) ( τxd ) τdt CREWES Rsrch Rport Volum 5 (003) 7

8 Gibso, Lmourux, d Mrgrv Thus, πit ( ) τ ξ ( τ x) g( t x) θ ϕ( τ t) dτdt (8) =, =,, πi( τ, t) ( ξ,ξ) T( xx, ) g( τ t) θ ϕ( τ t) dτdt = V gθ ϕ( x, x, ξ, ξ) M ϕ, θ = SV, gθ ϕ = V gs, θ ϕ Sic grlizd Koh-Nirbrg oprtors compss ll of L, giv rbitrry Gbor multiplir M o R, thr xists distributio σ S such tht σ ( X, D) =M By Thorm, this is quivlt, i trms of Schwrtz krls, to th qutio which c b solvd for σ i trms of to yild T_F σ = V gs, (9) σ = F T V g S (0) I cotrst, thr is o obvious wy to solv qutio (9) for i trms of σ, sic ithr of th oprtors V or S g is ijctiv Idd, dtrmiig th Gbor symbol tht corrspods to giv Koh-Nirbrg symbol sms to b much mor itrict problm Idticl Gussi widows Lt us spciliz to th cs of widow pir cosistig of idticl Gussis, gt () = () t = π t, whr w us th ottio t = t t for t R Th spcil structur of ths widows ms tht w c obti ltrt formul for th Schwrtz krl of y Gbor multiplir bsd o thm Th ky poit i our proof of Thorm, t which w c tk dvtg of our prst choic of Gussi widows, is th itgrl (8) Th itgrd ivolvs th product 8 CREWES Rsrch Rport Volum 5 (003)

9 Rprsttio of lir oprtors π( τx) π( tx) ( τ xgt ) ( x) = Thus, i trms of th tmprd chg of vribls w s tht ( τ x) gt ( x) hs th form π( τ+ t x) π ( tτ) = ( vw, ) = (, t τ ), τ + t ( τ x) gt ( x) = G( v xg ) ( w), () whr V ϕ g V () v π G v θ, w obti = d ( ) w G w = π Substitutig () ito th xprssio (8) for πit ( τ) ξ VgϕV θ = G( vx) G( w) θ ϕ( τ, t) dτdt = G ( vx) G ( w) θ ϕ ψ ( v, w) dtj dvdw πiw ξ ψ T =, ψθ ϕ, πiw ξ ( ) T ( ) ( x, 0) G G vw vwdvdw T πiw ξ ( ) T ( ) = ( x, 0) G G v, w ψθ ϕ v, w dv dw ( G = F G T ψθ ϕ ), () whr G () v G ( v) = d dots covolutio with rspct to th first vribls Rplcig F with F, d crryig F through th covolutio, w obti from () tht VgϕVθ = F ( G G FT ψθ ϕ) Procdig s i th proof of Thorm, this lds to th followig formul for th Schwrtz krl of M : K ( M ) = T N ψf F (3) G G Rcllig tht gt () = () t = π t, () v π G v =, d ( ) w G w = π, w hv Sic πt / = dt = ( ) πη π w G G η, w =, th formul (3) simplifis s i th xt propositio CREWES Rsrch Rport Volum 5 (003) 9

10 Gibso, Lmourux, d Mrgrv Propositio 5 Lt S gt () = () t = π t b idticl Gussis o R Th for y, th Schwrtz krl of M is K( M ) = T F N F, (4) g, ψ G whr G : R R is th Gussi τ + t vribls ψ ( τ, t) = (, t τ) (, w ) G( w) π η η, = d ψ is th tmprd chg of Ech of th oprtors T, F, N d F occurrig i th xprssio (4) for th ψ G Schwrtz krl of M is ijctiv Thrfor if M = σ ( X, D), w c, by comprig th Schwrtz krls of th two oprtors, xprss th Gbor symbol of M i trms of σ s (, y ) i y N whr H( y) π η π η = F F σ, η, = (5) H Morovr, for giv σ S, th oprtor σ ( X, D) c b xprssd s Gbor multiplir bsd o idticl Gussi widows oly if th xprssio (5) yilds wlldfid tmprd distributio (Not tht (5) cot b writt s covolutio, bcus th rpidly icrsig fuctio H dos ot hv Fourir trsform) Ar thr widow pirs ( ) othr th idticl Gussis for which o c obti rsults logous to ths? Th purpos of this ppr is to dscrib clss of widow pirs for which this is possibl MAIN RESULTS Dfiitio d xmpls of comptibl widow pirs Th xprssio (4) for th Schwrtz krl of Gbor multiplir bsd o idticl Gussis is th bsis for th formul (5) With y to obtiig mor grl rsults log ths lis, w ivstigt th clss of ll widow pirs for which formul logous to (4) xists I this rgrd th sstil proprty of Gussi is th fctoriztio rul (), so w focus o th clss of ll widow pirs tht oby such rul This lds dirctly to th followig ky tchicl otio Dfiitio 6 W sy tht widow pir ( ) o R is comptibl, d tht g d r comptibl widows, if thr xist fuctios G, G : R C d tmprd chg of vribls ( vw, ) = ψ ( τ, t) with w = t tu such tht for vry τ,, tx R, ( τ x) gt ( x) = G( v xg ) ( w) 0 CREWES Rsrch Rport Volum 5 (003)

11 Rprsttio of lir oprtors Not tht our drivtio of formul (3) i Sctio is vlid for rbitrry comptibl widows s w hv just dfid thm Hc w immditly hv th followig rsult Propositio 7 Lt ( ) b comptibl widow pir o Schwrtz krl of M is K ( M ) = T N ψf F, G G whr G, G d ψ r rltd to ( ) s i Dfiitio 6 R Th for y S If ( ) is comptibl th so is (, g) Th prcis coctio btw th ccompyig sts of fuctios s sttd blow my b sily vrifid Hr w writ ψ : R R for th chg of vribls ψ ( x, y) = ( y, x), d w us th ottio F ( x) = F( x) Propositio 8 Lt ( ) b comptibl widow pir o ( τ x) gt ( x) = G( v xg ) ( w), R, so tht i ccordc with Dfiitio 6 Th th rvrs widow pir (, g) is lso comptibl d stisfis th qutio, th whr g( τ x) ( t x) = G ( v x) G ( w), G = G, G = G, d ( v, w) = ( v ψ w), Tbl lists svrl fmilis of comptibl widow pirs, logsid thir corrspodig fctoriztios d chgs of vribls, s prscribd by Dfiitio 6 Th clss of comptibl widow pirs is sily s to b ivrit with rspct to umbr of lmtry oprtios: rscli trsltio d modultio of ithr widow, d lir chg of vribls pplid simultously to both widows This obsrvtio shows tht th list i Tbl is ot complt O th othr hd, th widow pirs listd i Tbl r rsobly rprsttiv For xmpl, it turs out tht vry ogtiv comptibl widow pir o R c b obtid by trslti dilti rvrsig or rsclig th giv xmpls, or by rvrsig th ordr of widow pir obtid this wy This is rthr ivolvd tchicl rsult tht will b prstd i forthcomig ppr [3] CREWES Rsrch Rport Volum 5 (003)

12 Gibso, Lmourux, d Mrgrv Tbl Exmpls of comptibl widows Th prmtrs m µ k κ 5) r positiv sclrs Th widow pirs of lis (-3) r dfid o widow pirs of lis (4,5) r dfid o R oly Comptibl widow pirs ( ) # gt () ( τ ) ( vw, ) = ψ ( τ, t) G () G ( ) S ( τ, t τ ) () v πξ i t S ( τ, t τ ) i v () v πξ i w 3 mt µτ mt+ µτ m ( m+µ ) v w ( m+ µ, t τ m ) µ + µ t 4 t τ τ log ( τt t ) + v +, tτ v cosh( w/ ) t 5 kt τ κτ ( ( ) ) ( ) ( τt + t+ log ), tτ ( ) ( k+ κ ) v v,,,, pprig i lis (3- ( ) + R for y ; th / k κ κ w k + w k+ κ k+ κ Th xmpl of li (), Tbl, which stisfis th dfiitio of comptibility i trivil wy, is ot dirctly of prcticl itrst Of cours th Gussi widows of li (3) r th prototyp for th otio of comptibility, d thy r stdrd widow choic for th short-tim Fourir trsform Th most itrstig xmpls i Tbl r th widows of li (4) d thir grliztios i li (5) Ths wr discovrd usig th ltrt chrctriztio of comptibility dvlopd i th xt sctio of this ppr; thir full drivtio will b prstd i [3] Lik Gussis, ths widows hv fudmtl rol i probbility thory d ris i coctio to vrit of th ctrl limit thorm Mor prcisly, lt E dot th fuctio g of li (4) i th spcil cs = : t E() t = Proprly scld ffi trsformtios of E, of th form ( t E b ) b, r th dsity fuctios of so-clld xtrm vlu probbility distributios, dscribd by Fishr d Tipptt i [] Bsd o this, w rfr to th widows of lis (4, 5) s xtrm vlu widows s Figur t CREWES Rsrch Rport Volum 5 (003)

13 Rprsttio of lir oprtors t t FIG Extrm vlu widows Th fuctio f () t =, bov, d its rvrsl f ( t), blow, r plottd for = 0 Not th f ( t) is clos to big cusl, i zro for gtiv t vlus of t This is bcus th doubl xpotil tds to th Hvisid fuctio with icrsig Not tht th Fourir trsform of xtrm vlu widow is proportiol to th gmm fuctio, rstrictd to li of costt rl prt: t kt k π i F ( ξ ) = Γ ξ Thus th Fourir trsform of xtrm vlu distributio is lytic, vr zro, d rpidly dcrsig I Sctio 34, w show tht xtrm vlu widows hv crti dvtg ovr Gussis with rspct to th rprsttio of lir oprtors A ltrt chrctriztio of comptibility Hr w dvlop ltrt chrctriztio of comptibility tht mks it sir to driv formuls d to prov som tchicl fcts tht will b dd i subsqut CREWES Rsrch Rport Volum 5 (003) 3

14 Gibso, Lmourux, d Mrgrv sctios Th rgumts giv hr r purly tchicl; thy build towrd our mi rsults which r prstd i th followig Sctio 33 Lt ( ) b comptibl pir o R with corrspodig fuctios G, G d tmprd chg of vribls ( vw, ) = ψ ( τ, t), s stipultd i Dfiitio 6 It c b dducd from th bsic qutio, ( τ xgt ) ( x) = G( vxg ) ( w) τ, tx, R, tht thr r svrl simplifyig ssumptios to b md cocrig G, G d ψ without y icumbt loss of grlity (6) Firstly, ot tht G (( v τ, t) x) G ( t τ) = ( τ x) g( t x) = (( τ t) ( xt)) g(0 ( x t)) = G (( v τ t, 0) ( xt)) G ( t τ ) = G (( v τ t, 0) + t x) G ( w) (7) This lds to th first of our simplifyig propositios Propositio 9 It my b ssumd without loss of grlity tht v( τ, t) = v( τ, t 0) +, t τ t R (8) Proof Suppos tht (8) fils to hold, d cosidr th w fuctio v, dfid by v ( τ, t) = v( τ, t 0) +, t τ t R Bsd o th fct tht ψ = ( vw, ) is tmprd chg of vribls, it my b vrifid tht th chg of vribls ψ = ( v, w) is tmprd lso Ad by costructio, Now, pplyig th idtity (7) yilds v ( τ, t) = v ( τ, t 0) +, t τ t R G ( v x) G ( w) = G ( v( τ t, 0) + tx) G ( w) = G( v x) G( w) Thrfor th dfiig qutio (6) rmis stisfid if w rplc ψ by th w chg of vribls ψ 4 CREWES Rsrch Rport Volum 5 (003)

15 Rprsttio of lir oprtors Propositio 0 It my b ssumd without loss of grlity tht v (0, 0) = 0 Proof Th qutio tht dfis comptibility, (6), rmis stisfid if G is rplcd by d v is rplcd by G ( y) = G ( y+ v(0, 0)) v ( τ, t) = v( τ, t) v(0, 0) Th modifid chg of vribls ψ = ( v, w) is tmprd, d by costructio v (0, 0) = 0 Propositio Without loss of grlity, G (0) = d G Proof Substitutig w = 0 (i, t = τ ) d x = 0 = g ito th qutio (6) yilds () tgt () = G( vtt (, )) G(0) (9) Sic 0 = g, it follows from (9) tht tht G (0) 0 Th qutio (6) vidtly rmis stisfid if G is rplcd by G (0) G d G is rplcd by G(0) G, so w my ssum G (0) = Assumig i dditio tht (8) holds d tht v (0, 0) = 0, w hv tht vtt (, ) = v(0, 0) + t= t It th follows from qutio (9) tht G () t = () t g () t It will strmli subsqut rgumts to ssum tht G, G d ψ hv th forms giv i Propositios 9, 0, d W ow itroduc th otio of product-trsltio commuttivity for widow pir Dfiitio Lt ( ) b widow pir o R W sy tht ( ) is producttrsltio commuttiv, or tht g d r product-trsltio commuttiv, if thr xist fuctios C : R C d ζ : R R such tht for vry w R, gt = C( w) T ( g ), (0) w ζ ( w) d such tht ( τ, t) ( tζ( t τ), tτ) is tmprd chg of vribls Although it is ot immditly obvious, product-trsltio commuttivity of widow pir is quivlt to comptibility This fct fcilitts th lysis of comptibility CREWES Rsrch Rport Volum 5 (003) 5

16 Gibso, Lmourux, d Mrgrv Thorm A widow pir ( ) is product-trsltio commuttiv if d oly if it is comptibl Morovr, if widow pir ( ) hs this proprty th, without loss of grlity, th trms occurrig i th rspctiv dfiig qutios, gt = C( w) T ( g )d ( τ x) g( t x) = G ( v x) G ( w), w ζ ( w) my b ssumd to b rltd by th formuls G = C, v( τ, t) = tζ( t τ)d ζ( y) =v( y, 0) Proof W prov th quivlc of Dfiitios 6 d ; th sttd formuls mrg i th cours of th proof Suppos tht ( ) is comptibl, with ssocitd fuctios G, G d ψ tht stisfy th proprtis i Propositios 9, 0, d Th bsic qutio of Dfiitio 6, qutio (6), c b writt symboliclly s T g = T T G G ( xx, ) ψ ( x, 0) = T g T( x, x) ψt( x, 0) G G g( τ, t) = G ( v( τ + x, t+ x) x) G ( w) () Dfi ζ ( y) =v( y, 0), so tht substitutig x = t ito () yilds g( τ, t) = G (( v τ t, 0) + t) G ( w) = ζ G ( t ( w)) G ( w) = ζ ζ gt ( ( w)) ( t ( w)) G( w) (by Propositio ) () Th idtity w= t τ implis tht g( τ, t) = gtw ( t) Thrfor () is quivlt to th qutio gt () t = G ( w) T ( g )() t (3) w ζ ( w) Not tht v( τ, t) = v( τ t, 0) + t = tζ( t τ), so th tmprd chg of vribls ψ = ( vw, ) is prcisly th mp ( τ, t) ( tζ( t τ), tτ) Togthr with (3), this vrifis tht th pir ( ) coforms to Dfiitio of product-trsltio commuttivity, with C = G Covrsly, suppos tht th widow pir ( ) is product-trsltio commuttiv, with ssocitd fuctios C, ζ s i Dfiitio St v( τ, t) = tζ( t τ ), so tht 6 CREWES Rsrch Rport Volum 5 (003)

17 Rprsttio of lir oprtors (( v τ, t), w( τ, t)) = ( tζ( t τ), t τ) is tmprd chg of vribls As bfor, g( τ, t) = gt ( t), so by dfiitio, w g( τ, t) = ( tζ( w)) g( tζ( w)) C( w) T( xx, ) g( τ, t) = ( tζ( w) x) g( tζ( w) x) C( w) = ( g)( v x) C( w) Thus ( ) coforms to Dfiitio 6, with ssocitd fuctios G ψ = ( vw, ) = g, G = C d W r ow i positio to driv two dditiol formuls cocrig comptibl widow pirs Rcll tht X : R R dots th idtity mp; X k dots its k -th compot W writ ( gx ) for th vctor ( gx ) ( ),, gx, d similrly ( ) ( ) gx, = gx,,, gx, A tild ovr fuctio idicts rflctio i th rgumt: F ( y) = F( y) Propositio 3 If ( ) is comptibl pir th, without loss of grlity, th ssocitd fuctio G is giv by th formul G g = (4) Ad th chg of vribls ψ c b xprssd s ψ ( τ, t) = ( tζ ( t τ), t τ), whr th rstrictio of ζ to poits w t which G ( ) 0 w is giv by th formul ( gx ) gx, ζ = g (5) Proof By Thorm, g d stisfy th qutio Itgrtig both sids ovr gt = G ( w) T ( g ) w ζ ( w) R d rrrgig trms yilds CREWES Rsrch Rport Volum 5 (003) 7

18 Gibso, Lmourux, d Mrgrv G ( w) = T gt ζ ( w) w ( g ) g ( w) = g g ( w) = Obsrv tht Also, ( gx ) ( w) = XgT = XG ( w) T ( g ) w ζ ( w) = G ( w) ( X + ζ ( w)) g g = G ( w), so, providd G ( ) 0 w, th right-hd sid of (5) bcoms G ( w) ( X + ζ( w)) g gx, gx, = ζ( w) + c gx, = ζ( w) G ( w) It turs out tht G c vr b zro, so tht i fct th formul (5) is vlid vrywhr, but w will ot prov this i th prst ppr (s [3]) Nxt w prov som tchicl rsults, i ticiptio of th comig sctio, whr w rlt th Gbor symbols of Gbor multiplirs bsd o comptibl widows to thir Koh-Nirbrg symbols Propositio 4 Lt ( ) b comptibl widow pir, with ssocitd fuctios G, G Th without loss of grlity G ˆ ˆ G = ( g ) ( g ) Proof This is just pplictio of Propositios d 3 8 CREWES Rsrch Rport Volum 5 (003)

19 Rprsttio of lir oprtors Rcll tht ψ dots th prticulr chg of vribls ψ ( τ, t) = ( τ, t τ) d tht T_ dots th corrspodig oprtor T ψ Lmm 5 Lt ( ) b comptibl widow pir, with ssocitd chg of vribls ψ ( τ, t) = ( tζ ( t τ), t τ) (s i Thorm ) Th ψ ( τ, t) = ( tζ ( t τ), t τ) Proof Writ ( f( xy) f( xy)) ψ ( xy),,, =, Th, =, =, ( x y) ψ ψ ( x y) ( f ζ( f f) f f) Combiig th bov xprssios for x d y yilds tht Thus, f = x y+ ζ ( y) ψ ψ ψ ζ ( x, y) = ( f, f ) = ( f, f f ) = ( x+ ( y) y, y) Propositio 6 Lt ψ ( τ, t) = ( tζ ( t τ), t τ) b th chg of vribls ssocitd to i comptibl widow pir ( ) Th FT T πϕ F, whr = ψ ϕ( η, y) = η ( y ζ( y)) Proof First ot tht T T T ψ ψ Lt F( τ, y) S b dummy fuctio, with τ ψ = srvig s Fourir dul vribl to η Th ( FT ) πτη i ψ ψf ( η, y) = Tψ ψf( τ, y) dτ πτη i F( τ ζ( y) y y) dτ (by Lmm 5) = +, =, πη i ( yζ( y)) πτη i F( τ y) d F ( ) πη i ( yζ( y)) = F η, y τ By dsity of S i S d cotiuity of th oprtors i qustio, it follows tht FT T = i πϕ F ψ CREWES Rsrch Rport Volum 5 (003) 9

20 Gibso, Lmourux, d Mrgrv Rltio btw th Gbor d Koh-Nirbrg symbols I Sctio 3 w drivd xplicit formul for th Gbor symbol of Gussi Gbor multiplir i trms of its Koh-Nirbrg symbol, d th usd this to costri th clss of Koh-Nirbrg oprtors tht c b rlizd s such Gbor multiplir W ow prst th corrspodig rsults i much mor grl stti mly, for rbitrry comptibl widows Thorm 3 Lt S d lt ( ) b comptibl widow pir o R Th M = σ ( X, D) if d oly if th symbols d σ r rltd by th qutio ˆ πϕ i ( g ˆ ) ( g ) F = F σ, (6) whr th phs fuctio ϕ is giv i trms of ( ) by th formul ( gx ) ( y) gx, ϕη (, y) = η y g ( y) Proof Comprig th Schwrtz krl of M, giv i Propositio 7, to th Schwrtz krl of σ ( X, D), giv i (5), w hv TF ψ N F = TF G σ G G G F = FT T ψ F σ (7) Applyig Propositio 4 to th lft-hd sid of (7), d Propositio 6 togthr with th formul (5) of Propositio 3 to th right-hd sid, yilds th dsird qutio It is prhps clrr to formult qutio (6) with th trms tht dpd o th widows combid ito sigl fuctio F, s follows: πϕ i FF= F σ, whr F = ( gˆ ˆ ) ( g ) (8) Of cours o c solv xplicitly for th Gbor symbol to yild = N/ F σ, F F (9) with F s i (8) For th cs whr g d r comptibl, qutio (9) costituts th compio rsult to qutio (0) i Sctio Th xprssios 0 CREWES Rsrch Rport Volum 5 (003)

21 Rprsttio of lir oprtors ( gˆ ˆ ) ( g )d πiϕ corrspodig to ch of th comptibl pirs i Tbl r workd out i Tbl 3, whr th cosqut rltios btw Gbor d Koh-Nirbrg symbols r listd Th followig chrctriztio of oprtors tht c b rprstd s Gbor multiplir bsd o giv widow pir rsts sstilly o Thorm 3 Tbl 3 Compriso of Gbor d Koh-Nirbrg symbols for comptibl widows Not tht, s i Tbl, m, µ,, k, κ dot positiv sclrs, d th widows of lis (4,5) r dfid o R oly Gbor vs Koh-Nirbrg symbols for comptibl widow pirs ( ) # gt () ( τ ) Rltio btw F ( η, y) d F σ ( η, y) S ( η) F = F σ π iξ t S (0) πξ i y ( ) ( ) η + ξ F = F σ ξ 3 mt µτ mµ π η y πm iη y m+ µ m+ µ m+ µ t 4 t τ τ t 5 kt τ κτ Γ F= F σ / y πη i πi y ( η Γ ) y ( ) σ / F = + F Γ (/ ) ( + ) + κ π k+ κ ( ) ( ) k+ p κ y k y k+ κ k+ κ + πη i y k i η Γ F= ( + ) F σ Thorm 4 Lt σ S Th Koh-Nirbrg oprtor σ ( X, D) c b rprstd s Gbor multiplir bsd o giv comptibl widow pir ( ) if d oly if σ GS, or quivltly, σ H S, whr G = ( gˆ ˆ ) ( g ), H = ( g) ( gˆ ˆ ) Proof Sic th chg of vribls ( τ, t) ( tζ( t τ ), t) ssocitd to ( ) is rquird to b tmprd, th fuctio hs th proprty tht h( η, y) = = πϕη i (, y) πη i ( yζ( y)) hs = (/ h) S = S Now, for fixd oly if σ S, th qutio (6) of Thorm 3 holds for som S if d CREWES Rsrch Rport Volum 5 (003)

22 Gibso, Lmourux, d Mrgrv ( gˆ ˆ ) ( g ) ˆ Fσ S ˆ ( ) ( ) (( ) ( ˆ ˆ = g g S σ g g) ) S h W c rd off from this rsult som bsic huristic pricipls cocrig th xistc of Gbor multiplir bsd o giv widows tht rprsts giv Koh- Nirbrg oprtor, s follows I ordr to rprst σ ( X, D) s Gbor multiplir bsd o th comptibl widow pir ( ), it hs to b th cs tht ση, ( y) dcys s fst s gˆ ˆ i th vribls η, d s fst s g i th vribls y Th symbol σ ( x, ξ ) itslf hs to b s smooth s g i th vribls x, d s smooth s ˆˆ g i th vribls ξ Thorm 4 shows tht for Gbor multiplirs bsd o giv widow pir ( ) to rprst wid clss of lir oprtors, th fuctio H should pproximt dlt fuctio, so tht H S S Not tht th scop of idticl widows g = is limitd by th ucrtity pricipl: it is ot possibl for g d ˆˆ g to both b highly loclizd, so th symbol σ ( x, ξ ) of oprtor rprstbl usig idticl widows must b slowly vryig i ithr x or ξ This costrit c b voidd by usig distict widows, providd o of is highly loclizd whil th Fourir trsform of th othr is highly loclizd I kpig with this ssrtio, w will s i Sctio 35 tht mximum grlity is chivd with th sigulr widow pir ( ) = (, δ ) d its rvrsl ( δ, ) Grlity of Gussi vrsus xtrm vlu widows With rspct to th rliztio of lir oprtors s Gbor multiplirs, th grlity of giv widow pir ( ) is rprstd by th st Rg (, ) = σ S S such tht σ( XD, ) = M Ad so, giv two widow pirs ( ), ( ) o grl th ( ) if R, w sy tht ( g, ) is mor or quivltly, if Rg (, ) Rg (, ), Rg (, ) Rg (, ) I trms to this ottio, Thorm 4 sys tht for comptibl widow pir o R, CREWES Rsrch Rport Volum 5 (003)

23 Rprsttio of lir oprtors ( ) ( ) ( ˆ ˆ) d ( ) ( ˆ R = g g S R = g ˆ ) ( g ) S, which llows us to compr th grlity of y two comptibl widow pirs For xmpl, th xt rsult shows tht xtrm vlu widows r mor grl th Gussis Propositio 7 Lt, m, µ > 0 b fixd O ( () ()) is mor grl th th widow pir R, th widow pir t t g t, t =, ( () ()) t t mt t g t, t = µ, Proof W crry out som computtios so s to b bl to pply Thorm 4 Writ From Tbl 3, K = ( g ) ( g ), K = ( g ) ( g ) π i ( η) πη i + y K ( η y) ( ) y g / ( ), Γ / + Not tht KS = KS, whr Ad KS = K S, whr, = Γ, mµ π η y m+ µ m+ µ K ( η, y) = πη i + η Γ (/ ) g, K (, y) = K ( η, y) π i ( η) =Γ y y / ( + ) K ( η, y) = K ( η, y) m π η y m+ µ m+ µ = µ W clim tht K / K P, from which it follows tht ( K / K ) S S To vrify th clim, it suffics to comput th rt of dcy of K d to s tht it is lss th tht of K By Stirlig s pproximtio this works out to b CREWES Rsrch Rport Volum 5 (003) 3

24 Gibso, Lmourux, d Mrgrv ( ) η ( ) y K η, y η π +, which is slowr th th Gussi dcy of K Now, lt ρ S b rbitrry, d cosidr th distributio K ρ : (( ) ρ) K ρ = K K / K K S, which provs tht KS KS, or quivltly KS KS By Thorm 4, this shows tht ( ) is mor grl th ( ) Sigulr widows Th clss of widow pirs stblishd i Dfiitio is usful bcus Gbor multiplirs bsd o thm r wll-dfid irrspctiv of th Gbor symbol W wr thus fr to crry out somwht grl lysis, ucumbrd by qustios of itrprttio or scop of vlidity of formuls But if o is itrstd, ot i grl lysis, but rthr i studyig prticulr oprtors such s, sy, spcific typ of psudodiffrtil oprtor, th thr is good rso to cosidr widows of mor grl typ This coms of cours t pric Gbor multiplirs bsd o mor grl widows my b wll-dfid oly for limitd clss of Gbor symbols, d o hs to crry out som sort of d hoc lysis to dtrmi this clss W will cosidr fw prticulr xmpls of such sigulr widows To bgi, w stblish th rlvt dfiitios Dfiitio 8 If S r such tht o of th xprssios or, g is wlldfid d o-zro, but ( ) is ot widow pir i th ss of Dfiitio, th w cll ( ) sigulr widow pir o R Dfiitio 9 W sy tht sigulr widow pir ( ) o R is comptibl, d tht g d r sigulr comptibl widows, if thr xist distributios G, G S d chg of vribls ( vw, ) = ψ ( τ, t) (ot cssrily tmprd) with w= t τ such tht for vry x R, th qutio holds i th ss of distributios T g = T T G G ( xx, ) ψ ( x, 0) Prhps th simplst wy to obti sigulr comptibl widows is s limits of rgulr comptibl widows For istc, if k δ is pproximt idtity, th th comptibl pirs (, k ), which hv cosistt chg of vribls, td to th limit (, δ ) Th pir (, δ ) is idd sigulr comptibl, with ccompyig fuctios G = δ, G = d chg of vribls ψ ( τ, t) = ( τ, t τ) Not tht, for vry ϕ, θ S, 4 CREWES Rsrch Rport Volum 5 (003)

25 Rprsttio of lir oprtors Thus πix ξ δ V ϕ( x, ξ) = ϕ( ξ), V θ( x, ξ) = θ( x) δ πix ξ M, ϕ, θ =, θ ϕ (30) Sic ix π ξ θ ϕ S, th qutio (30) shows tht th Gbor multiplir M is wlldfid for vry S,δ Th Schwrtz krl of It follows tht M my b computd dirctly from (30):,δ ix θ ϕ = F θ ϕ πix ξ π ξ = F T θ ϕ ( TF ) = θ ϕ K M = T F (3),δ ( ) (W could hv ltrtivly pplid th formul of Propositio 7 to obti th sm rsult) Th xprssio (3) is of cours th Schwrtz krl of th Koh-Nirbrg psudodiffrtil oprtor hvig symbol Tht is, chgig ottio slightly so tht σ dots th Gbor symbol i plc of, w hv: Propositio 0 For vry σ S, M = σ ( X, D), δ σ This rsult is of thorticl importc i tht it shows Gbor multiplirs to b compltly grl, t lst oc w llow sigulr widows Applyig Propositio 8 d th djoit formul of Propositio 4 yilds tht M M δ,, δ = for y S Thus ( δ, ) too is sigulr comptibl pir which givs ris to wlldfid Gbor multiplir irrspctiv of th symbol Although thy r gui xmpls of sigulr comptibl widows, th pirs (, δ ) d ( dlt, ) do ot rlly giv ythig w, sic Gbor multiplirs bsd o thm r just Koh-Nirbrg oprtors (or thir djoits) i disguis O th othr hd, if w lt td to i th comptibl pir o t kt, τ κτ R, w obti somthig w, mly th pir CREWES Rsrch Rport Volum 5 (003) 5

26 Gibso, Lmourux, d Mrgrv kt ( gt ( ), τ ( )) = H( t ) H( τ) κτ,, (3) whr H dots th Hvisid jump fuctio d k, κ > 0 r costts Th ssocitd ( ) fuctios () ( ) k+κ v G v = H v, kw ( kh ( w) κ H ( w)) w w 0 G ( w) = =, κ w w> 0 d th chg of vribls ( vw, ) = ψ ( τ, t) = (mx{ τ, t}, t τ), vrify tht (3) coforms to th dfiitio of sigulr comptibl widow pir By Propositio 7 th Gbor multiplir M bsd o th widows (3) hs krl K ( g, M ) = T N ψf F, (33) G G providd th lttr is wll-dfid A short clcultio yilds tht k + κ ( kh ( y) κ H ( y)) y G G ( η, y) = k+ κ πiη Not tht ( kh ( y) κ H ( y)) y is ot diffrtibl t y = 0, which implis tht if ( η, y) is sigulr t som poit ( η, 0) th th usul distributiol clculus brks dow for th right-hd sid of (33) I this cs th itrprttio of g, M rquirs dditiol clrifictio (which i prctic c b quit strightforwrd) W hv lrdy poitd out tht for y S, th pir (, ) is comptibl, d hc so is its rvrsl (, ) I dimsio =, limitig cs of th lttr is th pir (, ), ( t) = H( t) t (34) Th tructd xpotil i th pir (34) is dvtgous for coupl of rsos Not oly is it mbl to xplicit clcultio, but lso th sigulrity t 0 rdrs th pir (, ) mor grl th if wr Schwrtz clss fuctio, s w will ow show Not first tht (34) is sigulr comptibl with ssocitd fuctios G =, G =, ψτ (, t) = ( t, t τ) Ad th Gbor multiplir, M mks ss for rbitrry symbol S Furthrmor,, G πiη πϕ i ( y) G η, =, FT T ψ F = F, whr ϕ( η, y) = ηy Thus if Koh-Nirbrg oprtor σ ( X, D) c b rprstd s, Gbor multiplir M bsd o th widows (34), th by (7) th Gbor d Koh- Nirbrg symbols r rltd by th qutio 6 CREWES Rsrch Rport Volum 5 (003)

27 Rprsttio of lir oprtors F η, y = F σ η, y πη i πη i y ( ) ( ), ξ = πξ σ (35) ix x ( ) ( i ) π ξ Th qutio (35) imposs o rstrictios o σ for th xistc of corrspodig Gbor symbol I othr words, th widow pir (34) is compltly grl: t RH ( ( t ), ) = S SUMMARY Th strtig poit for th prst ppr ws th two-fold prmis tht (i) i prctic o oft coutrs lir oprtors i Koh-Nirbrg form, σ ( X, D), d tht (ii) it is of itrst both from th umricl d lyticl poits of viw to xprss such oprtor s Gbor multiplir M = V N V, for som choic of loclizig widows g W hv dtrmid clss of widow pirs, mly, comptibl pirs d sigulr comptibl pirs, for which thr xists xplicit formul by which to comput th Gbor symbol i trms of th giv Koh-Nirbrg symbol σ Mor prcisly, th rtio σ / of th Fourir trsforms of th symbols is giv i trms of th widows by th xprssio whr th phs fuctio ϕ is πϕ i ( gˆ ˆ ) ( g ), (36) ( gx ) ( y) gx, ϕη (, y) = η y g ( y) g (37), Extrm vlu widows r prticulrly itrstig xmpl of comptibl widows which r pproximtly cusl (or rvrs cusl), d r mor grl th Gussis with rspct to th rprsttio of lir oprtors Th clss of sigulr comptibl widows icluds widow pirs i trms of which vry lir oprtor my b rprstd s Gbor multiplir Withi this frmwork, th Koh-Nirbrg formultio itslf is xmpl of Gbor multiplir, bsd o th sigulr widow pir (, δ ) Ths rsults offr th prospct of ivstigtig (or vlutig) prticulr lir oprtors, for xmpl prtil diffrtil oprtors or osttiory filtrs, i trms of thir myrid rprsttios s Gbor multiplirs I coclusio w mtio coupl of op problms stmmig from th prst ppr Cocrig th formul (36) for th rtio of th trsformd symbols, it is kow to b vlid for comptibl d sigulr comptibl widows But r thr othr, ocomptibl widows for which it is lso vlid? Mor grlly, is thr brodr clss of widow pirs, for which formuls logous to (36) d (37) c b obtid? CREWES Rsrch Rport Volum 5 (003) 7

28 Gibso, Lmourux, d Mrgrv REFERENCES [] Hs G Fichtigr d Krysztof Nowk, A first survy of Gbor multiplirs, i Advcs i Gbor lysis, Hs G Fichtigr d Thoms Strohmr, ditors, Applid d Numricl Hrmoic Alysis, Birkhäusr, Bosto, 003 [] R A Fishr d L H C Tipptt, Limitig forms of th frqucy distributio of th lrgst or smllst mmbr of smpl, Procdigs of th Cmbridg Philosophicl Socity, XXIV Pt (98), [3] Ptr C Gibso d Ptr Zizlr, Comptibl widows i Gbor lysis (workig titl), i prprtio [4] Krlhiz Gröchi Foudtios of tim-frqucy lysis, Applid d Numricl Hrmoic Alysis, Birkhäusr, Bosto, 00 [5] Michl P Lmourux, Ptr C Gibso, Jff P Grossm d Gry F Mrgrv, A fst, discrt Gbor trsform vi prtitio of uity, 35pp, submittd to Th Jourl of Fourir Alysis d Applictios [6] Gry F Mrgrv d Robrt J Frguso, Wvfild xtrpoltio by osttiory phs shift, Gophysics, 64 (999), [7] Gry F Mrgrv, Robrt J Frguso, d Michl P Lmourux, 00, Approximt Fourir Itgrl Wvfild Extrpoltors for Htrogous, Aisotropic Mdi, Cdi Applid Mthmtics Qurtrly, i prss [8] Gry F Mrgrv, Michl P Lmourux, Jff P Grossm, d V Iliscu, Gbor dcovolutio of sismic dt for sourc wvform d Q corrctio, i 7d A Itrt Mt Soc of Expl Gophys, Expdd Abstrct Volum, (00), [9] Richrd Rochbrg d Kzuy Tchizw, Psudodiffrtil oprtors, Gbor frms, d locl trigoomtric bss, i Gbor lysis d lgorithms: Thory d pplictios, Hs G Fichtigr d Thoms Strohmr, ditors, Applid d Numricl Hrmoic Alysis, Birkhäusr, Bosto, 998 [0] Lurt Schwrtz, Théori ds oyux, Procdigs of th Itrtiol Cogrss of Mthmticis (950), CREWES Rsrch Rport Volum 5 (003)