A Donoho-Stark criterion for stable signal recovery in discrete wavelet subspaces

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A Donoho-Stark criterion for stable signal recovery in

discrete wavelet subspaces

Laurent Gosse

To cite this version:

Laurent Gosse.

A Donoho-Stark criterion for stable signal recovery in discrete wavelet

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re overy in dis rete wavelet subspa es

Laurent Gosse

IAC{CNR \Mauro Pi one" (sezionedi Bari) ViaAmendola 122/I -70126 Bari, Italy

Abstra t

We derive a suÆ ient onditionbymeans of whi h one an re over a s ale-limited signal fromtheknowledge of atrun ated versionof itina stablemannerfollowing the anvas introdu ed by Donoho and Stark [13℄. The proof follows from simple omputations involving the Zak transform, well-known in solid-state physi s. Ge-ometri harmoni s (in the terminology of [7℄) fors ale-limited subspa es of L

2 (R) arealsodisplayedforseveral test- ases.Finally,somealgorithmsarestudiedforthe treatment of zero-angleproblems.

Key words: Produ t oforthogonal proje tions,Hilbert-S hmidtoperator, geometri harmoni s,singularoperator with losedrange,gradientalgorithms. 1991 MSC: 47a52, 47b32, 65r20, 65t60, 94a11

1 Introdu tion

1.1 Preliminaries

Theproblemofsignalre overyandextrapolation anbeformalizedinthefollowingway: Asignalismodeledasafun tionsofthevariablet(usuallystandingfortime)belonging

toa ertain losedlinearsubspa eV ofa(separable)Hilbertspa eH ,ingeneralL 2

(R) . onlyafra tion rofsisobserved:thereisasetT (notne essarilyaninterval) su h that foreveryt2T,r(t)=0,expressingthefa tthatthe orrespondinginformationhasbeen lost.If

A

standsforthe hara teristi fun tionofthesetA,one anwriter=(1 T

)s. worse, theobservations an be orrupted byanoise, whi his nonethelessassumed to

besmallin L 2

(R). Inthislast ase,oneobserves~r=r+.

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ballofthe losedlinearsubspa eV.IfPisa ompa toperator,thenB=P(B)is ompa t, andthisimpliesthatV isnite-dimensionalwhi h istoorestri tive.

Wenowdedu ea\trun ationoperator",theproje tionT: T :L 2 (R)!L 2 (R) ; f 7!(1 T )f:

It is assumed that QP is a ompa t operator(but notne essarily TP). As s2 V, Ps=s,theobservationsrewrite:

r=Ts=TPs=(Id Q)Ps; ~r=TPs+ : (1.1) However,one anobservethat, thanksagaintotheassumptionPs=s,one hasmoreover: r=TPs=(Id Q)Ps=(Id QP)s; ~r=(Id QP)s+ : (1.2) A tually, fromthissimple al ulation,one anmakethefollowingimportantremarks: QP isa ompa toperatoronH=L

2

(R) aslongasjTjisnite 1

:hen eitsrangeisnot losedandzeroisana umulationpointin itsspe trum.Eigenvalues analsodisplaya verysharpde ayratedependingonthesmoothnessofthefun tionsinV (seee.g.[6,26℄). the operator Id QP dened on L

2

(R) is aFredholm operator with losed range and nite-dimensionalnull-spa e;itsrestri tionto V oin ideswithTP =(Id Q)P. TheFredholmalternativeappliedtoId QP:V !L

2

(R) ensuresthatitsrange,ran(Id QP), is losed in L

2

(R) and ran(Id QP) = ker(Id PQ) ?

. Moreover, the null-spa e ker(Id QP)isatmostofnitedimensionandin aseker(Id QP)=f0g,ran(Id QP)= L

2

(R) thusthe equations(1.2)are invertible.Hen e oneswit hes fromthe potentially ill-posedinverseproblemoftryingtosolvedire tlytheequationTPs=r(seee.g.[21℄)tothe stableone(Id QP)s=r.Morepre isely:

Theorem1 LetV andT besu hthattheoperator normkQPk<1:inthenoise-free ase, any s2V anbe fully re overedfrom r,i.e. ker(Id QP)=f0g and(Id QP)

1 r=s. In thenoisy ase, thestabilityestimate holds:

s (Id QP) 1 ~ r L 2 (R) kk L 2 (R) 1 kQPk : (1.3)

Theproof anbefoundin[13℄(Theorem4)and[49℄(Corollary1).Theestimate(1.3)shows thatthenoiseis atmostampliedbyafa tor(1 kQPk)

1

;itishen efortha onvenient strategytorelyontheFredholmoperatorId QPtoperformsignalre overy/extrapolation. However,fors62V, thesolutionof(Id QP)s=r andTPs=r will learly dier.Asa onsequen eof Theorem 1,(Id QP)

1

an be omputed(at least, theoreti ally)viaan iteratives heme,theso{ alledNeumannseries:

(Id QP) 1 = 1 X k =0 (QP) k : (1.4) 1

The niteness hypothesis for the measure of T an be understood through the simpleexampleofthe\slidingbumps":let'beaC

1

fun tionsupportedin[ 1;1℄, anddenethesequen e'

n

(t)='(t n)whi hisboundedine.g.anySobolevspa e H

s

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Proje tions(AP)method.Pra ti alperforman e anbeimprovedbyfollowingtheresultsin e.g.[5,8℄.Theso- alled\GeneralizedGer hberg-Papoulis"algorithmstudiedin[27,28,45,46℄ redu es tothe AlternatingProje tionsmethod with the hoi e ofagivenmulti-resolution subspa eV =V

J

ofs ale-limitedfun tions,forsomes aleparameterJ 2N. Corollary 1 Under the hypotheses of Theorem 1, let s

` = P ` k =0 (QP) k r. The following errorestimate holds:(linear onvergen e ofAlternatingProje tions)

ks s ` k L 2 (R) kQPk ` ks rk L 2 (R) : (1.5)

Proof. In the present ase, the approximation s `

satises the relation: s 0 = r, s k +1 = r+QPs k .Hen es k +1 s=(Id Q)s+QPs k s=QP(s k

s);theresult(1.5)follows.2 Similaralgorithmsarealsowidelyusedinthe ontextofirregularsampling,seeforinstan e [15,18,39℄, whi h addresses the related issue of re onstru ting a fun tion belonging to a subspa eV startingfrom a olle tion ofpointwiseobservations; these results anbeseen asanextremeexampleofthelarge-sievestabilityestimatesprovedin [12℄.

1.2 Obje tives andoutline ofthe paper

We are partlymotivated by the question raised in the paper [27℄ (bottom of page 229): \we assume that the observation of the signal f inside the interval [ T;T℄ an uniquely determinethevalueoff upto[ ;℄inthetimedomain.GivenT,thevaluewilldepend on the regularity of the signal and the s ale parameters J. The mathemati al relationship betweenthese parametersisstillopen".Tothebestofourknowledge,itisstillunanswered; inthepresentpaper,wethusproposetoestablishthatunderarathersimple riterionbased ontheHilbert-S hmidtoperatornormofthe ompositeoftwoorthogonalproje tions,[13℄, stablere overyispossiblebymeansoftheiterativete hniquespresentedin[45℄.

Thispaperis thereforeorganizedasfollows.Inx2,were allthesubspa esofL 2

(R) whi h will beusefulin thepaper,namely thePaley-Wienerspa e ofband-limited fun tions and themulti-resolutionanalysis;te hni alresultsaboutthe ompositionprodu toftwo orthog-onalproje tionsin Hilbert spa e are also re alled,in luding the hara terizationthrough theminimal anoni alanglebetweensubspa es.Inx3,wederiveourDonoho-Stark riterion forstablesignalre overyby omputingtheHilbert-S hmidtnormoftheprodu tof proje -tionsPQbytakingadvantageofthestru tureofReprodu ingKernelHilbertspa es;some onsequen esareobtainedbyusingabstra tresultsfrom[25℄;numeri alsimulations follow-ingoriginalideasfrom[9℄aredisplayedin x3.4.Inx4,weexploit thefa t that aFredholm operatorhasa losedrangetostudyiterativealgorithmsforsingularoperators(see[26,33℄) in the ontext of signal re overy with a zero-angle problem (that is, when kQPk = 1); non-uniqueness is resolved by working with the \minimum-norm least squares" solution. Final on lusionsare drawnin x5. Appendixes A andB ontainauxiliaryresultsabouta te hni allemmaandtheZaktransform.

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and seemingly unknownpaper ondu ting wavelet extrapolation and omparing it to the MNLSalgorithmsof[21℄is[14℄.Donoho-Stark riterionisstudiedinthe ontextof irregu-larsamplingin[16℄,seealso[39℄.SomeelementsdealingwithCompressedSensingandthe produ toftwoorthogonalproje tionsaregivenin [41℄andalso[20℄,espe ially x5.

2 Band-limitedand s ale-limitedextrapolations

2.1 Paley-Wiener spa e andMulti-ResolutionAnalysis(MRA)

Inthemajorityofappli ations(ex eptfor[36,42,44℄),V standsforaspa eofband-limited ors ale-limitedfun tions.Foranyf 2L

2

(R),wenormalizeitsFourier-Plan hereltransform F:L 2 (R)!L 2 (R ) as: 82R; [Ff℄()= ^ f()= Z R f(t)exp( 2it)dt:

It followsthat afun tion f is saidto be band-limitedassoon asthere exists ! >0su h that

^

f()=0forany withjj>!.We anthereforeintrodu ethePaley-Wienerspa e:

PW ! (R ) = n f 2L 2 (R) su hthat ^ f()=0forjj>! o :

ThePaley-Wienertheoremstatesthatfun tionsbelongingtoPW !

(R ) anbeextended to thewhole omplexplaneasentirefun tionsofexponentialtype;

f 2PW !

(R) )8z2C; jf(z)jsup t2R

jf(t)jexp(!j=(z)j):

Asa onsequen eofanalyti ontinuationtheoryforfun tionsofone omplexvariable,the knowledge ofsu hafun tion restri tedto anyarbitraryintervalofR allowsto dedu eall its remainingvalues in C. Thus band-limited extrapolation orresponds to the hoi e V =PW

!

(R). Next,weintrodu ebrie ythe on eptofMulti-ResolutionAnalysis(MRA): (seee.g.[30℄fordetails)

Denition1 A sequen e of nested subspa es V j

is alled a Multi-ResolutionAnalysis of L 2 (R) if: f0g V 1 V 0 V 1 L 2

(R): Moreover, the following properties musthold: for all f 2L 2 (R) , kP Vj f fk L 2 !0asj!+1also,P Vj f !0asj! 1. if f(t)2V j ,thenf(t=2)2V j 1

andfor allk2Z,f(t 2 j

k)2V j

. thereexistsashift-invariantorthonormalbaseofV

0

givenbythes alingfun tion n

(t)= (t n)for n2Z.

Inthisdenition,P Vj

standsfortheorthogonalproje torontothesubspa eV j

.Intuitively, it asks for the V

j

's to be linear subspa es of L 2

(R) with in reasing temporal resolution: whenj de reases,fun tionsin V

j

tendto be ome onstants.Oppositely,whenj in reases, they are allowed to os illate with high instantaneous frequen y. The wavelet spa es W

j are dened asthe orthogonal omplement of V

j

inside V j+1

, whi h means: for all j 2 Z, V j+1 =V j W j .From n ,thebaseofV 0

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j P V j f = X n2Z <f; j;n > j;n ; <f; j;n >= Z R f(t) j;n (t):dt; (2.2)

whi histhebest approximationoff in V j

intheleast-squaressense.

2.2 Composite of twoproje tionsinHilbert spa eandstable re overy

Inallthesequel,weshallusethefollowingnotationforthenormofanybounded operator T :H!H ,HbeingaseparableHilbertspa e,

kTk : =sup f2H kTfk H kfk H :

Moreover,kerT and ran(T) will stand for its null-spa e and its range, respe tively. Very general results about the stru ture of the omposition of two orthogonal proje tions in Hilbert spa earegivenin [34℄.

Lemma1 LetH bea Hilbertspa e andP A

,P B

betwoorthogonal proje tions ontoA, B whi h are two losedlinearsubspa esof H.Then thereholds:

kP A P B k=kP B P A k1: (2.3)

Indeed,theproofoftheLemma(forwardedintheAppendix)showsabitmore:wea tually havethat kP A P B k 2 =kP B P A k 2 =(P A P B P A )=(P B P A P B

),thespe tralradius. Lemma2 Under the hypotheses ofLemma1, thereholdsmoreover:

kP A P B k=sup f2B kP A fk H kfk H =sup f2A kP B fk H kfk H =kP B P A k: (2.4)

Proof. Bydenition,theoperatornormofP A P B :H!Hreads: kP A P B k=sup f2H kP A P B fk H kfk H : Sin eP B

isanorthogonalproje tion,one ansplitH=BB ? su hthatf =P B f+(Id P B )f andkfk 2 H =kP B fk 2 H +k(Id P B )fk 2 H .Thisyields: kP A P B k 2 =sup f2H kP A P B (P B f)k 2 H kP B fk 2 H +k(Id P B )fk 2 H ;

andthisexpressionis learlymaximized forf 2B.Thesamereasoning anbemadewith P

A

and Lemma1allowsto on lude.2

Atthislevel,itisof riti alimportan etobeabletoestimateasa uratelyaspossiblethe quantitykQPkwhi h ontrolsboththeinversibilityofId QPbutalsotheerrorestimates (1.3)and(1.5).Inboth ases,the onditionkQPk<1expressesthefa t thatthere exists nofun tion belongingto V whi h L

2

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kQPk= sup f2L 2 (R) kQPfk L 2 (R) kfk L 2 (R) = sup f2L 2 (R) kQPfk L 2 (R) kPfk L 2 (R) =sup g2V kQgk L 2 (R) kgk L 2 (R) :

Forinstan e, ifone onsiders s ale-limited extrapolationwith the so{ alleddis ontinuous Haar basis((t)= [0;1℄ (t)),then kQPk=1forT =[2 J k;2 J (k+1)℄,V =V J and any k2Zhen estable re overy annotbeperformed;see howeverthe omputationswith this s aling fun tion in [7℄.Insharp ontrast,ifthes aling fun tion is hosento be a band-limited fun tion (see [45℄, or the \prolate spheroidal wavelets" in [43℄), then kQPk < 1 be ause belongs to a Paley-Wiener spa e (see Theorem 4 in [45℄). In [32℄, the authors provedthefollowingresult(seeTheorem2,page340),whi hisa onsequen eof[13,49℄: Theorem2 (see[32℄)LetV =PW

!

for agiven!>0andT anarbitrarymeasurableand boundedsetof R; thenthe equation(1.2) isalwaysinvertible, that is, kQP

! k<1. We annotexpe tsu h astrongresultin aseV =V

j

,amoregeneralsubspa ebelonging toaMRAofL

2

(R );espe ially,assoonasthes aling fun tionhas ompa tsupport and jTjisbigenough,itispossibletondnon-trivialfun tionsf 2V

j

su hthatkQfk=kfk.

2.3 Geometri interpretationof ompositionof proje tions

Denition2 Let A;B be two linear subspa es in a Hilbert spa e H ; the number 0 (A;B)

2

is alled theminimal anoni al angle betweenA andB andsatises:

os(A;B)= sup a2A;b2B j(a;b)j kakkbk : = os(A;B): (2.5)

In parti ular, os(A;B) = 1 when A B whi h is pre isely the situation one wants to absolutely avoid in the ontext of an extrapolation problem be ause it means that, as they stand, the la unary and possibly noised observations perfe tly t into the spa e of fun tions ontaining the original signal.In this ase, there is no hopefor re overyby meansofalternatingproje tionsbe auseranP

A ranP B impliesP A P B =P B P A =P A and kP A

k=1.Now, we an givea smallresult on erningan interpretation ofthe quantities involvedinLemma1asthe osineoflinearsubspa esinageneralsetting:

Lemma3 Under thehypotheses ofLemma1, thereholdskP A P B k 2 = os 2 (A;B). Proof. Thanksto(3)intheproofofLemma1,wehavethat kP

A P B k 2 =(P B P A P B )and sin eP B P A P B

isself-adjoint,thisimplies:

(P B P A P B ) =kP B P A P B k =sup u2H;kuk1 j(P B P A P B u;u)j =sup u2H (PAPBu;PBu) kuk 2 H =sup u2H (P 2 A P B u;P B u) kuk 2 H =sup b2B (P A b;P A b) kbk 2 H =sup b2B sup a2A (a;b) 2 kak 2 H kbk 2 H = os 2 (A;B) WeusedthatP A ,P B

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[11,49℄.Partofitisprovedinthestandardtextbook [4℄,pages21-22.

Theorem3 (see[11,40℄) LetA;B betwo losedlinear subspa esof aHilbertspa eH ;the followingstatements areequivalent:

(1) os(A;B)<1

(2) A+B is losedinH , i.e. A+B=A+B andA\B =f0g.

(3) There existsC>0su hthat foralla;b2AB,kak+kbkCka+bk.

Clearly,statement(1)impliesthatA\B=f0g:otherwiseitwouldsuÆ etopi kv2A\B thusP A P B v=vand1iseigenvalueofP A P B

.Inthe ontextofband-limitedextrapolation, the onditionA\B =f0ghasarather learmeaning:sin eA standsforthesubspa eof fun tionssupportedonT andBfortheoneofband-limitedfun tions,bythePaley-Wiener theorem,itisequivalenttothestatementthatnonon-zeroanalyti fun tion anvanishon apositivemeasureintervalofR.

3 Reprodu ing kernel Hilbertspa e approa hto estimate kQPk

Westartwitha lassi aldenition(see[2,48℄formoredetails):

Denition3 A (separable) Hilbert spa e H is alled a Reprodu ing kernelHilbertspa e (RKHS)offun tionsR!Rifforanyt2R, thereexistsa ontinuousfun tionK(:;t)2H , alledthe reprodu ingkernel,whi h satises:

8t2R; K(:;t)2H ; 8(t;f)2RH ; f(t)=<f;K(:;t)>= Z

K(s;t)f(s):ds:

Inotherwords,pointevaluationf 7!f(t)is ontinuousasanappli ationH!R.TheRiesz representationtheorem guarantees, for everyt2R, thatthe fun tion K(:;t) isunique. Themain pointhereisthat, under mildassumptionsand forH=L

2

(R), thespa es V of interestforband-limitedands ale-limitedextrapolationareRKHS.

Theorem4 (see[29,48℄) 1. For any ! 2 R

+

, the Paley-Wiener subspa e PW ! (R) of L 2 (R ) is a RKHS with the Shannonkernel:K ! (s;t)= R ! ! exp(2i(t s)):d= sin2!(s t) (s t) . 2. If j(t)jC(1+jtj) 1 2 "

for " >0, any multi-resolution subspa e V j is a RKHS with kernel:K j (s;t)= P n2Z j;n (s) j;n (t)=2 j P n2Z (2 j s n)(2 j t n).

3.1 Cal ulation ofthe Hilbert-S hmidtnormofQP V

j

:Donoho-Stark riterion

At this point, one observes that for K 2 L 2

(R 2

) ontinuous and any t 2 R, f(t) = R

R

K(s;t)f(s)ds isaHilbert-S hmidtoperator;hen eforband-limitedextrapolation,

P ! Qf(t)=(Qf;K ! (:;t))= Z R K ! (s;t) T (s)f(s):ds;

and asimilarexpressionholds fors ale-limitedextrapolation.Thus,on theonehandit is well-knownthatinthis ase,kP

! QkkP ! Qk HS =kK ! T k L 2 (R 2 )

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! !

normwhi h ontrolstheerrorestimate(1.5).Weswit hnowtoMRAsubspa es:

Theorem5 ForV j

beingsomeMRA subspa eofL 2

(R) asso iatedwitha ontinuous s al-ingfun tion satisfying kk

L 2 (R) =1andj(t)jC(1+jtj) 1 2 " ,thereholds kP V j Qk 2 HS = Z 2 j T Z d jj 2 (0;s):ds; (3.1)

withZ(f)(t;)standingfor the Zaktransformofthe fun tion f ( f. Appendix B).

Proof. We wantto omputekP V j Qk 2 HS = I j (T) = R T R R j P n2Z j;n (s) j;n (t)j 2 :ds:dt for anyj 2 Z.First,bya simpleres alingargument,we getthat I

j (T)=I 0 (2 j T): hen ewe on entrateonthetaskof omputingI

0

(T)whi hissplitintoseveralsteps.

(1) First,foranys2R, wedenethefun tionk s

:t7! P

n2Z

(s n)(t n)andwedo theFouriertransforminthetvariable:

^ k s ()= X n2Z (s n)exp( 2in) ^ ()= ^ ()Z(s; ):

ThePlan herelequalityallowsustorewriteI 0 (T)= R T R R j()j 2 jZ(s; )j 2 dds. (2) WeknowthatZ(s; )=exp( 2is)Z

^ (;s)andthatZ ^ (;s+1)=Z ^ (;s)for anys( f.[50℄,p.161{163),soweget: I 0 (T)= Z T Z R j()j 2 jZ ^ (;s)j 2 dds= Z T Z 1 0 jZ ^ (;s)j 2 X k 2Z j ^ (+k)j 2 | {z } =1 dds;

thanksto thepropertiesofthes alingfun tion generating aMRA(see[10℄,p.173). In aseT =[0;1℄,this is alreadyenough to on ludethat I

0 ([0;1℄) = R R j ^ ()j 2 d = kk 2 L 2 (R)

.Moregenerally,ifT =[a;b℄witha;b2Z 2 ,thenI j ([a;b℄)=2 j jTjkk 2 L 2 (R) . (3) Toestimate R 1 0 jZ ^ (;s)j 2

d,wemustusethefollowingfa t( f. [50℄,p.165): Z 1 0 Z ^ (;s)Z ^ (;s)d= X k 2Z ^ ; ^ (: n) exp( 2iks)=Z ^ (0;s); where()= ^ ( )= ^

()be auseisreal-valued.Hen etheexpressionredu esto Z 1 0 Z ^ (;s)Z ^ (;s) d=Z ^ ^ (0;s)=Z d jj 2 (0;s):

ItremainstointegrateonT toobtain(3.1).

2

As an onsequen eof(3.1), one anre overpartof theresultestablished by Donohoand Stark (Lemma2in [13℄)forinstan e in aseT =[0;n℄, n2N, and (t)=

sin!t t ,! =2 j : sin es7!Z d jj 2

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kP ! Qk 2 HS =n d jj 2 (0)=jTjkk 2 L 2 (R) =jTjk [ ! 2 ; ! 2 ℄ k 2 L 2 (R) =!jTj=2 j jTj: Thethirdequality omesfrom Plan herelidentity.

3.2 Equivalent formofthe Hilbert-S hmidtnormkQP Vj

k HS

Theestimate(3.1)isdiÆ ulttousewhenT hasmany onne ted omponents,orevenifT isanintervalwithnon-integerextremities;thefollowingresultxesthisissue:

Corollary 2 Underthe hypothesesof Theorem5, thereholds: kQP V j k 2 =kP V j Qk 2 kP V j Qk 2 HS = X k 2Z jj 2 2 j T (k): (3.2)

Proof. Thisisadire t onsequen eof thePoissonsummationformula: R T Z d jj 2 (0;s)ds= R T P k 2Z d jj 2 (k)exp( 2iks)ds = R T P k 2Z jj 2 (k s)ds = P k 2Z R R jj 2 (k s) T (s)ds = P k 2Z jj 2 T (k) 2

OnFig.1,wedisplaythesquaresofseveralstandards alingfun tionstobeusedin(3.2). Similarlyto Theorem 10 in [13℄,one anquestionthesharpness ofthe suÆ ient riterion (3.2) and wonder whether it is possible to nd sets T su h that kQP

V j

k < 1 and (3.2) isn't satised.A tually, this is possiblefor bandlimited extrapolation; however, a ru ial ingredientin theproofof Theorem 10 in [13℄ lies in the fa t that thereprodu ing kernel forthe Paley-Wienerspa e PW

!

is thefun tion \sin !

(t s)"whi h de ays whenjt sj grows.This isnotthe asewhenV

j

isaMRAsubspa ein thesense ofDenition1asone hasonlythefollowingsimpleestimate,

jK 0 (s;t)j X n j(s n)jj(t n)j C p 1+js tj ;

whi his a onsequen eof thede ayassumptionon andtheinequalityfor anyx;y 2R: (1+jxj)(1+jyj)1+jx yj.Indeed,letus onsiderT =T

1 [T

2

:the oreoftheproofin [13℄istoestablishthat,withstraightforwardnotation,<Q

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−2

−1

0

1

2

3

4

5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Fig.1. S alingfun tionsjj 2

:Daube hies4 (top, left),Daube hies6 (top, middle), Coi et 5 (top, right), Symmlet 10 (bottom, left), Meyer 3 (bottom, middle) and sin (bottom,right).

Thisquantityisthes alarprodu tinL 2 (R)ofP V 0 (f T 1 )andP V 0 (f T 2 ):itdoesn'tde rease if T 1 and T 2

arefar from ea h other. Hen e the ondition (3.2) is probablysharperthan it analogue for bandlimited extrapolation studied in [13℄. Moreover, it doesn't seem that analoguesofthe\largesieve"estimatesstudied in[12℄ allowto improve(3.2)in aseT is theunionofmanydisjointintervals.

Lemma6 LetT R andV bea losedlinear subspa eofL 2

(R ) su hthatthe orthogonal proje tions P andQ satisfy kQPk<1. Then, for any x 2V =ran(P), <Tx;x >=0 if andonly ifx=0.

Proof. Anyx2ran(P)rewritesx=Pf forsomef 2L 2 (R) ,so <Tx;x>=<Tx;Px>=<PTPf;Pf >=<T 2 Pf;Pf>=kTPfk 2 : Hen e, assuming that kTPfk

2

= 0 yields that Pf = QPf. But, from the ontents of Lemma2,thisimpliesthatforsu hanx,onehas,

1=

kQPfk kPfk

=kQPk;

whi h ontradi tsthehypothesis.2

Corollary 3 UnderthegeneralhypothesesofTheorem5,assoonasthesuÆ ient ondition P k 2Z jj 2 2 j T

(k)<1ismet,the following hold: (1) kQP

Vj k=kP

Vj

Qk<1and os ranP Vj

;ranQ

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(2) ran(P V

j

)\ran(Q)=f0gandran(P V j )ran(Q)is losedinL (R) , (3) V j = ran(P V j T); espe ially ran(P V j T) and ran(TP V j

) are losed and the operator TP Vj isnot ompa t, (4) ran(P V j +Q) = ran(P V j

) ran(Q); in parti ular, the orthogonal proje tion onto ran(P

V j

)ran(Q)reads (Id Q)(Id P V j Q) 1 P V j +(Id P V j )(Id QP V j ) 1 Q. (5) ran((Id P Vj )Q) andran(Q(Id P Vj ))are losed.

Proof. Points(1)and(2)followfromTheorem3.For(3),theproperty\ran(P V

j

(Id Q)) losed"isa onsequen eofLemma2.4in[25℄assoonasran(P

V j )+ran(Q)is losedin L 2 (R); learly,ran(P Vj T)V j

.Inordertoprovethe onverse,itsuÆ estoobservethatker(TP Vj

)= f0gfromtheproofofLemma6;moreover,ran(TP

V j

)is losedbe auseran(P V

j

T)is losed. Theorem II.19 in [4℄ allows to on ludethat P

Vj

T is onto. Points (4) and (5) also ome fromLemma 3.4of[25℄.2

A ording to [27℄ (see also [45℄), ran(P Vj

T) ispre isely the spa e U j

written in Theorem 1in the ontextof s ale-limitedextrapolation.Takinginto a ountforthenon-zeroangle hypothesisallowsto renetheirresultbyshowingthatU

j =V j aslongaskQP Vj k<1for general s aling fun tions inside an orthogonalwavelet framework.In these former works, thepropertyU

j =V

j

wasprovedonlyforband-limited s alingfun tions. Remark1 Here, we letP be any orthogonal proje tion L

2

(R ) !ran(P): from Corollary 3.2 in [25℄, it omes that both onditions kPQk < 1 and k(Id P)Tk < 1 imply that L

2

(R) =ran(P)ran(Q) be ausethe se ondoneensures that Id (Id P)(Id Q)

is invertible. Unfortunately we aren't able to present a situation for whi h kP

V j Qk<1 and k(Id P V j )Tk<1holdforV j

aMRAsubspa eandameasurablesetT.However,itisrather easytovisualizetheirmeaning:therst onditionexpressesthefa tthat,apartfromzero,no fun tion supportedonT belongstoV

j

,andthese ond,thatnofun tion supportedonRnT (thatis,the measurementsin(1.1))belongstothedire tsumofwaveletsubspa es

`>j W ` ; fun tionsbelongingto `>j W `

generallyhavea ertainnumberofvanishingmoments,[10℄.

3.3 Relation withMinimum-Norm(MN) solution

Point(2) in Corollary3has aninteresting onsequen e; namely, onsidering theso- alled Minimum-Norm (MN) solution as proposed in Theorem 1 in [28℄, it is shown that the iterationlimitsof(1.4)admitsthefollowingminimization formulation:

ksk L 2 (R) = inf f2V n kfk L 2 (R) su hthatTf =Tsfors2V o : (3.3)

Thisisoneofthe\bestapproximationproblems" onsideredin[1℄,se tions5and6.First,as soonastheinvertibility onditionkQPk<1ismet(andinparti ular,foranyband-limited extrapolationproblem, see[32℄),thisformulationisnotrelevant.Inthespe ial asewhere onedealswitha\zero-angle"problemforwhi h os(ran(P);ran(Q))=1,thedimensionof ker(Id QP)isstri tlypositiveandonemustrestri tequation(1.2)tor2ran(Id QP)= ker(Id PQ)

?

thus satisfyinga nite numberof orthogonality onditions;see espe ially theCommentandCorollary2in[49℄ (page698).

Letusbeginbyre allingaresultfrome.g.[3℄: Lemma4 Let Hbea Hilbert spa e andM

1 ;M

2 ;:::;M

K

be a family of losed linear sub-spa es of H ; if M : =\ K i=1 M i

denotes the ( losed) interse tion of the M i 's andP M i isthe orthogonal proje tionon M i

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havethatr=s QPsandthe orrespondings2V de omposesintos=(Id Q)s+Qs= r+QPs,insidewhi hone anplugagainthede ompositions=r+QPsinordertoobtain: s=r+QP(r+QPs)=r+QPr+(QP) 2 s.Denotings (k ) thek th

iterateof(1.4),onegets:

s= k X i=0 (QP) i r+(QP) k +1 s,s (k ) =s (QP) k +1 s:

Theorthogonalproje tionsQandPsatisfythehypothesesofLemma4,hen ewe andedu e thats=lim k !+1 s (k ) =s P ran(P)\ran(Q) (s)whereP ran(P)\ran(Q)

standsfortheorthogonal proje tionontotheinterse tionofV andthesubspa eoffun tionssupportedinT.Atthis level, one observesthat the onditionkQPk<1 impliesthat ran(P)\ran(Q) =f0g,so lim

k !+1 s

(k )

=s. For azero-angleextrapolationproblem, this propertydoesn'thold and thelimits anbe hara terizedbytheminimalpropertyofanyorthogonalproje tion,

ksk L 2 (R) =ks P ran(P)\ran(Q) (s)k L 2 (R) = inf f2V\ran(Q) ks fk L 2 (R) :

Asa onsequen eofboththepre edingequalityandtheorthogonalde ompositionL 2

(R)= ran(Q)ran(Q)

?

,thatpi kinganyg2V su hthat(Id Q)g=(Id Q)syieldss g= Q(s g)2V\ran(Q)whi hleadstoksk

L 2 (R) =inf g2V ks (s g)k L 2 (R) =inf g2V kgk L 2 (R) . ThisMN solutionemerging from(1.4)isunstable intheverygeneralsituation onsidered in[49℄.Herewelimitourselvestoasomewhatsimpler aseforwhi hQPis ompa twhi h yieldsthatId QP isaFredholmoperatorwith losedrange.Hen e aslightperturbation r+ of r2ran(Id QP) willstill belong tothe rangeof Id QP if is smallenough. However,thisnotionofsolutiondoesn'tallowtotreatproblemslike(1.2)forwhi hkQPk= 1andradmitsanorthogonalproje tionontoker(Id PQ);theyareinthenextse tion.

3.4 Singular ValueDe omposition(SVD)andeigenfun tions ofQP V

j Q

Withpreviousnotations,letusnowlookatA=QP asaboundedoperatordened onthe Hilbert spa e H=L

2

(R) whi his assumedto be ompa tand non-self-adjoint.Itis easy tosee thatran(P)

?

=ran(Id P)ker(QP).Thus,ifwedene, A=QP:V =ran(P)!ran(Q); A

=PQ:ran(Q)!V =ran(P); (3.5) thestandardSVDtheoryfor ompa toperatorsinHilbert spa esgives:

A A k =PQP k = k k ; AA ' k =QPQ' k = k ' k ; k 0; k2N; (3.6) whereA

=PQistheadjointofA.Moreover, A k =QP k = p k ' k ; A ' k =PQ' k = p k k : Clearly, denition(3.5) impliesthat A

A=PQP: V !V, AA

:ran(Q)!ran(Q) are self-adjointandtherefore:

( k

) k 2Z

isanorthonormalbase ofran(PQP)=ran(PQ), ('

k )

k 2Z

isanorthonormalbase ofran(QPQ)=ran(QP), Thesingularvalues

k

aresmallerthan1sin eAisa ompositionof2orthogonal proje -tions;there alsoholdskAk=

p

0

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the olle tionoffun tions(' k ; k ),normalized su hask k k L 2 (R)

=1is alledtheProlate SpheroidalWavefun tions(PSWF). Theyhavebeenstudied indetailbySlepian, Landau and Pollak; see [38℄ and [7℄,the book [19℄ andthe surveys[20,31℄for moreonthis topi . Thefun tion

0

isanextremalfun tionofthetypestudied intheSe tion4in[1℄. Lemma7 If QP isa ompa t operator,thenitssingularfun tions(3.6) satisfy:

8k2N; P' k p k = k : (3.7) Proof. FromQPQ' k = k ' k

,wegetfrom(3.6)that: k P' k =PQ(PQ' k )=PQP( p k ' k )= k p k ' k : 2

Here,wetryto omputenumeri allytheanaloguesofthefun tions' k

whenP isassumed tobetheorthogonalproje tionontoaMRAsubspa ewithagivenindexj2Zforvarious hoi esofthes alingfun tion.OnFigs.2and3,wedisplaytherst10fun tionssatisfying QP Vj Q' k = k ' k

splitting betweenthe even and odd ones. One an easily see that the shapeof thes aling fun tion appearsvery learlyin these eigenfun tionswhi h are quite dierentfromoneanothera ordingtothe hoi eofthes alingfun tion:seein parti ular theonesemergingfromtheDaube hies4 omparedtotheSymmlet10.Theeigenfun tions omingout of theCoi et 5 s aling fun tions havealso aparti ular shape. The behavior of theeigenvalues

k

ispresentedforea h hoi eof thes aling fun tion;however,evenif wedisplayedonlythe10rsteigenfun tions( orrespondingtoeigenvaluesvery loseto1), we hose to showthe whole set ofnumeri aleigenfun tions. Possibleina ura iesmaybe present be ausethe linearsystem is ill- onditioned and diÆ ultto diagonalizeeÆ iently. Matri es were 256256or 512512 and the s ale index j = 4 orj = 5; the dis rete wavelettransformisinvolvingaperiodizationof thesignal.Thesenumeri alresultsfollow early omputationsdisplayedin [9℄. Classi alPSWFwere omputed using thealgorithms proposedin[47℄with512pointsandtheSlepianparameter =13.

4 Case kQPk=1: Minimum-NormLeastSquares (MNLS) solution

Insharp ontrastwithband-limitedextrapolationforwhi hthepropertykQPk<1 gener-allyholds(whi hisanotherformoftheanalyti extensionprin ipleforfun tionsbelonging tothePaley-Wienerspa ePW

!

),itiseasytoseethatforMRAsubspa es,the\bad ase" kQP

V j

k=1 anhappen.Indeed, pi k T =[ a;a℄and as alingfun tion j;n of ompa t support: learlykQP Vj k=1for2ajsupp( j;0

)j.Throughoutthisse tion,weshallwrite P insteadofP

V j

forsimpli ity.

4.1 LeastSquaressolutionandnormal equations

Inthe ontextofband-limitedextrapolation,MNLSsolutionshavebeenstudiednumeri ally in [21℄whota kleddire tlyequations(1.1).

Denition4 LetF :X !Y beaboundedlinear operator withX,Y twoHilbert spa es: 2

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kFx yk Y = inf z2X n kFz yk Y o : (2) x y

2 X is alled \best approximate solution" if it is a Least Squares solution with minimumnorm, thatis:

kx y k X = inf z2X n kzk X

with z isaLeastSquaressolution ofFx=y o

:

It is well-known that x is aleast squares solutionsif and onlyif it satises the so{ alled \normal" equation F

Fx = F

y; in innite dimension, this modied problem may have no solution. However,in ase ran(F) is losed, the set of all least squares solutions is a nonempty onvexsetwhi hthereforeadmitsauniqueelementofminimumnorm.Hen ein this ontext,itmakessense tospeakabout\the best approximate solution"x

y

ofFx=y whi hisalsoreferredtoasitsMinimum-NormLeastSquares(MNLS)solution.

Dealing with operators with a losed range brings many advantages when it omes to solving equations like (1.1); however, ex ept in ase ran(TP) is nite dimensional (thus losed, whi h is anassumptionin [21℄), theoperator A=TP is ompa tandits rangeis generallynot losed (see [26℄for moredetails). Thus it forbidstospeak abouttheMNLS of(1.1)withoutsupplementaryassumptions.Itisthereforeinterestingtoon eagainswit h totheformulation(1.2)involvingaFredholmoperator,whi hmaybesingularinthesense thatker(Id QP)6=f0g,but forwhi hran(Id QP)isalways losed.

Thenormalequationsfor(1.2)read:

(Id PQ)(Id QP)s=(Id [PQ+QP PQP | {z } PQÆ(Id P)+QÆP

℄)s=(Id PQ)r: (4.1)

Clearly,sin eker(Id PQ) ?

=ran(Id QP),theright-handsidesatises: (Id PQ)r=(Id PQ) r P ker(Id PQ) r =(Id PQ)P ran(Id QP) r:

Be ause of the hypothesis kQPk = kPQk = 1, the formal Neumann series for inverting (4.1) may not onvergesin e for any x 2 ran(P), [PQ+QP PQP℄x = QPx and for x 0 2ran(P) ? , [PQ+QP PQP℄x 0 =PQx 0

.However,this formal seriesis equivalent to thefollowingiteratives heme,

s 0 =(Id PQ)r; s k +1 =(Id PQ)r+[PQ+QP PQP℄s k ; whi hisaspe ial aseofthefollowing\steepestdes ent"algorithm,

s k +1 =s k k (Id PQ) (Id QP)s k r ; (4.2)

withtheparti ular hoi e k

1ands 0

=(Id PQ)r.Byits onstru tion,alltheiterates ofthealgorithm(4.2)belongtoran(Id QP)

assoonastheinitialvalues 0

does:thisfa t isusedin[21℄inordertoredu e omplexityforband-limitedextrapolationwhenjTjisbig. Indeed,oneseesthat startingfrom z

0

=randthendening anauxiliarysequen e, z k +1 =z k k (Id QP)(Id PQ)z k r ; onere oversanyofthes

k

valuesin (4.2)forsomek2N by omputings k

=(Id PQ)z k

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General onvergen eresultsofgradientalgorithmsforsingularoperatorswith losedrange in Hilbert spa es have been proved in [33,24℄: we are about to adapt them now to our parti ularextrapolation/re overyproblem.

Theorem8 Consider theFredholmoperatorId QP:ran(P)!ran(T)withkQPk=1: the sequen e(s k ) k 2N generatedby (4.2)with q k =(Id PQ) (Id QP)s k r ; k = kq k k 2 k(Id QP)q k k 2 ; onverges inL 2

(R) towardaleastsquaressolutionswhi h dependson the initial value: s=s y +P ker(Id QP) s 0 ; whereP ker(Id QP)

standsfor theorthogonalproje tion ontothekernelofId QP.In ase the initial valuesatises

s 0 2ran(Id PQ)=ker(Id QP) ? ; the sequen e(s k ) k 2N

onverges towardthe MNLS s y

of the equation(Id QP)s=r. The on lusionofTheorem8stillholdsforthesimpliedversionofthealgorithmobtained byxing a onstantvalueof

k

aslongasit issmallerthan 2=kId QPk; learly, kId QPk=k(Id Q)P+(Id P)kk(Id Q)Pk+kId Pk=2.Hen e the ase

k 1is admissibleandtheformalNeumannseries omingfrom thenormalequationis onvergent:

s y = X k 0 [PQ+QP PQP℄ k (Id PQ)r:

Remark2 (1) The \ losed range hypothesis" is ru ial here: for some results valid in aseitis bypassed, seee.g. [23℄. They allow to inverseequationsinvolving a ompa t operator but theobtainedsolutionisunstable.

(2) The expression of P

ker(Id QP)

an be made more expli it by observing that ker(Id QP)=ran(P)\ran(Q)6=f0g(seeLemma2.2in[25℄).Sin etheserangesare losed,

ran(Id PQ) =ker(Id QP) ? = ran(P)\ran(Q) ? ran(P) ? +ran(Q) ? = ran(Id P) ? \ran(Id Q) ? ? =ran(Id P)+ran(Id Q) =ker(P)+ker(Q):

A ording to Corollary 3, the ondition kQPk < 1 ensures that ran(P)\ran(Q) = f0g= ker(P)+ker(Q)

?

, buthereitdoesn'thold thusran(Id PQ)6=L 2

(R). (3) The MNLS s

y

also belongsto ker(Id QP) ?

=ker(P)+ker(Q) , whi h ontains all theiteratess k assoon ass 0 2ker(Id QP) ? . (4) Stability of the MNLS s y

in the presen e of additive noise (t) in the observations is ensuredbytheboundednessoftheso{ alledMoore-Penrosegeneralizedinverseof(4.1); seee.g.[21,23{26℄ for details.

(5) The issueof the ontinuous dependen e of s y

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than the standard Ger hberg-Papoulis iterates. Hen e it makes sense to speed it up by setting up a Conjugate Gradient (CG) routine as follows: let s

0

2 ran(P) and ompute v 0 = p 0 = (Id PQ) (Id QP)s 0 r . If p 0 6= 0, then s 1 = s 0 0 p 0 with 0 = kv 0 k 2 =k(Id QP)v 0 k 2

likeintheformeralgorithm. Now,fork2N,

k 1 = <v k 1 ;p k 1 > k(Id QP)p k 1 k 2 ; v k =v k 1 k 1 (Id PQ)(Id QP)p k 1 ; (4.3) andaslongasv k 6=0ofkv k

k"with"asmallpositivenumber, ompute

k 1 = <v k ;(Id PQ)(Id QP)p k 1 > k(Id QP)p k 1 k 2 ; p k =v k k 1 p k 1 : (4.4) Whenkv k k<", itremainstosets k =s 0 P k `=0 ` p `

.Alongwiththe omputationofthe iterates,itisinterestingto omputethefollowingfun tion,

g(s k )=<v k ;s k s y >; s y theMNLSof(1:2); asitsatisesthefollowingrelation(see[24℄):

g(s k )=g(s k 1 ) k 1 kv k 1 k 2 :

At last, we dene the twopositive numbers 0 < m M as the spe tral bounds of the operatorR whi h isdenedastherestri tionof(Id PQ)(Id QP)toran(Id PQ):

8x2ran(Id PQ); mkxkkR xkMkxk: We anadaptthe onvergen eresultof[24℄toour ontextasfollows: Theorem9 Lets

0

2 ran(P), the iterates (4.3)-(4.4) generatesa sequen e(s k

) k 2N

whi h onvergesmonotoni allytowardaleastsquaressolutionof(1.2)s=s

y +P ker(Id QP) s 0 .In ases 0

2ran(Id PQ)=ker(P)+ker(Q), onehas the de ayestimate:

ks k s y k r g(s 0 ) m M m M+m k :

Ingeneral,onegivesaninitial values 0

2ran(T); fors ale-limitedextrapolation,P =P V j andker(P Vj )=ran(Id P Vj )= `>j W `

,thewaveletsubspa essatisfyingV j+1

=V j

W j and ontainingfun tionswithvanishingmoments.Thesimplest hoi e isof ourses

0 =0. Corollary 4 LetV

j

beaMRA subspa e andT su hthatkQP V j k=1:the MNLS solution s y of(1.2) satises: s y 2ran(Id P Vj Q)= `>j W ` +ran(T): In parti ular, it an happen that s

y 62V j andTP V j s y =(Id Q)P V j s y 6=(Id QP V j )s y . Proof. Thisismainly a onsequen eof Point(2)inRemark2.2

In[24℄,the onvergen eofthealgorithm(4.3)-(4.4)in aseonedealswithaboundedsingular operatorwithnon- losedrangeisalsoestablishedundersomesupplementaryhypotheseson rands

0

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forband-limitedextrapolationin [21℄andfors ale-limitedextrapolationin [28℄.

5 Con lusion

Wepresentedin this paperarather simpleandexpli it riterion allowingto estimatethe operator norm kQPkwhi h ontrols thestability of theextrapolation pro ess in the par-ti ular aseP istheorthogonalproje torontooneofthenestedsubspa esofaMRA.The hoi e of the s aling fun tion doesn't appear in the omputation, and it is not required that the \hole"T should be anintervalof R. Geometri harmoni s for several hoi es of thes alingfun tionsarealsodisplayedtogetherwiththeir orrespondingeigenvalueswhi h showa sharp de ay from nearly 1 to zero beyond a ertain level. These results allow to give a pre ise answer to a question raisedin [27℄ and also [17℄ in the ontext of a pe u-liarappli ation.Con erningtheextensionofthese Donoho-Starktype riteria,the aseof nite-dimensionalproblemsandsparseCompressedSensingsituationshavebeentreatedin [20,41℄.Moreelaborateintegraltransformsarestudiedin[42℄andPSWFforthefra tional Fouriertransformare omputedin [36,44℄towhi hasimilarapproa hmightbeapplied.

A Proof ofLemma1 Clearly,kP A P B kkP A kkP B

k1;wesplittheproofintoseveralsteps:

(1) For any bounded operator T on H , let (T) stand for the spe tral radius of T, i.e. (T)=supjjforthesu hthatT Iisnotinvertible.Then(T)=limsupkT

n jj

1=n whenn!+1.Thisyieldsinparti ular(T)jjTjj.

(2) ForT self-adjointonH , onehasjTfj 2 =(Tf;Tf)=(f;T 2 f)jfjjT 2 fjjfj 2 kT 2 k, hen ekTkkT 2 k 1=2 ,and kTkkT n k 1=n for n=2 p

,whi h implieskTk(T)and bythepre edingstep,kTk=(T).

(3) IfT, U are self-adjointand invertible,kTUk= (UTTU) 1=2 , kUTk=(TUUT) 1=2 . ButTUUT =TU(UTTU)(TU) 1

,soTUUT etUTTU aresimilarandhavethesame spe trum(thus samespe tralradius).ThisimpliesthatkTUk=kUTk.

(4) Thepre eding step is still valid when T, U are self-adjoint andlimits of self-adjoint and invertible operators;this is the ase for anytwo orthogonalproje tionsP

A and P

B

whi h anbeapproximatedbythemselvesplus"Idwith">0asmallrealnumber andIdtheidentitymapping.

2

B The Zaktransform:denition and properties

Inthisappendix se tion,welimitourselvestore allsomebasi fa ts abouttheZak trans-formoriginallyintrodu edin the ontext ofsolid-statephysi s.Following[50℄,wehave: Denition5 Letf bea ontinuousfun tionde ayingatleastlikeC(1+jtj)

1 "

with">0 asjtj!+1. TheZaktransformof f isdenedas

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Zf(t;+1)=Zf(t;); Zf(t+1;)=exp(2i)Zf(t;):

Hen eZmapsafun tiondenedonRtoanotherwhi hisfullydeterminedbyitsrestri tion to thetorus T=[0;1℄

2

in the time/frequen y plane.Let ^ f()=

R R

f(t)exp( 2it)dtbe theFouriertransformoff 2L

2 (R ), thereholds:8t;, Zf(t;)=exp(2it)Z ^ f( ;t); Z ^ f(;t)=exp(2it)Z ^ f(t; ): (B.2) Moreover,thereareinversionformulas:

f(t)= Z 1 0 Zf(t;)d; ^ f()= Z 1 0 exp( 2it)Zf(t;)dt: (B.3)

TheZaktransformisanisometryfromL 2

(R ) ontoL 2

(T)be auseforanyf;g, Z T Zf(t;)Zg(t;)dt:d= Z R f(t)g(t):dt: Referen es

[1℄ Tsuyoshi Ando, Saburou

Saitoh,Restri tionsofreprodu ing kernelHilbertspa es tosubsets,Preliminary reports,SuriKaisekiKenkyuJo,KoukyuRoku743(1991)164{187;availableat

http://www.kurims.kyoto-u.a .jp/ kyodo/kokyuroku/ ontents/pdf/0743-18.pdf [2℄ N.Aronszajn,Theoryofreprodu ing kernels,Trans.Amer.Math.So .68(1950)

pp.337{404

[3℄ H.H. Baus hke, F. Deuts h, H. Hundal,S.H. Park, A elerating the method of alternating proje tions, Trans.Amer. Math. So . 355 (2003)pp. 3433-3461. [4℄ H.Brezis, Analyse fon tionnelle: theorie et appli ations, Masson. [5℄ Ch.Byrne,Aunied treatmentofsomeiterativealgorithmsin signalpro essing

and image re onstru tion, InverseProb. 20 (2004)103{120.

[6℄ C.H. Chang,C.W. Ha,Sharp inequalities of singular values of smooth kernels, IntegralEquationsand OperatorTheory 35 (1999) 20{27.

[7℄ R.R. Coifman, S. Lafon, Geometri harmoni s: a novel tool for multis ale out-of-sample extension of empiri al fun tions, Applied and Computational Harmoni Analysis 21(2006) 31{52.

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