The abc-Problem for Gabor Systems

116 Pages·2016·1.064 MB·English

by  Xin-rong Dai, Qiyu Sun

MEMOIRS
of the
American Mathematical Society
Volume 244 • Number 1152 • Forthcoming
The abc-Problem for Gabor Systems
Xin-Rong Dai
Qiyu Sun
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 244 • Number 1152 • Forthcoming
The abc-Problem for Gabor Systems
Xin-Rong Dai
Qiyu Sun
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Names: Dai,Xin-Rong,1971-—Sun,Qiyu,1966-
Title: Theabc-problemforGaborsystems/Xin-RongDai,QiyuSun.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2016.
|Series: Memoirsofthe AmericanMathematicalSociety, ISSN 0065-9266; volume 244,number
1152|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2016031123(print)|LCCN2016035445(ebook)|ISBN9781470420154
(alk. paper)|ISBN9781470435042
Subjects: LCSH:Wavelets(Mathematics)|Gabortransforms. |Matrices.
Classification: LCCQC20.7.W38D352016(print)|LCCQC20.7.W38(ebook)|
DDC515/.2433–dc23
LCrecordavailableathttps://lccn.loc.gov/2016031123
DOI:http://dx.doi.org/10.1090/memo/1152
Memoirs of the American Mathematical Society
Thisjournalisdevotedentirelytoresearchinpureandappliedmathematics.
Subscription information. Beginning with the January 2010 issue, Memoirs is accessible
from www.ams.org/journals. The 2016 subscription begins with volume 239 and consists of six
mailings, each containing one or more numbers. Subscription prices for 2016 are as follows: for
paperdelivery,US$890list,US$712.00institutionalmember;forelectronicdelivery,US$784list,
US$627.20institutional member. Uponrequest, subscribers topaper delivery ofthis journalare
also entitled to receive electronic delivery. If ordering the paper version, add US$10 for delivery
withintheUnitedStates;US$69foroutsidetheUnitedStates. Subscriptionrenewalsaresubject
tolatefees. Seewww.ams.org/help-faqformorejournalsubscriptioninformation. Eachnumber
maybeorderedseparately;please specifynumber whenorderinganindividualnumber.
Back number information. Forbackissuesseewww.ams.org/backvols.
Subscriptions and orders should be addressed to the American Mathematical Society, P.O.
Box 845904, Boston, MA 02284-5904USA. All orders must be accompanied by payment. Other
correspondenceshouldbeaddressedto201CharlesStreet,Providence,RI02904-2294USA.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2)
service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink.
Sendreq[e-mail].
Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases,
requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).
Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe
firstpageofeacharticlewithinproceedingsvolumes.
MemoirsoftheAmericanMathematicalSociety (ISSN0065-9266(print);1947-6221(online))
ispublishedbimonthly(eachvolumeconsistingusuallyofmorethanonenumber)bytheAmerican
MathematicalSocietyat201CharlesStreet,Providence,RI02904-2294USA.Periodicalspostage
paid at Providence, RI.Postmaster: Send address changes to Memoirs, AmericanMathematical
Society,201CharlesStreet,Providence,RI02904-2294USA.
(cid:2)c 2016bytheAmericanMathematicalSociety. Allrightsreserved.
(cid:2)
ThispublicationisindexedinMathematicalReviewsR,Zentralblatt MATH,ScienceCitation
Index(cid:2)R,ScienceCitation IndexTM-Expanded,ISI Alerting ServicesSM,SciSearch(cid:2)R,Research
(cid:2) (cid:2) (cid:2)
AlertR,CompuMathCitation IndexR,Current ContentsR/Physical, Chemical& Earth
Sciences. ThispublicationisarchivedinPortico andCLOCKSS.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 212019181716
Contents
Preface vii
Chapter 1. Introduction 1
1.1. Outlines 3
Chapter 2. Gabor Frames and Infinite Matrices 11
2.1. Gabor frames and uniform stability of infinite matrices 13
2.2. Maximal lengths of consecutive twos in range spaces of infinite
matrices 15
2.3. Uniform stability and null spaces of infinite matrices 16
Chapter 3. Maximal Invariant Sets 21
3.1. Maximality of invariant sets 23
3.2. Explicit construction of maximal invariant sets 27
3.3. Maximal invariant sets around the origin 28
3.4. Gabor frames and maximal invariant sets 33
3.5. Instability of infinite matrices 34
Chapter 4. Piecewise Linear Transformations 37
4.1. Hutchinson’s construction of maximal invariant sets 38
4.2. Piecewise linear transformations onto maximal invariant sets 39
4.3. Gabor frames and covering of maximal invariant sets 40
Chapter 5. Maximal Invariant Sets with Irrational Time Shifts 43
5.1. Maximal invariant sets with irrational time shifts 45
5.2. Nontriviality of maximal invariant sets with irrational time shifts 48
5.3. Ergodicity of piecewise linear transformations 53
Chapter 6. Maximal Invariant Sets with Rational Time Shifts 57
6.1. Maximal invariant sets with rational time shifts I 62
6.2. Maximal invariant sets with rational time shifts II 63
6.3. Cyclic group structure of maximal invariant sets 68
6.4. Nontriviality of maximal invariant sets with rational time shifts 72
Chapter 7. The abc-problem for Gabor Systems 81
7.1. Proofs 84
Appendix A. Algorithm 91
Appendix B. Uniform sampling of signals in a shift-invariant space 95
Bibliography 97
iii

Abstract
A longstanding problem in Gabor theoryis toidentify time-frequency shifting
lattices aZ × bZ and ideal window functions χ on intervals I of length c such
I
that {e−2πinbtχ (t−ma) : (m,n) ∈ Z×Z} are Gabor frames for the space of
I
all square-integrable functions on the real line. In this paper, we create a time-
domain approach for Gabor frames, introduce novel techniques involving invariant
sets of non-contractive and non-measure-preserving transformations on the line,
and provide a complete answer to the above abc-problem for Gabor systems.
ReceivedbytheeditorMarch25,2014.
ArticleelectronicallypublishedonJune17,2016.
DOI:http://dx.doi.org/10.1090/memo/1152
2010 MathematicsSubjectClassification. Primary42C15,42C40;Secondary37A05,94A20.
Key words and phrases. abc-problem for Gabor systems, Gabor frames, infinite matrices,
piecewiselineartransformation,ergodictheorem,sampling,shift-invariantspaces.
Xin-Rong Dai’s affiliation: School of Mathematics, Sun Yat-sen University, Guangzhou,
510275,People’sRepublicofChina;email: [email protected].
QiyuSun’s affiliation: DepartmentofMathematics,UniversityofCentralFlorida,Orlando,
Florida32816;email: [email protected].
(cid:2)c2016 American Mathematical Society
v

Preface
A Gabor system generated by a window function φ and a rectangular lattice
aZ×bZ is given by
G(φ,aZ×bZ):={e−2πinbtφ(t−ma): (m,n)∈Z×Z}.
Gabor theory could date back to the completeness claim in 1932 by von Neumann
and the expansion conjecture in 1946 by Gabor. Gabor theory has close links to
Fourier analysis, operator algebraand complex analysis, and it has beenapplied in
a wide range of mathematical and engineering fields.
OneoffundamentalproblemsinGabortheoryistoidentifywindowfunctionsφ
andtime-frequencyshift latticesaZ×bZ suchthatG(φ,aZ×bZ)areGabor frames
for the space L2(R) of all square-integrable functions on the real line R. Denote by
R(φ) the set of density parameter pairs (a,b) such that G(φ,aZ×bZ) is a frame
for L2(R). The range R(φ) is an open domain on the plane for window functions
φ in Feichtinger algebra, but that range is fully known surprisingly only for small
numbers of window functions, including the Gaussian window function and totally
positive window functions.
The ranges R(φ) associated with general window functions φ, especially out-
side Feichtinger algebra, are almost nothing known and Janssen’s tie suggests that
theycouldbearbitrarilycomplicated. Idealwindowfunctionsχ onintervalsI are
I
importantexamplesofsuchwindowfunctionsandtheyhavereceivedspecialatten-
tions. Inthispaper,weanswerthatrangeproblembyprovidingafullclassification
of triples (a,b,c) for which G(χ ,aZ×bZ) generated by the ideal window function
I
χ on an interval I of length c is a Gabor frame for L2(R), i.e., the abc-problem
I
for Gabor systems. For an interval I of length c, we show that the range R(χ ) of
I
density parameter pairs (a,b) is neither open nor path-connected, and it is a dense
subset of the open region below the equilateral hyperbola ab=1 and on the left of
the vertical line a=c.
To study the range R(χ ) of density parameter pairs (a,b) associated with
I
ideal window function χ , we normalize the interval I to [0,c) and the frequency
I
parameterbto1. Thisreducestheabc-problemforGaborsystemstofindingoutall
pairs (a,c) of time-spacing and window-size parameters such that G(χ ,aZ×Z)
[0,c)
are Gabor frames.
Denote by B0 the set of all binary vectors x := (x(λ))λ∈Z with x(0) = 1 and
x(λ)∈{0,1} for all λ∈Z, and let D contain all real numbers t for which there
a,c
exists a binary solution x∈B0 to the following infinite-dimensional linear system
(cid:2)
χ (t−μ+λ)x(λ)=2, μ∈aZ.
[0,c)
λ∈Z
vii
viii PREFACE
Wecreateatime-domainapproachtoGaborframesandshowthatG(χ ,aZ×Z)
[0,c)
is a Gabor frame if and only if D =∅.
a,c
WedonotapplytheaboveemptysetcharacterizationofGaborframesdirectly,
instead we introduce another set S of real numbers t for which there exists a
a,c
binary solution x∈B0 to another infinite-dimensional linear system
(cid:2)
χ (t−μ+λ)x(λ)=1, μ∈aZ.
[0,c)
λ∈Z
The set S is a supset of D and conversely D can be obtained from S
a,c a,c a,c a,c
by some set operations. Most importantly, S is a maximal set that is invariant
a,c
under the transformation R and that has empty intersection with its black hole
a,c
[max(c +a−1,0),min(c −a,0)+a)+aZ, where c = c−(cid:5)c(cid:6) is the fractional
0 0 0
part of the window size, and
⎧
⎨ t+(cid:5)c(cid:6) if t∈[min(c −a,0),0)+aZ
0
R (t):= t+(cid:5)c(cid:6)+1 if t∈[0,max(c +a−1,0))+aZ
a,c ⎩ 0
t if t∈[max(c +a−1,0),min(c −a,0)+a)+aZ.
0 0
The piecewise linear transformation R is non-contractive on the whole line
a,c
anditdoesnotsatisfystandardrequirementforHutchinson’sremarkableconstruc-
tion of maximal invariant sets. In this paper, we show that Hutchinson’s construc-
tion works for the maximal invariant set S of the transformation R , and even
a,c a,c
more surprisingly it requires only finite iterations, i.e.,
S =(R )D(R)\([max(c +a−1,0),min(c −a,0)+a)+aZ)
a,c a,c 0 0
forsomenonnegativeintegerD,wheneveritisnotanemptyset. Thereforecomple-
ment of the set S is a periodic set with its restriction on one periodconsisting of
a,c
finitely many holes (left-closed right-open intervals). So we may squeeze out those
holes on the line and then reconnect their endpoints. This holes-removal surgery
yields an isomorphism from the set S to the line with marks (image of holes).
a,c
Moreimportantly,restrictionofthenonlineartransformationR ontothesetS
a,c a,c
becomes a linear transformation on a line with marks, and interestingly the set of
marks forms a cyclic group for a∈Q.
After exploring deep about locations and sizes of holes, we show that hole-
removal surgery is reversible and the set S can be obtained from the real line by
a,c
puttingmarksatappropriatepositionsandtheninsertingholesofappropriatesizes
at marked positions. The above delicate and complicated augmentation operation
leads to parametrization of the set S via two nonnegative integers for a(cid:7)∈Q and
a,c
via four nonnegative integers for a∈Q. This parametrization yields our complete
answer to the abc-problem for Gabor systems, see Figure 2 on Page 82.
The piecewise linear transformation R is non-measure-preserving on the
a,c
whole line, but certain ergodic theorem could be established. As it involves four-
teencases(andfewmoresubcases)forfullclassificationoftriples(a,b,c)suchthat
G(χ ,aZ×bZ) is a Gabor frame for L2(R), an algorithm is proposed for that
[0,c)
intricate verification. The abc-problem for Gabor systems has also close link to the
stable recovery problem of rectangular signals f in the shift-invariant space
(cid:6) (cid:2) (cid:2) (cid:7)
V (χ ,Z/b):= d(λ)χ (t−λ): |d(λ)|2 <∞
2 [0,c) [0,c)
λ∈Z/b λ∈Z/b
fromtheirequally-spacedsamplesf(t +μ),μ∈aZ,witharbitraryinitialsampling
0
position t .
0
PREFACE ix
TheauthorswouldliketothankProfessorsAkramAldroubi, HansFeichtinger,
DeguangHanandCharlesMicchellifortheirremarksandsuggestions. Theproject
is partially supported by the National Science Foundation of China (No. 10871180
and 11371383), NSFC-NSF (No. 10911120394), and the National Science Founda-
tion (DMS-1109063 and DMS-1412413). The authors are partially supported by
Computational Science Innovation Team of Guangdong Province Key Laboratory
of Computational Science.
Xin-Rong Dai and Qiyu Sun