Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture

transcript

University of Newcastle, AUSTRALIA

September 23, 2010

Wayne Lawton

Department of Mathematics

National University of Singaporematwml@nus.edu.sg

http://www.math.nus.edu.sg/~matwml

Frames and Riesz Sets

.: HZkfS k

Consider a complex Hilbert space

;,|||||,||||| 22

222 Hggfgg kZk

;,),(2

Zj jj babaZH

.)(,|||||||||||| 222

22 Zccfcc

Definition S is a frame ; Riesz set if

dxxhxfhfRLH )()(,),(2

1 give Parseval frames =http://en.wikipedia.org/wiki/POVM ;give orthonormal sets.

Relation to Kadison-Singer[KS59, Lem 5] A pure state on a max. s. adj. abelian subalgebra

iff uniquely extends to))(( 2 ZB Ais pavable. No for

[CA05] Feichtinger’s Conjecture Every frame (with norms of its elements bounded below) is a finite union of Riesz sets.

[KS59] R. Kadison and I. Singer, Extensions of pure states, AJM, 81(1959), 547-564.

[CA05] P. G. Casazza, O. Christiansen, A. Lindner and R. Vershynin, Frames and the Feichtinger conjecture, PAMS, (4)133(2005), 1025-1033.

B))(( 2 ZBB ).(ZAopen for]),1,0([LA

[CA06a, Thm 4.2] Yes answer to KSP equiv. to FC.

[CA06a] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics and engineering, PNAS, (7) 103 (2006), 2032-2039.

[CA06b] Multitude of equivalences.

[CA06b] P. G. Casazza, M. Fickus, J. Tremain, and E. Weber, The Kadison-Singer problem in mathematics and engineering, Contemp. Mat., 414, AMS, Providence, RI, 2006, pp. 299-355.

F.C. for Frames of Translates

)(},:)()({ 2 RLfZkkxfxfS k

xx xxf

4.0sin)3,0[

sin)1,0[ 0.4sinc,,sinc,

give OSxsinc,)1,0[ f

give frames

but only

ZjFFFFZ j 3,321 However

}:)({ )3,0[ jFkkx are OS

}:)k-(x0.4sinc{ jFk are RS

Fourier Tricks for the Upper Frame Bound222 |)(ˆ)(ˆ||)()(||,|

kZk Rx

dyxfygdxxfxgfg

222 |)(ˆ)(ˆ||)(ˆ)(ˆ|

ZjZk Ry

iky duejufjugdyexfyg

dujufjugu Zj

|)(ˆ)(ˆ|

h juhjuhu )(ˆ)(ˆ)(

||||||||||||||||)( min

g gduu

duuu fg

)()(]1,0[

where the Grammian

Fourier Tricks for the Riesz Boundsduuucdyyfycfc fuRyZk kk )(|)(ˆ||)(ˆ||)(ˆ||||| 2

whereuki

kkZk k eueecc 2)(,ˆ therefore

kf is a Riesz set if and only if

for almost all ]1,0[u

1sinc,)1,0[ fxf 2

210)3,0[ || eeef f

]1,8.0[]2.0,0[]2.0,2.0[0.4sinc fxf

Translations by Arithmetic Sequencesduuudfc n

funZjk kk )(|)(ˆ||||| 2

where )()(;1

nfnnfnkjk uucd

nZjkfk : is a Riesz set if and only if

for almost all ]1,0[u

H. Halpern, V. Kaftal, and G. Weiss,The relative Dixmier property in discrete crossed products, J. Funct. Anal. 69 (1986), 121-140. Matrix pavings and Laurent operators, J. Operator Theory 16#2(1986), 355-374.

HKW86 If kff is Riemann integrable then

satisfies Feichtinger’s conjecture with each RS of the form nZjkfk : and approx. orthogonal.

[CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54.

CCK01 Fails if Bf with B a Cantor set.

Feichtinger’s Conjecture for Exponentials

).(,|||||)(| 22 FPffdttfBt

is a Riesz Pair if

such that

),(,),( FBZFTBorelB 0

..),(,1

PRFBFZ j

j j B satisfies FCE if

FCE : Every B satisfies FCE or equivalently

every Cantor set B satisfies FCE.

FCE FC for frames of translates.

Quadratic Optimization)(,)(ˆ)(ˆ)(ˆ|)(|

2 FPfkfjkjfdttfFkj

).(,|||||)(| 22 FPffdttfBt

the maximum 0 that satisfies

)][(specmin FF RBR where

FR is the restriction ).()(: 22 FZRF and

])(ˆ[),]([ jkkjB B ))((][ 2 ZBB

Theorem

is the Toeplitz matrix

has a bounded inverse.

..),( PRFB iff

))((][ 2 FBRBR FF

Equivalences

W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1, November 30, 2009.

Bulletin Malaysian Mathematical Sciences Society (2) 33 (2), (2010) 169-176.

(L,Lemma 1.1) ),( FB

}:{ Fkek

is a Riesz pair

is a Riesz basis for ),(2 BL

}\:{ FZkek is a frame for ),\]1,0([2 BL

hBh \]1,0[)ˆ(supp can be ‘robustly

reconstructed from samples . }\:)({ FZkkh

Lower and Upper Beurling

Properties of Integer Sets

|),(|minlim)( 1 kaaFFDRakk

and Separation

|),(|maxlim)( 1 kaaFFDRakk

Lower and Upper Asymptotic

|),(|mininflim)( 21 kkFFd

|),(|minsuplim)( 21 kkFFd

||min)( 2121

Characterizing Riesz Pairs

[MV74] Corollary 2 ..),())(/1,( PRFBBFaa

[MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82.

[BT87,SS09] Res. Inv. Thm. ..),(and 0)( PRFBFdF

[BT87] J. Bourgain and L. Tzafriri, Invertibility of "large" submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Mathematics, (2) 57 (1987),137-224.

[BT91] Theorem 4.1

|||)(ˆ|),1,0( 2 kkZk

B satisfies FCE (e.g by removing open intervals with exp. decr. lengths)

[LT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., {\bf 420}(1991),1-43.

[SS09] D. A. Spielman and N. Srivastava, An elementary proof of the restricted invertibility theorem, arXiv:0911.1114v1 [math.FA] 5 Nov 2009.

[LA09] Corollary 1.1 )(meas)(..),( BFDPRFB

[BT91] M. Ledoux and M. Talagrand, Probability in Banach Spaces, 15.4 Invertibility of Submatrices, pp. 434-437. Springer, Heidelberg, 1991.

Syndetic Sets and FCE

is syndetic if

W. Lawton, Minimal sequences and the Kadison-Singer problem, http://arxiv.org/find/grp_math/1/au:+Lawton_W/0/1/0/all/0/1, November 30, 2009.

Bulletin Malaysian Mathematical Sciences Society (2) 33 (2), (2010) 169-176.

Theorem (L,Paulsen) ]1,0[B

),( FB

Verne Paulsen, Syndetic sets, paving, and the Feichtinger conjecture, http://arxiv.org/abs/1001.4510 January 25, 2010.

V. I. Pausen, A dynamical systems approach to the Kadison-Singer problem, Journal of Functional Analysis 255 (2008), 120-132.

satisfies FCE

if and only if there exists a syndetic

ZF 0L .1,...,2,1,0 ZLF

ZF such that is a Riesz pair.

Research Problem 1.The Cantor set

]),(),(),[(\],[ 7223

)()(limweak 21

11 nn xxxxn

constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so

where 2),32(, 121

1 nyy nnn

121 )2cos()(ˆ

j jB ykk and

|||)(ˆ|),1,0( 2 kkZk

B Riesz pair ),( FB

with syndetic .ZF COMPUTE IT !

Polynomials

k wcwwcwL )()(

CZRTf /:

Laurent

iktkecteLtf 2

1 )()(

trigonometric

Jensen

|||)(|logexpj

jcdttf

CCf }0{\:

Spectral Envelopes

)()( FSTM

)(, FPZFcompact and convex. Extreme points are

set of trigonometric

polynomials f whose frequencies are in F.

kcdttff 21

22 |||)(|||||

.1||||),(:||closureweak 2 fFPff

(Banach-Alaoglu) The set if probability measures

)()( TCTM with the weak*-topology is.t

spectral envelope of F

Symbolic DynamicsThere is a bijection between integer subsets and points in the

ZFFZ }1,0{

has the product topology and the shift homeomorphism

Bebutov (symbolic) dynamical system

M. V. Bebutov, On dynamical systems in the space of continuous functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).

.,),1())(( Zkbkbkb Orbit closures are closed shift invariant subsets

}.:{closure)( ZkbbO k A point b is recurrent if for every open bU there exists a nonzero Zk with .Ubk

Research Problem 2.

Theorem If

is recurrent thenTheorem If

Proof Follows from the Riemann-Lebesque lemma.

)( FG O is convex.

then ).()( FSGS

F )(FSFurthermore, if F is nonempty then F is infinite and the set

)(FSe of extreme points consists of limits of squared moduli of

polynomials whose coefficients converge uniformly to zero.

What is )(FS and how is it related to the dynamical and

ergodic properties of the shift dynamical system

?)( FO

Sample Result for RP 2.

Theorem. Let be a shift invariantergodic measure on ),( FOX and

),(2 XLH

).0()(, bbfHf Then the positive

ffkpRZp k ,)(,: definite function

is the Fourier transform of ).(FS

Example If CZF }1,0{:is wide sense stationary then

).(|])),1[((#|lim 2

FSenFnFkk

Spectral Envelopes

]),([ NMSg}:{],[ NkMZkNMF integer interval

Fejer-Riesz

01)(,0,]),([ dttggMNNMPg

Corollary )()(]),0([ TMZSS Proof First observe that for

every 1N the

Fejer kernel ]),0([|)1(| 2

NSeNK k

hence )(0 TMK weakN

so for )(TM.

weakNN Kg

satisfies TtttNNtKN ),2/sin(/)2/)1sin(()1()( 1

Also ]),,([ NNPgN

.]),0([1)(,01

0NSgdttgg NNN

http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf

Spectral Envelopes

Corollary ]),([ NMS is convex.

Lemma ]),([ NMSg e

http://en.wikipedia.org/wiki/Choquet_theory

Choquet Every

dteeetrr

cos2112

2 |)(2||)()||1(| 21

)||1/(||2,1||,|| 22 re is

]),([ NMSgrepresented by a measure on the extreme points.

Example

.)(roots Tg

Syndetic Sets and Minimal Sequences

b is a minimal sequence if

is a minimal closed shift-invariant set.

[GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.

[G46] W. H. Gottschalk, Almost periodic points with respect to transformation semigroups, Annals of Mathematics, 47 (1946), 762-766.

nFFZ 1

[G46] b is a minimal sequence iff for every open bU the set }:{ UbZk k is syndetic.

[F81] Theorem 1.23 If then some

contains minimal sequence piecewise syndetic.0b

[F81] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.

Thue-Morse Minimal Sequence 010110011010011010010110 = b 101 bbb

The Thue–Morse sequence was first studied by Eugene Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent a rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw.

http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence

isconstructed for 0n1. through substitutions 001,110

2. through concatenations 00|1 0|1|10 0|1|10|1001

3. 2mod ofexpansion 2 base in the s1' of # nbn 4. solution of Tower of Hanoi puzzle http://www.jstor.org/pss/2974693

,F ,...}7,4,2,1,2,3,5,8{..., F

Thue-Morse Spectral Measure

121 )(limweak)(

kkebFS

041 )2(sin2limweak

S. Kakutani, Strictly ergodic symbolic dynamical systems. In Proc. 6th Berkeley Symp. On Math. Stat. and Prob., eds. Le Cam L. M., Neyman J. and Scott E. El., UC Press, 1972, pp. 319-326.

can be represented using a Riesz product

[KA72] 2nd term is purely singular continuous with dense support.

Thue-Morse Spectral Measure

1 2 4 7 8 11 13 14 16 19 21 22 25 26 28 31 32 35 37 38 41 42 44 47 49 50 52 55 56 59 61 62

Morse F=

Bohr Minimal Sets and Sequences

Let and define

and let

QR \ ZZ :

,,|)(| 21 Zkkk

),0( 41 and define the Bohr set

}.|)(|:{ kkZkB

Theorem B is a minimal sequence and

and )ˆ(supp)( BBBS Zk

kkk ))(,()(ˆ

is positive definite on .2Z

Research Problem 3.

Group Theory Z discrete group D, T extreme pos. def. functions on D that = 1 at identity, or T compact group G and Z matrix entries of irred. representations of G.

Generalize

Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture

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