Spectral Envelopes, Riesz Pairs, and Feichtinger’s Conjecture
University of Newcastle, AUSTRALIA
September 23, 2010
Wayne Lawton
Department of Mathematics
National University of [email protected]
http://www.math.nus.edu.sg/~matwml
Frames and Riesz Sets
0
e.g.
.: HZkfS k
Consider a complex Hilbert space
;,|||||,||||| 22
222 Hggfgg kZk
;,),(2
Zj jj babaZH
.)(,|||||||||||| 222
22
22 Zccfcc
Zk kk
Definition S is a frame ; Riesz set if
H
and
Rx
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
xx xxf
4.0
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||)()(||,|
Zk Ry
kZk Rx
kkZk
dyxfygdxxfxgfg
2
]1,0[
222 |)(ˆ)(ˆ||)(ˆ)(ˆ|
Zk u
iku
ZjZk Ry
iky duejufjugdyexfyg
dujufjugu Zj
2
]1,0[
|)(ˆ)(ˆ|
Zj
h juhjuhu )(ˆ)(ˆ)(
||||||||||||||||)( min
22
]1,0[
fff
u
g gduu
duuu fg
u
)()(]1,0[
where the Grammian
Fourier Tricks for the Riesz Boundsduuucdyyfycfc fuRyZk kk )(|)(ˆ||)(ˆ||)(ˆ||||| 2
]1,0[
2222
whereuki
kkZk k eueecc 2)(,ˆ therefore
)(uf
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
]1,0[
22
where )()(;1
1 n
nfnnfnkjk uucd
so
)(unf
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
n
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
BBt
Since
).(,|||||)(| 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
Rakk
|),(|minsuplim)( 21 kkFFd
Rakk
||min)( 2121
F
F
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
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
7219
7219
7223
121
121
21
21 B
)()(limweak 21
21
21
21
21
11 nn xxxxn
B
constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so
where 2),32(, 121
247
1 nyy nnn
hence
121 )2cos()(ˆ
j jB ykk and
|||)(ˆ|),1,0( 2 kkZk
B Riesz pair ),( FB
with syndetic .ZF COMPUTE IT !
Polynomials
Zk j
jmk
k wcwwcwL )()(
CZRTf /:
Laurent
Zk
iktkecteLtf 2
1 )()(
trigonometric
Jensen
1
01||
|||)(|logexpj
jcdttf
CCf }0{\:
Spectral Envelopes
)()( FSTM
)(, FPZFcompact and convex. Extreme points are
set of trigonometric
polynomials f whose frequencies are in F.
Fk
kcdttff 21
0
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
where
}.:{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
],1[
21
FSenFnFkk
n
Spectral Envelopes
]),([ NMSg}:{],[ NkMZkNMF integer interval
Fejer-Riesz
1
01)(,0,]),([ dttggMNNMPg
Corollary )()(]),0([ TMZSS Proof First observe that for
every 1N the
Fejer kernel ]),0([|)1(| 2
021
NSeNK k
N
kN
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
rtsi
cos2112
21
0
2)(21
21
2 |)(2||)()||1(| 21
21
)||1/(||2,1||,|| 22 re is
]),([ NMSgrepresented by a measure on the extreme points.
is
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
)(bO
[G46] b is a minimal sequence iff for every open bU the set }:{ UbZk k is syndetic.
[F81] Theorem 1.23 If then some
jF)(
jFO
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
22
121 )(limweak)(
k
k
xi
kkebFS
1
0
241
041 )2(sin2limweak
n
k
k
nx
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
