Spectral Envelopes of Integer Subsets 2 nd Asian Conference on Nonlinear Analysis and Optimization Patong beach, Phuket, THAILAND September 9-12, 2010.
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Spectral Envelopes of Integer Subsets
2nd Asian Conference on
Nonlinear Analysis and Optimization
Patong beach, Phuket, THAILAND
September 9-12, 2010
Wayne Lawton
Department of Mathematics
National University of Singaporematwml@nus.edu.sg
http://www.math.nus.edu.sg/~matwml
Characters
ZnTTen ,:
CRZ integer, real,
ZRT / circle group (real)
Tttkin ete ,)( 2
characters (complexexponential functions)
complex numbers
1||: wCwT circle group (complex)
Polynomials
Fk j
jmk
k wcwwcwL )()( CCL }0{\:
ZF finite subset
CTf :
Laurent polynomial
Fk
iktkecteLtf 2
1 )()(
trigonometric polynomialwhose frequencies are in F
Theorem 1 (Jensen)
1
01||
|||)(|logexpj
jcdttf
Spectral Envelopes
)()( FSTM
)(, FPZFis compact and convex. Extreme points are
set of trigonometric
polynomials f whose frequencies are in F.
Fk
kcdttff 21
0
22 |||)(|||||
.1||||),(:||closureweak 2 fFPff
Theorem 2 (Banach-Alaoglu) The set if probability
)()( TCTM with the weak*-topologymeasures.t
spectral envelope of .F
Spectral Envelopes
]),([ NMSg}:{],[ NkMZkNMF integer interval
Theorem 3 (Fejer-Riesz)
1
01)(,0,]),([ dttggMNNMPg
Corollary 1 )()(]),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 2 ]),([ NMS is convex.
Lemma 1 ]),([ NMSg is an extreme point
all the roots of its Laurent polynomial .T
http://en.wikipedia.org/wiki/Choquet_theory
Theorem 4 (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 1
Feichtinger’s Conjecture for Exponentials
).(,|||||)(| 22 FPffdttfSt
is a Riesz Pair if such that
ZFTBorelSFS ),(),,(Definition
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.
0
FCE )(TBorelSTheorem 5. (Lawton-Paulsen)
.),(..1n
j jj FZFSPR
)(TBorelS
syndetic. with ),(.... FFSPRPR
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.
Quadratic Optimization)(,)(ˆ)(ˆ)(ˆ|)(|
,
2 FPfdekfjkjfdttfFkj
SSt
Since
).(,|||||)(| 22 FPffdttfSt
the maximum 0 that satisfies
)][(eiginf FF RSR where
FR is the restriction ).()(: 22 ZFRF and
])(ˆ[),]([ jkkjS S ))((][ 2 ZBS
Theorem 6
is the Toeplitz matrix
has a bounded inverse.
),( FS is a Riesz Pair iff
))((][ 2 FBRSR FF
Numerical ExperimentsClearly the only candidate counterexamplesare Fat Cantor sets such as the set
]),(),(),[(\],[ 7223
7219
7219
7223
121
121
21
21 S
)()(limweak 21
21
21
21
21
11 nn xxxxn
S
constructed like Cantor’s ternary set but whose lengths of deleted open intervals are halved, so
where 2),32(, 121
247
1 nxx nnn hence
121 ).2cos()(ˆ
j jS ykk
Numerical Experiments
function A = cantor(N,M)% function A = cantor(N,M)y(1) = 7/24;for j = 2:M
y(j) = .5*(2^(-j-1)+3^(-j));endk = 0:N;A = 0.5*cos(2*pi*y(1)*k);for j = 2:M A = A.*cos(2*pi*y(j)*k);end
Background[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 bounded frame can be written as a finite union of Riesz sequences.
[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.
Lower and Upper Beurling
Densities of
|),(|minlim)( 1 kaaDRakk
and Separation
|),(|maxlim)( 1 kaaDRakk
Lower and Upper Asymptotic
|),(|mininflim)( 21 kkd
Rakk
|),(|minsuplim)( 21 kkd
Rakk
||min)( 2121
Z
Fat Cantor SetsSmith–Volterra–Cantor set (SVC) or the fat Cantor set is an example of a set of points on the real line R that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after themathematicians Henry Smith, Vito Volterra and Georg Cantor.
http://en.wikipedia.org/wiki/File:Smith-Volterra-Cantor_set.svg
http://www.macalester.edu/~bressoud/talks/AlleghenyCollege/Wrestling.pdf
The Smith–Volterra–Cantor set is constructed by removing certain intervals from the unit interval [0, 1].
The process begins by removing the middle 1/4 from the interval [0, 1] to obtain
The following steps consist of removing subintervals of width 1/22n from the middle of each of the 2n−1 remaining
intervals. Then remove the intervals (5/32, 7/32) and (25/32, 27/32) to get
Riesz Pairs.),/( ZZRTBorelS
))(( EPS
orTLhhPhPPS ),(||,||||||0.1 211
is a Riesz basis for its span.Definition ),( S is a Riesz Pair (RP) if
[LA09, Lem 1.1] RP),( S iff
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.
Bull. Malysian Mathematical Society (2) 33 (2), (2010) 169-176.
PPS ;)(2 TL onto }).:{)(();()( 22 neEspanTLx inx
Sorth. proj. of
orTLhhhPhP ZS ),(||,||||||||||0.2 22\2
).(||,||||||0.3 2\3\\3 TLhhPhPP STSTZ
[LA09, Cor 1.1] RP),( S ).()( SD
H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967), 37-52.
Riesz Pairs
[MV74] RP),())(/1,( SSaa
[MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc., (2) 8 (1974), 73-82.
[CA01, Thm 2.2] TSRPmnZS nn
n },,,0{),( 11 (never the case if S is a fat Cantor set)
[CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates, Collect. Math., 52(2001), 35-54.
[BT87, Res. Inv. Thm.] RPSdS ),(0)(,
[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.
[LA09, Thm 2.1] RPnot),(setBohr S
Stationary Setsa setDefinition For a discrete group G
is
[MV74]Feichtinger’s Conjecture holds forZ
SpaceHilbertGghg }:{).(,, 1
11 abhhhh
abba if
G stationary
stationary Bessel sets iff for every fatCantor set TS there exist a partition
.,...,1,RP),(1 mjSZ jm
J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew. Math., 420 (1991),1-43.
}:)({ 2 Znex inxS [BT91,Thm 4.1]
satisfies FC if .|||)(ˆ|)1,0( 2
kkZk
S
Syndetic Sets and Minimal Sequences
is syndetic if there exists a positive integerZ n with
.,...,2,1 Zn
Z1,0 is a minimal sequence if its orbit closure
These are core concepts in symbolic topological dynamics [GH55]
is a minimal closed shift-invariant set.
[GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc., Providence, R. I., 1955.
Symbolic Dynamics Connection
1.
[LA09, Thm 1.1] These conditions are equivalent:There exists a partition
2.
3. min. seq. and
.,...,1,RP),(1 mjSZ jm
RP.),( S
V. Paulson, Syndetic Sets, Pavings and the Feichtinger Conjecture,
http://arxiv.org/abs/1001.4510 January 25, 2010.
[VP10] gives powerful extensions of this result.
V. Pauson, A dynamical systems approach to the Kadison-Singer problem, J. Functional Analysis, 225 (2008), 120-132.
.RP),(syndetic SZ Z
W. Lawton, Frames and the Kadison-Singer Problem, Wavelets and Appli- cations Conference, Euler Institute, St. Petersburg, Russia, June 14-20, 2009.
W. Lawton, Extending Pure States on C*-algebras and Feichtinger’s Con- jecture, Special Program on Operator Algebras, 5th Asian Mathematical Con- ference, Putra World Trade Center, Kuala Lumpur, Malaysia, June 22-26, 2009.
Power Spectral Measure
Theorem (Khinchin, Wiener, Kolmogorov)
Definition A function
exist.
k
kkkh hnhnR
)()(lim)( 2
1
is wide sense stationary if
Since
CZh :
k
kkkh hm
)(lim 2
1 and
22
21 )(limweak
k
k
xikk
h ehS
on ThR is positive definite the Bochner-Herglotz Theorem
such that
.),()(1
0
2 ZnxdSenR hinx
h implies there exists a positive measure
hS
W. Lawton, Riesz Pairs and Feichtinger’s Conjecture, International Conf. Mathematics and Applications, Twin Towers Hotel, Bangkok, Thailand,
December 17-19, 2009.
New Result Theorem If is
is wide sense stationary and
Zis a fat Cantor set and ifTS
0 there exists a closed set TS such that
)\( STm and TSS
then ),( S is not a RP.
Proof SySTyTSS )(
Define 1,)()( )(2
21
kexPk
k
yxi
kk
then dxxPS k
k
2|)(|lim
0
m
and for all
and
mdxxPkk
1
0
2|)(|lim
such that
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
can be constructed for nonnegative n1. 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
Thue-Morse Spectral Measure
22
21 )(limweak
k
k
xikk
b ebS
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] Theorem 2nd term is purely singular continuous and
has dense support.
Corollary Let ,...}7,4,2,1,2,3,5,8{...,)(support bTM
For every 0 there exists a fat Cantor set S such that
1)( Measure Lebesgue S and ),( TMS is not a RP.
Volterra Iteration
x n
k
k
ndyyxF
0
1
0
2 )2(sin2limweak)(
that approximates the cumulative distribution
is given by
21
2
0
221
1 0,)()]2/(sin2[)( xyFdyxFx
nn
1),1(1)( 21
11 xxFxF nn
and is a weak contraction with respect to the total variation norm [BA08] and hence it converges uniformly to
M. Baake and U. Grimm, The singular continuous diffraction measure of the Thue-Morse chain, J. Phys. A: Math. Theor. 41 (2008) 422001 (6pp) , arXiv:0809.0580v2
,10,)(1 xxxF
.F
MATLAB CODEfunction [x,F] = Volterra(log2n,iter)% function [x,F] = Volterra(log2n,iter)%n = 2^log2n;dx = 1/n;x = 0:dx:1-dx;S = sin(pi*x/2).^2;F = x;for k = 1:iter
dF = F - [0 F(1:n-1)];P = S.*dF;I = cumsum(P);F(1:n/2) = I(1:2:n);F(n/2+1:n) = 1 - F(n/2:-1:1);
end
Thue-Morse Distribution 20 iterations
Thue-Morse Spectral Measure
Spline Approximation AlgorithmIs obtained by replacing
is given by
21
2
021
1 0,)()()( xyFdyAxFx
an
an
1),1(1)( 21
11 xxFxF an
an
also converges uniformly to an approximation
10,)(1 xxxF a
)1,[on 1),,0[on 1)()2/(sin2 21
21
21
212 yAy
aF to .F
Spline Approx. Distribution (20 iterations)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Spline Approx. Spectral Measure
Distribution Comparison
Binary Tree Model
)(2 1adF
0
21
21 1,1 ba
21
1
041
41
21
)(4 2adF a b
)(8 3adF
0 81
81
41
aa ab41
83
83
21
bb ba
babbaabbabbbabaabbaabaaadF a )(16 48536.01464.08536.09749.41464.08536.01464.00251.0
Binomial Approximation
),0( 21
mnmba
For every and 2n the intervals that
contribute 21
1 )intervals ofunion (andF
are those with m a’s and (n-m) b’s with hence
4660650.303390072lnln2ln
2lnln2/)(lnln
cn
mbb
bnb
so the fraction of these dyadic intervals is
dtttcn
ncnn
k
n cncnncn
k
n
21
0
][1][][
0)1(
][])[(2
)2741940.03865546exp())(exp())/][((exp2 2212
21 ncnncnnn
Hausdorff-Besicovitch Dimension d]1,0[ dimensional H. content of a subset ]1,0[S
j j
jdi
dH ISLSC :inf)(
0)(:0inf)(dim SCdS dHH
S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions". Mathematische Annalen 101 (1929). S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions". J. London Mathematical Society 12 (1937).F. Hausdorff (March 1919). "Dimension und äußeres Maß". Mathematische Annalen 79 (1–2): 157–179.
Theorem For the approximate support S of adFndn
n
dH nSC
2)2741940.03865546exp(2lim)(
])2ln2741940.038655462(ln[explim dnn
therefore 5598940.94423195)(dim SH
Thickness of Cantor Sets
101100 ,, AACAOAAAC
[AS99] S. Astels, Cantor sets and numbers with restricted partial quotients, TAMS, (1)352(1999), 133-170.
Thickness
111101010000 , AOAAAOAA
111001002 AAAAC 0
j
jCC
||
||,
||
||mininf)(
10
O
A
O
AC
Ordered Derivation
|||||,||| 10 OOOO
))(1/()()( CCC
[AS99] Thm 2.4 Let kCC ,...,1
kCCCC 111 1)()( contains an interval.
be Cantor sets. Then
)})()(,1min{/11ln(/2ln)(dim 111 CCCC kH
Research Questions1.Clearly fat Cantor sets have Hausdorff dim =1 and thickness = 1. What are these parameters for approximate supports of spectral measures of the Thue-Morse and related sequences?
3. How are these parameters related to the Riesz properties of pairs ?),( S
M. Keane, Generalized Morse sequences, Z.
Wahrscheinlichkeitstheorie verw. Geb. 10(1968),335-353
4. What happens for gen. Morse seq. [KE68]?
2. How are these properties related to multifractal properties of the TM spectral measure [BA06]?
Zai-Qiao Bai, Multifractal analysis of the spectral measure of the
Thue-Morse sequence: a periodic orbit approach, J. Phys. A: Math. Gen.
39(2006) 10959-10973.
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