Transcript Slide 1

Spectral Envelopes, Riesz Pairs,
and Feichtinger’s Conjecture
University of Newcastle, AUSTRALIA
September 23, 2010
Wayne Lawton
Department of Mathematics
National University of Singapore
[email protected]
http://www.math.nus.edu.sg/~matwml
Frames and Riesz Sets
Consider a complex Hilbert space H
2
a b ; H  L ( R),  f , h   
e.g. H   ( Z ),  a, b   
jZ j j
2
and
xR
S   fk : k  Z  H.
f ( x) h( x) dx
Definition S is a frame ; Riesz set if  0    
 || g || 
2
2
 || c || 
2
2
 |  g , f  |   || g || , g  H ;
2
2
2
||  kZ ck f k ||2   || c ||2 , c   ( Z ).
2
kZ
k
2
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
2
2
uniquely
extends
to
A  B( (Z ))
B  B( (Z )) iff B
is pavable. No for
A  L ([0,1]),
open for
A   (Z ).
[CA05] Feichtinger’s Conjecture Every frame (with norms
of its elements bounded below) is a finite union of Riesz sets.
[CA06a, Thm 4.2] Yes answer to KSP equiv. to FC.
[CA06b] Multitude of equivalences.
[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.
[CA06a] P. G. Casazza and J. Tremain, The Kadison-Singer problem in mathematics
and engineering, PNAS, (7) 103 (2006), 2032-2039.
[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
S  { f k ( x)  f ( x  k ) : k  Z }, f  L ( R)
2
f  [0,1) , sincx  sinxx , [0,3) , sinc0.4x  sin0.04.4xx give frames
but only f  [0,1) , sinc x give OS
However
Z  F1  F2  F3 , Fj  j  3Z
{ [0,3) ( x  k ) : k  Fj } are OS
{ sinc 0.4 (x - k) : k  Fj } are RS
Fourier Tricks for the Upper Frame Bound
2
|

g
,
f

|
 |

k
kZ
2
g
(
x
)
f
(
x
)
dx
|
 |
k

kZ xR
kZ yR
  |  gˆ ( y ) fˆ ( x) e 2ikydy | 2   |
kZ

kZ
yR

ˆ ( x) dy | 2
ˆ
g
(
y
)
f
k


 2iku
2
ˆ


ˆ
g
(
u

j
)
f
(
u

j
)
e
du
|



j

Z
u[ 0 ,1] 

|  gˆ (u  j ) fˆ (u  j ) | 2 du 
u[ 0,1] jZ

 g (u )  f (u ) du
u[ 0 ,1]
where the Grammian  h (u )   hˆ(u  j ) hˆ(u  j )
jZ


    g (u )du  ||  f ||   || g ||22 ||  f ||    min  ||  f || 


 u[ 0,1]

Fourier Tricks for the Riesz Bounds
||  kZ ck f k ||22  
yR
| cˆ( y) | 2 | fˆ ( y) |2 dy  
u[ 0,1]
| cˆ(u) | 2  f (u) du
2 i k u
ˆ
where c   kZ ck e k , ek (u )  e
therefore
 f k  is a Riesz set if and only if
   f (u)   for almost all u [0,1]
f  [0,1) , sincx   f  1
f  [0,3)   f | e0  e1  e2 | 2
f  sinc0.4x   f  [ 0.2,0.2]  [0,0.2]  [0.8,1]
Translations by Arithmetic Sequences
||  k j  nZ ck f k ||22  
u[ 0,1]
| dˆ (u) | 2  nf (u) du
where d k  c j  nk ;  (u )    1  f (u  n )
so  f k : k  j  nZ  is a Riesz set if and only if
n
   f (u)   for almost all u [0,1]
HKW86 If  f is Riemann integrable then  f k 
satisfies Feichtinger’s conjecture with each RS
of the form  f k : k  j  nZ and approx. orthogonal.
CCK01 Fails if  f   B with B a Cantor set.
n
f
1
n
n
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.
[CA01] P. G. Casazza, O. Christiansen, and N. Kalton, Frames of translates,
Collect. Math., 52(2001), 35-54.
Feichtinger’s Conjecture for Exponentials
B  Borel(T ), F  Z , ( B, F ) is a Riesz Pair if
2
2
  0 such that tB | f (t ) | dt   || f || , f  P( F ).
n
B satisfies FCE if Z   j 1 F j , ( B, Fj ) R. P.
FCE : Every B satisfies FCE or equivalently
every Cantor set B satisfies FCE.
FCE FC for frames of translates.
Quadratic Optimization
 fˆ ( j ) ˆ (k  j ) fˆ (k ), f  P( F )
the maximum   0 that satisfies
2
2
 | f (t ) | dt   || f || , f  P(F ).
Since  tB
| f (t ) | 2 dt 
j , kF
B
tB
 minspec( RF [ B ] R ) where [ B ] B( (Z ))

F
2
is the Toeplitz matrix [ B]( j, k )  [ ˆ B (k  j ) ]
2
2
and RF is the restriction RF :  (Z )   ( F ).

F
Theorem ( B, F ) R.P. iff RF [ B ] R  B( ( F ))
has a bounded inverse.
2
Equivalences
(L,Lemma 1.1) ( B, F ) is a Riesz pair
{ek : k  F} is a Riesz basis for L ( B),
2
{ek : k  Z \ F} is a frame for L ([0,1] \ B),
2
 supp (hˆ)  [0,1] \ B  h can be ‘robustly
reconstructed from samples {h(k ) : k  Z \ F} .
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.
Properties of Integer Sets
Lower and Upper Beurling

D ( F )  lim 1k min | F  (a, a  k ) |

k 
aR
k 
aR
D ( F )  lim 1k max | F  (a, a  k ) |
Lower and Upper Asymptotic

d ( F )  lim inf

k 
d ( F )  lim sup
and Separation
k 
1
2k
min | F  (k , k ) |
1
2k
min | F  (k , k ) |
aR
aR
( F )  min | 1  2 |
1  2 F
Characterizing Riesz Pairs

(
B
,
F
)
R
.
P
.

D
( F )  meas( B)
[LA09] Corollary 1.1
[MV74] Corollary 2 (a, a  1 / ( F ))  B  ( B, F ) R.P.
[BT87,SS09] Res. Inv. Thm. F  d  ( F )  0 and( B, F ) R.P.
[BT91] Theorem 4.1   (0,1),
2

ˆ
|

(
k
)
|
|
k
|

 B
kZ
B satisfies FCE (e.g by removing open intervals with exp. decr. lengths)
[MV74] H. L. Montgomery and R. C. Vaughan, Hilbert's inequality, J. London Math. Soc.,
(2) 8 (1974), 73-82.
[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.
[LT91] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. reine angew.
Math., {\bf 420}(1991),1-43.
[BT91] M. Ledoux and M. Talagrand, Probability in Banach Spaces, 15.4 Invertibility of
Submatrices, pp. 434-437. Springer, Heidelberg, 1991.
[SS09] D. A. Spielman and N. Srivastava, An elementary proof of the restricted
invertibility theorem, arXiv:0911.1114v1 [math.FA] 5 Nov 2009.
Syndetic Sets and FCE
F  Z is syndetic if  L  0  F  0,1,2,...,L 1   Z .
Theorem (L,Paulsen) B  [0,1] satisfies FCE
if and only if there exists a syndetic F  Z
such that ( B, F ) is a Riesz pair.
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.
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.
Research Problem 1.
The Cantor set
B  [ , ] \ [( , )  ( , )  ( , ) ]
1
2
1
2
23
72
1 1
12 12
19
72
19
72
23
72
constructed like Cantor’s ternary set but whose
lengths of deleted open intervals are halved, so
 B  weak lim ( 12   x  12  x )  ( 12   x  12  x )
1
2

1
n
where y1 
ˆ B (k ) 
1
2
7
24
1
 n1
, yn  (2

1
2
n
n
 3 ), n  2 hence

cos(
2

k
y
)
j
j 1
  (0,1), | ˆ B (k ) |2 | k |    
and
Riesz pair ( B, F )
kZ
with syndetic F  Z .
n
COMPUTE IT !
Polynomials
Laurent
f : C \ {0}  C
L(w)   ck w  cw
k
kZ
trigonometric
m
 (w   )
j
j
f :T  R / Z  C
f (t )  L  e1 (t )   ck e
2ikt
kZ
Jensen exp 0 log | f (t ) | dt  c  |  j |
1
| j |  1
Spectral Envelopes
(Banach-Alaoglu) The set if probability measures
M (T )  C (T ) with the weak*-topology is
compact and convex. Extreme points are  t .
F  Z , P( F ) 
set of trigonometric
polynomials f whose frequencies are in F.
|| f ||   | f (t ) | dt   | ck |
1
2
2
0
M (T )  S ( F )

2
kF
spectral envelope of

F

 weak  closure | f | : f  P( F ),|| f ||  1 .
2
Symbolic Dynamics
There is a bijection between integer subsets and points in the
Bebutov (symbolic) dynamical system
Z
F
where
has the product topology and the shift homeomorphism
Z  F      {0,1}

( b)(k )  b(k  1), b  , k  Z .
Orbit closures are closed shift invariant subsets
O (b)  closure{ b : k  Z}.
A point b   is recurrent if for every open U  b
k
there exists a nonzero k  Z with  b  U .
k
M. V. Bebutov, On dynamical systems in the space of continuous
functions, Bull. Mos. Gos. Univ. Mat. 2 (1940).
Research Problem 2.
Theorem If
Theorem If
G  O (  F ) then S (G)  S ( F ).
 F is recurrent then S (F ) is convex.
Furthermore, if
F
is nonempty then
F
is infinite and the set
Se (F ) of extreme points consists of limits of squared moduli of
polynomials whose coefficients converge uniformly to zero.
Proof Follows from the Riemann-Lebesque lemma.
What is
S (F ) and how is it related to the dynamical and
ergodic properties of the shift dynamical system
O ( F ) ?
Sample Result for RP 2.
Theorem. Let  be a shift invariant
ergodic measure on X  O (  F ),
and f  H , f (b)  b(0). Then the positive
definite function p : Z  R, p(k )   f   k , f
is the Fourier transform of   S (F ).
H  L2 ( X ,  )
Example If  F : Z  {0,1}  C
is wide sense stationary then
lim | (# ( F  [1, n]))
n 
 12

ek | 2    S ( F ).
kF [1, n ]

Spectral Envelopes
F  [M , N ]  {k  Z : M  k  N} integer interval
Fejer-Riesz g  S ([M , N ]) 
1
g  P([M  N , N  M ]) , g  0,  g (t )dt  1
0
Corollary S ([0, ])  S (Z )  M (T )
Proof First observe that for every N  1 the
N

2
K

|
(
N

1
)
e
|
Fejer kernel N
 k 0 k  S ([0, N ])
satisfies K N (t )  ( N 1)1 sin((N 1)t / 2) / sin(t / 2),t T
weak
K

hence N   0  M (T ) so for   M (T )
g N  K N   weak
 . Also g N  P([ N , N ]),
1
2

1
g N  0,  g N (t )dt  1  g N  S ([0, N ]).
0
http://people.virginia.edu/~jlr5m/Papers/FejerRiesz.pdf
Spectral Envelopes
Corollary
S ([M , N ]) is convex.
Lemma g  Se ([M , N ])  roots( g )  T .
Choquet Every g  S ([M , N ]) is
represented by a measure on the extreme points.
Example   |  | e
2 is
, |  |  1, r  2 |  | /(1 |  | )
2
1
| (1 |  | ) (e1   ) |   | 2 (e1  e
2  12
2
0
 12
2 i ( s  t )
2
1

r
)|
dt
2
1r 2r cost
2
http://en.wikipedia.org/wiki/Choquet_theory
Syndetic Sets and Minimal Sequences
b
is a minimal sequence if
O (b)
is a minimal closed shift-invariant set.
[G46]
b
the set
is a minimal sequence iff for every open
U b
{ k  Z :  b U } is syndetic.
k
Z  F1  Fn then some O (  F )
j
contains minimal sequence b  0  Fj piecewise syndetic.
[F81] Theorem 1.23 If
[G46] W. H. Gottschalk, Almost periodic points with respect to transformation
semigroups, Annals of Mathematics, 47 (1946), 762-766.
[GH55] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc.,
Providence, R. I., 1955.
[F81] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,
Princeton University Press, Princeton, New Jersey, 1981.
Thue-Morse Minimal Sequence
b = b1  b0b1   10010110 0110100110
010110
  F , F  {...,8,5,3,2,1,2,4,7,...} isconstructed for n  0
1. through substitutions 001,110
2. through concatenations 00|1 0|1|10  0|1|10|1001 
3. bn # of 1's in thebase 2 expansionof n mod2
4. solution of Tower of Hanoi puzzle http://www.jstor.org/pss/2974693
http://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence
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.
Thue-Morse Spectral Measure
can be represented using a Riesz product
S ( F )    weak lim

k 
1
2 k 1

n 1
 k
  k

b() e
2 i  x
2
  0  weak lim  2 sin (2  x)
1
4
1
4

n
2
k

k 0
[KA72] 2nd term is purely singular continuous with dense support.
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.
Thue-Morse Spectral Measure
Morse F=
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
Bohr Minimal Sets and Sequences
Let   R \ Q and define  : Z  Z
| k   (k ) |  , k  Z ,
1
2
and let
  (0, )
1
4
and define the Bohr set
B  { k  Z :| k   (k ) |   }.
Theorem
 B
is a minimal sequence and
  S (B )  supp (ˆ )  B  B and
2
 ˆ (k )   is positive definite on Z .
kZ
(k ,
 ( k ))
Research Problem 3.
Generalize
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.