On Rearrangements of
Fourier Series
On the distribution of
values of orthonormal
systems with restricted
supports
Mark Lewko
Orthonormal System
© := f Á1 ; Á2 ; : : : ; Án g
Ái : T ! C
hÁi ; Ái i = 1
hÁi ; Áj i = 0
i6
= j
Meta-Question
Given:
© := f Á1 ; Á2 ; : : : ; Án g
- µ [N ]
What can we say about:
X
f (x) :=
an Án (x)
n2 -
Random
versus
- µ [N ]
A lot of work to
marginally beat trivial
estimates
Sharp (or nearly sharp) results
•
Stochastic processes
•
Metrical entropy
Explicit
•
•
•
•
Analytic number theory (circle
method)
Combinatorics (sum-product
estimates)
Fourier restriction theory (multilinear estimates)
? ABC Conjecture / PFR
conjecture / Faltings theorem ?
Theorem (Bourgain)
© := f Án : n 2 [N ]g
X
jj
jjÁi jj L 1 ¿ 1
j- j À NÃp=2
X
an Án jj p ¿
n2 -
n
!
1=2
jan j 2
(Random)
No explicit examples (unless p is even integer)
Finite Field Restriction Estimates
© := f e(n ¢x) : n 2 F3p g
- := f e(n) : n 2 Pg
P := f (n1 ; n2 ; n21 + n22 ) : n1; n2 2 F2p g
¯¯
¯¯X
¯¯
¯¯
¯¯
n2 P
¯¯
¯¯
¯¯
an e(n ¢x) ¯¯
¯¯
¿ jFj 1¡
3=p
³X
jan j 2
Lp
Conjectured for p ¸ 3 and ¡ 1 not a square.
´ 1=2
Compressed Sensing
© := f Án : n 2 [N ]g
X
X
j
ai Ái (x)j 2 =
i
x 2 [N ]
X
jai j 2
i 2 [N ]
- µ [N ]
X
X
j
y2 -
i
j- j X
ai Ái (x)j »
N
2
i 2 [N ]
jai j 2
Compressed Sensing II
Theorem (Candes-Tao / Rudelson-Vershynin)
© := f Án : n 2 [N ]g
A = (a1 ; a2; : : : ; an )
j- j X
(1 ¡ ²)
N
i 2 [N ]
2
X
X
j
jai j ·
y2 -
i
jjA jj 0 = R
j- j X
ai Ái (x)j · (1 + ²)
N
2
r log4 (n)
¿ j- j
2
²
i 2 [N ]
jai j 2
Rearrangements of Fourier Series
© := f Á1 ; Á2 ; : : :g
X
X
f (x) =
an Án (x)
f (x) a.e.
= lim
`! 1
an Án (x)
n· `
Kolmogorov (1920’s)
Does there exist a ¼: N ! N such that:
X
f (x) = lim
a.e.` ! 1
a¼(n)Á¼( n ) (x)?
n· `
Rearrangements of Fourier Series II
X
© := f Á1 ; Á2 ; : : :g
f (x) =
an Án (x)
¯
¯
¯
¯
¯X
¯
¯
M f (x) = max ¯ an Án (x) ¯
¯
` ¯
¯
n· `
Rademacher-Menshov
X
jjM f jj L 2 ¿ log(N )(
jan j 2 ) 1=2
Bourgain
¯
¯
¯
¯
¯X
¯
¯
M ¼f (x) = max ¯ a¼( n ) Á¼( n ) (x) ¯
¯
` ¯
¯
n· `
X
jjM ¼f jj L 2 ¿ loglog(N )(
jan j 2 ) 1=2
Rearrangements of Fourier Series III
¯
¯
¯
¯
¯X
¯
¯
M ¼f (x) = max ¯ a¼( n ) Á¼( n ) (x) ¯
¯
` ¯
¯
n· `
¯
¯
¯X
¯
¯
¯
¯ a¼( n ) Á¼( n ) (x) ¯
¯
¯
n2 I
¯
¯
¯X
¯
¯
¯
a
Á
(x)
¯
¯
¼( n ) ¼( n )
¯
¯
n2 -
j- j = jI j
Thank You!