On Rearrangements of Fourier Series Mark Lewko On the distribution of values of orthonormal systems with restricted supports
Dec 25, 2015
On Rearrangements of Fourier Series
Mark Lewko
On the distribution of values of orthonormal systems with restricted supports
Random versus Explicit
µ [N ]Sharp (or nearly sharp) results
• Stochastic processes
• Metrical entropy
A lot of work to marginally beat trivial estimates • Analytic number theory
(circle method)
• Fourier restriction theory (multi-linear estimates)
• Combinatorics (sum-product estimates)
• ? ABC Conjecture / PFR conjecture / Faltings theorem ?
Theorem (Bourgain)
©:= fÁn : n 2 [N ]g jjÁi jjL 1 ¿ 1
jjX
n2
anÁn jjp ¿
ÃX
n
jan j2! 1=2j j À N p=2
(Random)
No explicit examples (unless p is even integer)
Finite Field Restriction Estimates
P := f (n1;n2;n21+n22) : n1;n2 2 F
2pg
¯¯¯¯¯
¯¯¯¯¯
X
n2P
ane(n ¢x)
¯¯¯¯¯
¯¯¯¯¯L p
¿ jFj1¡ 3=p³ X
jan j2´1=2
©:= fe(n ¢x) : n 2 F3pg
:= fe(n) : n 2 Pg
Conjectured for p¸ 3 and ¡ 1 not a square.
Compressed Sensing
µ [N ]
X
y2
jX
i
aiÁi (x)j2
X
x2[N ]
jX
i
aiÁi (x)j2=X
i2 [N ]
jai j2
»j jN
X
i2 [N ]
jai j2
©:= fÁn : n 2 [N ]g
Compressed Sensing II
X
y2
jX
i
aiÁi (x)j2 · (1+ ²)j jN
X
i2 [N ]
jai j2(1¡ ²)j jN
X
i2 [N ]
jai j2 ·
A = (a1;a2; : : : ;an) jjA jj0 =R
r log4(n)²2
¿ j j
Theorem (Candes-Tao / Rudelson-Vershynin)
©:= fÁn : n 2 [N ]g
Rearrangements of Fourier Series©:= fÁ1;Á2; : : :g
f (x) = lim`! 1
X
n· `
anÁn(x)a.e.
Kolmogorov (1920’s)
f (x) = lim`! 1
X
n· `
a¼(n)Á¼(n)(x)?a.e.
Does thereexist a ¼: N ! N such that:
f (x) =XanÁn(x)
Rearrangements of Fourier Series II©:= fÁ1;Á2; : : :g
M f (x) =max`
¯¯¯¯¯¯
X
n· `
anÁn(x)
¯¯¯¯¯¯
jjM f jjL 2 ¿ log(N )(Xjan j2)1=2
f (x) =XanÁn(x)
Rademacher-Menshov
Bourgain
M ¼f (x) =max`
¯¯¯¯¯¯
X
n· `
a¼(n)Á¼(n)(x)
¯¯¯¯¯¯
jjM ¼f jjL 2 ¿ loglog(N )(Xjan j2)1=2
Rearrangements of Fourier Series III
M ¼f (x) =max`
¯¯¯¯¯¯
X
n· `
a¼(n)Á¼(n)(x)
¯¯¯¯¯¯
¯¯¯¯¯
X
n2I
a¼(n)Á¼(n)(x)
¯¯¯¯¯
¯¯¯¯¯
X
n2
a¼(n)Á¼(n)(x)
¯¯¯¯¯
j j = jI j