Post on 25-Dec-2015
transcript
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