On Rearrangements of Fourier Series

Mark Lewko

On the distribution of values of orthonormal systems with restricted supports

Orthonormal System

©:= fÁ1;Á2; : : : ;Áng

hÁi ;Áj i = 0

Ái : T ! C

i 6= jhÁi ;Ái i = 1

Meta-Question

f (x) :=X

n2­

anÁn(x)

©:= fÁ1;Á2; : : : ;Áng

­ µ [N ]

Given:

What can we say about:

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

Thank You!