ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Shift-Invariant Spaces with Extra Invariance
Introduction
Examples
Michael Northington V1
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Department of Mathematics
Vanderbilt University
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Under the supervision of Alexander Powell and Douglas Hardin
1
[email protected]
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
1 Introduction
Examples
2 Uncertainly Principles
Balian-Low Theorem
BLT for SIS
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
3 Sobolev Spaces
Fourier Transform and Fractional Sobolev Spaces on Rd
4 Generators of SIS
Generators of
SIS
5 Sobolev Spaces of Periodic Functions
Sobolev
Spaces of
Periodic
Functions
6 Proof of Main Theorem
Proof of Main
Theorem
Shift Invariant Spaces
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• L2 (Rd ) = {f : Rd |f (x)|2 dx < ∞}
• Tk , defined by Tk f (x) = f (x − k) is called a Shift
R
Operator.
• A closed subspace V ⊂ L2 (Rd ) is called a Shift Invariant
Space if for any f ∈ V and any k ∈ Zd , Tk f ∈ V . In other
words, V is closed under shifts by integers.
ShiftInvariant
Spaces
with Extra
Invariance
Given F ⊂ L2 (Rd ), we can build a SIS, V (F ).
Michael
Northington V
• Let T (F ) = {Tk fj : k ∈ Zd , fj ∈ F }.
Introduction
• Then let V (F ) be the closed span of T (F ), i.e.
Examples
L2 (Rd )
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
V (F ) = span(T (F ))
.
• The elements of F are called generators of V (F ).
• If F is finite, V (F ) is called finitely generated.
• If F = {f } then V (F ) = V (f ) is called singly generated
or principal.
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
• SIS are often used as approximation spaces in numerical
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
analysis.
• For algorithms involving singly generated SIS, only one
function needs to be stored.
• Many properties of V (F ) can be traced back to properties
of F .
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• Given a lattice Zd ⊂ Γ ⊂ Rd , V (F ) is said to be
Γ-invariant if f ∈ V (F ) implies that Tγ f ∈ V (F ) for all
γ ∈ Γ.
• V (F ) is said to be translation invariant if f ∈ V (F )
implies that Tt f ∈ V (F ) for all t ∈ Rd , i.e if V (F ) is
Rd -invariant.
Riesz Bases and Frames
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
• A collection {hn }∞
n=1 in a Hilbert space H is a Riesz Basis
for H if there exist constants 0 < A ≤ B < ∞ such that
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
∀c = (cn ) ∈ l2 (N), Akck2l2 (N) ≤ k
∞
X
cn hn k2 ≤ Bkck2l2 (N) .
n=1
• A collection {hn }∞
n=1 in a Hilbert space H is a frame for
H if there exist constants 0 < A ≤ B < ∞ such that
∀h ∈ H,
Akhk2H ≤
∞
X
n=1
|hh, hn i|2 ≤ Bkhk2H .
Examples of SIS
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• Paley Wiener Space -
{g ∈ L2 (Rd ) : gb(ξ) = 0 for |ξ| > 12 }.
• Splines with integer knots contained in L2 (R) - Snn−1
• Gabor Spaces -
span{f (x − k)e2πinx : n, k ∈ Z}.
• Wavelet Spaces -
span{a−k/2 f (a−k x − nb) : n, k ∈ Z}.
Examples of Singly Generated SIS
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• Paley Wiener Space -
{g ∈ L2 (Rd ) : gb(ξ) = 0 for |ξ| > 12 }.
• Splines with integer knots contained in L2 (R)
• Gabor Spaces -
span{f (x − k)e2πinx : n, k ∈ Z}.
• Wavelet Spaces -
span{a−k/2 f (a−k x − nb) : n, k ∈ Z}.
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Paley Wiener Space, P W , or the Space of Bandlimited
Functions
• g ∈ P W ⇐⇒ the Fourier Transform of g given by,
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Z
gb(ξ) =
g(x)e2πiξx dx
R
is zero outside of [− 21 , 21 ].
• Let f = sinc(x) = sin(πx)
πx .
• Then P W is generated by f , or
P W = V (f ).
ShiftInvariant
Spaces
with Extra
Invariance
1.0
Sinc
0.8
Michael
Northington V
0.6
Introduction
Examples
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
0.4
0.2
Sobolev
Spaces
0.0
Balian-Low
Theorem
BLT for SIS
Sinc(x)
Uncertainly
Principles
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
−0.2
Generators of
SIS
−5
0
x
5
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
• If g ∈ P W , and λ ∈ R, then Tλ g(x) = g(x − λ) is also in
PW.
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• This is true since
2πiλξ
d
T
gb(ξ).
λ g(ξ) = e
• Thus, P W is translation invariant.
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Spline Spaces - Snn−1
• f ∈ Snn−1 ⇐⇒ f restricted to [k, k + 1] is a polynomial
of degree n, f ∈ L2 (R), and f ∈ C n−1 (R).
• Let β0 be the characteristic function of [0, 1], and
iteratively define
Z
βn−1 (x − t)β0 (t)dt.
βn (x) =
R
• Then Snn−1 is a SIS generated by βn , or
Snn−1 = V (βn ).
ShiftInvariant
Spaces
with Extra
Invariance
1.0
B−splines
0.8
Michael
Northington V
Introduction
Examples
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
y
0.4
Sobolev
Spaces
0.2
Balian-Low
Theorem
BLT for SIS
0.6
Uncertainly
Principles
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
0.0
Generators of
SIS
−3
−2
−1
0
x
1
2
3
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• For some f ∈ Snn−1 consider shifting f by a non-integer
λ ∈ R.
• This has the effect of shifting the knots into the interior of
the intervals.
• Clearly, we can find some f ∈ Snn−1 where Tλ f ∈
/ Snn−1 .
• Thus, Snn−1 has no extra invariance.
Key Points
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• P W is translation invariant, and the generator decays
slowly (like x1 ).
• Snn−1 has compactly supported generators, and it has no
extra invariance.
• It turns out that this kind of behavior holds in general and
not just for these two examples.
ShiftInvariant
Spaces
with Extra
Invariance
Heisenberg Uncertainty Principle
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• An uncertainty principle in harmonic analysis is a result
which constrains how well-localized a function f or its
Fourier transform fb can be.
• A famous uncertainty principle is given by the
d-dimensional Heisenberg inequality: ∀f ∈ L2 (Rd ),
Z
Z
d2
2
2
2 b
2
|x| |f (x)| dx
|ξ| |f (ξ)| dξ ≥
kf k4L2 (Rd ) .
16π 2
Rd
Rd
ShiftInvariant
Spaces
with Extra
Invariance
Balian-Low Theorem
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• The Balian-Low Theorem for Gabor Systems is another
example of an Uncertainty Principle.
• Given f ∈ L2 (R) the associated Gabor system
G(f, 1, 1) = {fm,n }m,n∈Z is defined by
fm,n (x) = e2πimx f (x − n).
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Balian-Low Theorem)
Let f ∈ L2 (R). If G(f, 1, 1) is an orthonormal basis for L2 (R),
then
Z
Z
|ξ|2 |fb(ξ)|2 dξ = ∞.
|x|2 |f (x)|2 dx
R
R
BLT for SIS
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
• In fact, Balian-Low type theorems exist for generators of
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Shift Invariant Spaces.
Theorem (Aldroubi, Sun, Wang-2010)
Suppose that f ∈ L2 (R) and that T (f ) is a Riesz Basis for
V (f ). If V (F ) is n1 Z-invariant then for all > 0,
Z
|x|1+ |f (x)|2 dx = ∞.
R
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Tessera, Wang-2013)
Let f ∈ L2 (Rd ) and Zd ⊂ Γ is a lattice. Suppose that T (f ) is
a frame for V (f ). If V (f ) is Γ-invariant then for any > 0,
Z
|x|d+ |f (x)|2 dx = ∞.
Rd
Main Result
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Hardin, Northington, Powell)
Fix a lattice Zd ⊂ Γ ⊂ Rd and f ∈ L2 (Rd ). Suppose that
T (f ) is a frame for V (f ). If V (f ) is Γ-invariant then
Z
|x| |f (x)|2 dx = ∞.
Rd
Smoothness vs Decay
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Consider the following for f ∈ L2 (R).
Z
fb0 (ξ) =
f 0 (x)e2πixξ dx
d
R
Z
= (2πiξ)
f (x)e2πixξ dx
Rd
(2πiξ)fb(ξ).
Fractional Sobolev Spaces
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• For s > 0, the Sobolev space H s (Rd ) consists of all
measurable functions f defined on Rd such that
f ∈ L2 (Rd ) and
Z
kf kḢ s (Rd ) =
Rd
2s
|ξ| |fb(ξ)|2 dξ
1/2
< ∞.
(1)
Main Result
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Hardin, Northington, Powell)
Fix a lattice Zd ⊂ Γ ⊂ Rd and f ∈ L2 (Rd ). Suppose that
T (f ) is a frame for V (f ). If V (f ) is Γ-invariant then
Z
|x| |f (x)|2 dx = ∞.
Rd
In other words, the generator satisfies fb ∈
/ H 1/2 (Rd ).
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
• Given a lattice Γ ⊂ Rd and f ∈ L2 (Rd ), the Periodization
of f is defined as
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
PΓ (f )(x) =
X
|f (x + γ)|2 .
γ∈Γ
• The following calculation shows that the inner products of
elements of T (f ) are encoded in PZd (fb)
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
0
hTk f,Tk0 f iL2 (Rd ) = he−2πiξ·k fb, e−2πiξ·k fbiL2 (Rd )
Z
0
=
e−2πiξ·(k−k ) fb(ξ)fb(ξ)dξ
Rd
XZ
0
e−2πiξ·(k−k ) fb(ξ − l)fb(ξ − l)dξ
=
l∈Zd
Z
=
[0,1]d
[0,1]d
0
e−2πiξ·(k−k ) PZd (fb)(ξ)dξ
\
= PZd (fb)(k − k 0 )
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Proposition (de Boor, DeVore, Ron)
Given f ∈ L2 (Rd ), T (f ) = {Tk f : k ∈ Zd } forms a Riesz basis
for V (f ) if and only if there exists s ≥ 1 such that
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
s−1 ≤ PZd (fb)(ξ) ≤ s
a.e. x ∈ [0, 1]d .
(2)
Proposition (Bownik)
T (f ) forms a frame for V (f ) if and only if there exists s ≥ 1
such that for almost every x ∈ [0, 1]d ,
s−1 PZd (fb)(ξ) ≤ (PZd (fb)(ξ))2 ≤ sPZd (fb)(ξ).
(3)
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Aldroubi, Cabrelli, Heil, Kornelson, Molter)
Let Zd < Γ be lattices in Rd . Let R ⊂ Zd be a collection of
representatives of the quotient Zd /Γ∗ so that
X
PZd (fb)(ξ) =
PΓ∗ (fb)(x + k), a.e. x ∈ Rd .
k∈R
The space V (f ) is Γ-invariant if and only if for almost every
x ∈ Rd , at most one of the terms in the right hand sum is
nonzero.
Fourier coefficients
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
The Fourier coefficients of f ∈ L2 ([0, 1]d ) are defined ∀ξ ∈ Zd ,
Z
b
f (ξ) =
f (x)e−2πix·ξ dx.
[0,1]d
Also recall Parseval’s theorem
Z
X
|f (x)|2 dx =
|fb(ξ)|2
[0,1]d
ξ∈Zd
(4)
Periodic Sobolev Space
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Given s > 0, define the Sobolev space
H s ([0, 1]d ) = {f ∈ L2 ([0, 1]d ) : kf kḢ s ([0,1]d ) < ∞}, where
kf k2Ḣ s ([0,1]d ) =
X
ξ∈Zd
|ξ|2s |fb(ξ)|2 .
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Theorem (Hardin, Northington, Powell)
Let 0 < s < 1. If f ∈ H s (Rd ) and PZd (f ) is bounded, then
PZd (f ) ∈ H s ([0, 1]d ).
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
Lemma (Rd Proof by Bourgain, Brezis, Nirenberg)
If g ∈ H 1/2 ([0, 1]d ) and for almost every x ∈ [0, 1]d , either
g(x) = 0, or g(x) ≥ C > 0 then either g(x) = 0 a.e. or
g(x) ≥ C a.e.
ShiftInvariant
Spaces
with Extra
Invariance
Michael
Northington V
Introduction
Examples
Uncertainly
Principles
Balian-Low
Theorem
BLT for SIS
Sobolev
Spaces
Fourier
Transform and
Fractional
Sobolev Spaces
on Rd
Generators of
SIS
Sobolev
Spaces of
Periodic
Functions
Proof of Main
Theorem
• Assume fb ∈ H 1/2 (Rd ).
• Then PZd (fb), PΓ∗ (fb) ∈ H 1/2 ([0, 1]d ).
• Since T (f ) forms a frame, PZd (fb) is either zero, or
bounded away from zero.
P
• Since PZd (fb)(ξ) = k∈R PΓ∗ (fb)(x + k), we must have
that PΓ∗ (fb) is zero on a large set, and bounded away from
zero on a large set.
• This contradicts the previous lemma.