Download Wiener`s Lemma: Theme and Variations

Wiener’s Lemma: Theme and Variations
Karlheinz Gröchenig
European Center of Time-Frequency Analysis
Faculty of Mathematics
University of Vienna
http://homepage.univie.ac.at/karlheinz.groechenig/
ISTA, September 2009
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
1 / 31
Outline
1
Wiener’s Lemma — Classical
Wiener’s Lemma
Convolution Operators
Inverse-Closedness
2
Variations
Weights
Matrix Algebras
Absolutely Convergent Series of Time-Frequency Shifts
Time-Varying Systems and Wireless Communications
Convolution operators on groups
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
2 / 31
Wiener’s Lemma — Classical
Wiener’s Lemma
Wiener’s Lemma — Innocent Version
f possesses an absolutely convergent Fourier series, f ∈ A(Td ) ,
P
if f (t) = k ∈Zd ak e2πik ·t with coefficients in a ∈ ℓ1 (Zd ).
Norm:
kf kA = kak1 =
X
|ak |
k ∈Zd
Fact: A is a Banach algebra under pointwise multiplication.
Theorem (Classical Formulation)
d ) and f (t) 6= 0 for all t ∈ Td , then also 1/f ∈ A(Td ), i.e.,
If f ∈ A(TP
1/f (t) = k ∈Zd bk e2πik ·t with b ∈ ℓ1 (Zd ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
3 / 31
Wiener’s Lemma — Classical
Wiener’s Lemma
Wiener’s Lemma — Innocent Version
f possesses an absolutely convergent Fourier series, f ∈ A(Td ) ,
P
if f (t) = k ∈Zd ak e2πik ·t with coefficients in a ∈ ℓ1 (Zd ).
Norm:
kf kA = kak1 =
X
|ak |
k ∈Zd
Fact: A is a Banach algebra under pointwise multiplication.
Theorem (Classical Formulation)
d ) and f (t) 6= 0 for all t ∈ Td , then also 1/f ∈ A(Td ), i.e.,
If f ∈ A(TP
1/f (t) = k ∈Zd bk e2πik ·t with b ∈ ℓ1 (Zd ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
3 / 31
Wiener’s Lemma — Classical
Wiener’s Lemma
Wiener’s Lemma — Innocent Version
f possesses an absolutely convergent Fourier series, f ∈ A(Td ) ,
P
if f (t) = k ∈Zd ak e2πik ·t with coefficients in a ∈ ℓ1 (Zd ).
Norm:
kf kA = kak1 =
X
|ak |
k ∈Zd
Fact: A is a Banach algebra under pointwise multiplication.
Theorem (Classical Formulation)
d ) and f (t) 6= 0 for all t ∈ Td , then also 1/f ∈ A(Td ), i.e.,
If f ∈ A(TP
1/f (t) = k ∈Zd bk e2πik ·t with b ∈ ℓ1 (Zd ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
3 / 31
Wiener’s Lemma — Classical
Wiener’s Lemma
Wiener’s Lemma — Innocent Version
f possesses an absolutely convergent Fourier series, f ∈ A(Td ) ,
P
if f (t) = k ∈Zd ak e2πik ·t with coefficients in a ∈ ℓ1 (Zd ).
Norm:
kf kA = kak1 =
X
|ak |
k ∈Zd
Fact: A is a Banach algebra under pointwise multiplication.
Theorem (Classical Formulation)
d ) and f (t) 6= 0 for all t ∈ Td , then also 1/f ∈ A(Td ), i.e.,
If f ∈ A(TP
1/f (t) = k ∈Zd bk e2πik ·t with b ∈ ℓ1 (Zd ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
3 / 31
Wiener’s Lemma — Classical
Convolution Operators
Input-Output Relations
“system” A
x
b
Wiener’s Lemma is statement about quality of input-output relations.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
4 / 31
Wiener’s Lemma — Classical
Convolution Operators
Discrete Time-Invariant Systems and Convolution Operators
• A commutes with translations (Tm x)(k) = x(k − m):
ATm = Tm A
• A must be a convolution operator with “impulse response” a:
Ca x = a ∗ x
X
(a ∗ b)(k ) =
a(l)b(k − l)
k ∈ Zd
l∈Zd
Fact: If a ∈ ℓ1 (Zd ), then Ca is bounded on ℓp for 1 ≤ p ≤ ∞.
σℓp (Ca )
. . . spectrum of Ca as an operator acting on ℓp (Zd ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
5 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators I
Theorem
If a ∈ ℓ1 (Zd ) and Ca is invertible on ℓ2 (Zd ), then Ca−1 = Cb for some
b ∈ ℓ1 (Zd ).
Proof.
P
• Fourier series b
a(t) = k ∈Zd ak e2πikt for a ∈ ℓ2 (Zd ) or ℓ1 (Zd ).
b
• (Ca b)b = (a ∗ b)b = b
ab
• Ca corresponds to multiplication operator and is invertible if and
only if inft |b
a(t)| > 0
b = 1/b
• Let b be the sequence with Fourier series b
a. By Wiener’s
1
d
Lemma b ∈ ℓ (Z ).
• Since a ∗ b = b ∗ a = δ, we have Cb Ca = Id.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
6 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators II
Corollary
Assume that a ∈ ℓ1 (Zd ). TFAE:
(i) Ca is invertible on ℓ2 (Zd ).
(ii) Ca is invertible on ℓp (Zd ) for some p ∈ [1, ∞].
(iii) Ca is invertible on ℓp (Zd ) for all p ∈ [1, ∞].
Corollary
If a ∈ ℓ1 (Zd ), then
σℓp (Ca ) = σℓ2 (Ca ) = b
a(T)
∀p ∈ [1, ∞] .
Spectrum of convolution operator does NOT depend on the space ℓp
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
7 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators II
Corollary
Assume that a ∈ ℓ1 (Zd ). TFAE:
(i) Ca is invertible on ℓ2 (Zd ).
(ii) Ca is invertible on ℓp (Zd ) for some p ∈ [1, ∞].
(iii) Ca is invertible on ℓp (Zd ) for all p ∈ [1, ∞].
Corollary
If a ∈ ℓ1 (Zd ), then
σℓp (Ca ) = σℓ2 (Ca ) = b
a(T)
∀p ∈ [1, ∞] .
Spectrum of convolution operator does NOT depend on the space ℓp
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
7 / 31
Wiener’s Lemma — Classical
Convolution Operators
Wiener’s Lemma for Convolution Operators II
Corollary
Assume that a ∈ ℓ1 (Zd ). TFAE:
(i) Ca is invertible on ℓ2 (Zd ).
(ii) Ca is invertible on ℓp (Zd ) for some p ∈ [1, ∞].
(iii) Ca is invertible on ℓp (Zd ) for all p ∈ [1, ∞].
Corollary
If a ∈ ℓ1 (Zd ), then
σℓp (Ca ) = σℓ2 (Ca ) = b
a(T)
∀p ∈ [1, ∞] .
Spectrum of convolution operator does NOT depend on the space ℓp
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
7 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Abstract Intermezzo — Wiener Pairs
Naimark’s insight: Wiener’s Lemma is a result about the relation
between two Banach algebras, namely A(Td ) and C(Td ).
Condition “f (t) 6= 0, ∀t” means that f is invertible in C(Td ).
Definition
Let A ⊆ B be two (involutive) Banach algebras with common identity.
Then A is called inverse-closed in B, if
a ∈ A and a−1 ∈ B
=⇒
a−1 ∈ A .
• Wiener’s Lemma ⇔ A(Td ) is inverse-closed in C(Td ).
• in the large algebra there are more invertible elements and it may
be easier to verify invertibility
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
8 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Abstract Intermezzo — Wiener Pairs
Naimark’s insight: Wiener’s Lemma is a result about the relation
between two Banach algebras, namely A(Td ) and C(Td ).
Condition “f (t) 6= 0, ∀t” means that f is invertible in C(Td ).
Definition
Let A ⊆ B be two (involutive) Banach algebras with common identity.
Then A is called inverse-closed in B, if
a ∈ A and a−1 ∈ B
=⇒
a−1 ∈ A .
• Wiener’s Lemma ⇔ A(Td ) is inverse-closed in C(Td ).
• in the large algebra there are more invertible elements and it may
be easier to verify invertibility
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
8 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Abstract Intermezzo — Wiener Pairs
Naimark’s insight: Wiener’s Lemma is a result about the relation
between two Banach algebras, namely A(Td ) and C(Td ).
Condition “f (t) 6= 0, ∀t” means that f is invertible in C(Td ).
Definition
Let A ⊆ B be two (involutive) Banach algebras with common identity.
Then A is called inverse-closed in B, if
a ∈ A and a−1 ∈ B
=⇒
a−1 ∈ A .
• Wiener’s Lemma ⇔ A(Td ) is inverse-closed in C(Td ).
• in the large algebra there are more invertible elements and it may
be easier to verify invertibility
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
8 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Abstract Intermezzo — Wiener Pairs
Naimark’s insight: Wiener’s Lemma is a result about the relation
between two Banach algebras, namely A(Td ) and C(Td ).
Condition “f (t) 6= 0, ∀t” means that f is invertible in C(Td ).
Definition
Let A ⊆ B be two (involutive) Banach algebras with common identity.
Then A is called inverse-closed in B, if
a ∈ A and a−1 ∈ B
=⇒
a−1 ∈ A .
• Wiener’s Lemma ⇔ A(Td ) is inverse-closed in C(Td ).
• in the large algebra there are more invertible elements and it may
be easier to verify invertibility
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
8 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Abstract Intermezzo — Wiener Pairs
Naimark’s insight: Wiener’s Lemma is a result about the relation
between two Banach algebras, namely A(Td ) and C(Td ).
Condition “f (t) 6= 0, ∀t” means that f is invertible in C(Td ).
Definition
Let A ⊆ B be two (involutive) Banach algebras with common identity.
Then A is called inverse-closed in B, if
a ∈ A and a−1 ∈ B
=⇒
a−1 ∈ A .
• Wiener’s Lemma ⇔ A(Td ) is inverse-closed in C(Td ).
• in the large algebra there are more invertible elements and it may
be easier to verify invertibility
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
8 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Babylonian Confusion
•
•
•
•
•
•
•
•
•
•
A inverse-closed in B (Barnes)
(A, B) is a Wiener pair (Naimark)
A is a spectral subalgebra of B (Palmer)
A is a local subalgebra of B (K-theory, Blackadar)
A is a full subalgebra of B (translations from Russian)
A is a natural subalgebra of B (Dales-Davie)
A is invariant under holomorphic calculus in B (Connes)
Spectral invariance, spectral permanence (Arveson)
...
...
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
9 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Spectral Invariance
Spectrum in Banach algebra A (with unit e)
σA (a) = {λ ∈ C : a − λe is not invertible in A}
Spectral radius
1/n
rA (a) = max{|λ| : λ ∈ σA (a)} = lim kan kA
n→∞
Lemma
A ⊆ B with common unit e. TFAE:
(i) A is inverse-closed in B.
(ii) σA (a) = σB (a) for all a ∈ A.
(iii) rA (a) = rB (a) for all a = a∗ ∈ A.
(iii) ⇒ (i) is Hulanicki’s Lemma
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
10 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Spectral Invariance
Spectrum in Banach algebra A (with unit e)
σA (a) = {λ ∈ C : a − λe is not invertible in A}
Spectral radius
1/n
rA (a) = max{|λ| : λ ∈ σA (a)} = lim kan kA
n→∞
Lemma
A ⊆ B with common unit e. TFAE:
(i) A is inverse-closed in B.
(ii) σA (a) = σB (a) for all a ∈ A.
(iii) rA (a) = rB (a) for all a = a∗ ∈ A.
(iii) ⇒ (i) is Hulanicki’s Lemma
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
10 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Spectral Invariance
Spectrum in Banach algebra A (with unit e)
σA (a) = {λ ∈ C : a − λe is not invertible in A}
Spectral radius
1/n
rA (a) = max{|λ| : λ ∈ σA (a)} = lim kan kA
n→∞
Lemma
A ⊆ B with common unit e. TFAE:
(i) A is inverse-closed in B.
(ii) σA (a) = σB (a) for all a ∈ A.
(iii) rA (a) = rB (a) for all a = a∗ ∈ A.
(iii) ⇒ (i) is Hulanicki’s Lemma
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
10 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Functional Calculus
Riesz functional calculus (holomorphic functional calculus)
f analytic on open neighborhood O of σB (a) and γ ⊆ O is contour of
σB (a). Define
Z
1
f (a) =
f (z)(ze − a)−1 dz
2πi γ
Corollary
If A is inverse-closed and continuously embedded in B, then the Riesz
functional calculi for A and B coincide.
• =⇒ square roots, powers, pseudoinverse
• Theorem of Wiener-Levy
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
11 / 31
Wiener’s Lemma — Classical
Inverse-Closedness
Functional Calculus
Riesz functional calculus (holomorphic functional calculus)
f analytic on open neighborhood O of σB (a) and γ ⊆ O is contour of
σB (a). Define
Z
1
f (a) =
f (z)(ze − a)−1 dz
2πi γ
Corollary
If A is inverse-closed and continuously embedded in B, then the Riesz
functional calculi for A and B coincide.
• =⇒ square roots, powers, pseudoinverse
• Theorem of Wiener-Levy
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
11 / 31
Variations
Weights
Variation I — Weights
Weights: quantify decay conditions
Weighted
convergent Fourier series: f ∈ Av (Td ), if
P absolutely
2πik
·t with norm
f (t) = k ∈Zd ak e
kf kAv = kakℓ1v =
X
|ak |v(k)
k ∈Zd
Assumptions on v
• v is submultiplicative: v(k + l) ≤ v(k)v(l) for k, l ∈ Zd
• v is even: v(−k) = v(k)
Fact: If v is submultiplicative, then Av is a Banach algebra with respect
to pointwise convolution.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
12 / 31
Variations
Weights
Variation I — Weights
Weights: quantify decay conditions
Weighted
convergent Fourier series: f ∈ Av (Td ), if
P absolutely
2πik
·t with norm
f (t) = k ∈Zd ak e
kf kAv = kakℓ1v =
X
|ak |v(k)
k ∈Zd
Assumptions on v
• v is submultiplicative: v(k + l) ≤ v(k)v(l) for k, l ∈ Zd
• v is even: v(−k) = v(k)
Fact: If v is submultiplicative, then Av is a Banach algebra with respect
to pointwise convolution.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
12 / 31
Variations
Weights
Variation I — Weights
Weights: quantify decay conditions
Weighted
convergent Fourier series: f ∈ Av (Td ), if
P absolutely
2πik
·t with norm
f (t) = k ∈Zd ak e
kf kAv = kakℓ1v =
X
|ak |v(k)
k ∈Zd
Assumptions on v
• v is submultiplicative: v(k + l) ≤ v(k)v(l) for k, l ∈ Zd
• v is even: v(−k) = v(k)
Fact: If v is submultiplicative, then Av is a Banach algebra with respect
to pointwise convolution.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
12 / 31
Variations
Weights
Variation I — Weights
Weights: quantify decay conditions
Weighted
convergent Fourier series: f ∈ Av (Td ), if
P absolutely
2πik
·t with norm
f (t) = k ∈Zd ak e
kf kAv = kakℓ1v =
X
|ak |v(k)
k ∈Zd
Assumptions on v
• v is submultiplicative: v(k + l) ≤ v(k)v(l) for k, l ∈ Zd
• v is even: v(−k) = v(k)
Fact: If v is submultiplicative, then Av is a Banach algebra with respect
to pointwise convolution.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
12 / 31
Variations
Weights
Wiener’s Lemma — Weighted Version
b
Standard weights: v(k) = ea|k | (1 + |k|)s for a, b, s ≥ 0, b ≤ 1
Theorem
TFAE:
(i) Wiener’s Lemma holds for Av (Td ), i.e. if f ∈ Av (Td ) and f (t) 6= 0 for
all t, then 1/f ∈ Av (Td ).
(ii) v satisfies the GRS-condition (Gelfand-Raikov-Shilov)
lim v(nk)1/n = 1
n→∞
∀k ∈ Zd .
b
• The weight v(k) = ea|k | (1 + |k|)s satisfies GRS, if and only if
0 ≤ b < 1.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
13 / 31
Variations
Weights
Wiener’s Lemma — Weighted Version
b
Standard weights: v(k) = ea|k | (1 + |k|)s for a, b, s ≥ 0, b ≤ 1
Theorem
TFAE:
(i) Wiener’s Lemma holds for Av (Td ), i.e. if f ∈ Av (Td ) and f (t) 6= 0 for
all t, then 1/f ∈ Av (Td ).
(ii) v satisfies the GRS-condition (Gelfand-Raikov-Shilov)
lim v(nk)1/n = 1
n→∞
∀k ∈ Zd .
b
• The weight v(k) = ea|k | (1 + |k|)s satisfies GRS, if and only if
0 ≤ b < 1.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
13 / 31
Variations
Weights
Wiener’s Lemma — Weighted Version
b
Standard weights: v(k) = ea|k | (1 + |k|)s for a, b, s ≥ 0, b ≤ 1
Theorem
TFAE:
(i) Wiener’s Lemma holds for Av (Td ), i.e. if f ∈ Av (Td ) and f (t) 6= 0 for
all t, then 1/f ∈ Av (Td ).
(ii) v satisfies the GRS-condition (Gelfand-Raikov-Shilov)
lim v(nk)1/n = 1
n→∞
∀k ∈ Zd .
b
• The weight v(k) = ea|k | (1 + |k|)s satisfies GRS, if and only if
0 ≤ b < 1.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
13 / 31
Variations
Weights
Counter-Example
If v violates GRS, then there is k ∈ Zd and a > 0 such that
v(nk) ≥ ean
for n ≥ n0 .
(A non-GRS weight grows exponentially on a subgroup)
Fix δ ∈ (0, a] and set
f (t) = 1 − e−δ e2πik·t ∈ Av (Td )
Then f (t) 6= 0 for all t ∈ Td and
∞
X
1
= (1 − e−δ e2πik·t )−1 =
e−δn e2πink·t ,
f (t)
n=0
but
kf kAv =
∞
X
k =0
Karlheinz Gröchenig (EUCETIFA)
e
−δn
v(nk) ≥
∞
X
e−δn ean = ∞
k =n0
Wiener’s Lemma
Sept. 2009
14 / 31
Variations
Weights
Counter-Example
If v violates GRS, then there is k ∈ Zd and a > 0 such that
v(nk) ≥ ean
for n ≥ n0 .
(A non-GRS weight grows exponentially on a subgroup)
Fix δ ∈ (0, a] and set
f (t) = 1 − e−δ e2πik·t ∈ Av (Td )
Then f (t) 6= 0 for all t ∈ Td and
∞
X
1
= (1 − e−δ e2πik·t )−1 =
e−δn e2πink·t ,
f (t)
n=0
but
kf kAv =
∞
X
k =0
Karlheinz Gröchenig (EUCETIFA)
e
−δn
v(nk) ≥
∞
X
e−δn ean = ∞
k =n0
Wiener’s Lemma
Sept. 2009
14 / 31
Variations
Weights
Counter-Example
If v violates GRS, then there is k ∈ Zd and a > 0 such that
v(nk) ≥ ean
for n ≥ n0 .
(A non-GRS weight grows exponentially on a subgroup)
Fix δ ∈ (0, a] and set
f (t) = 1 − e−δ e2πik·t ∈ Av (Td )
Then f (t) 6= 0 for all t ∈ Td and
∞
X
1
= (1 − e−δ e2πik·t )−1 =
e−δn e2πink·t ,
f (t)
n=0
but
kf kAv =
∞
X
k =0
Karlheinz Gröchenig (EUCETIFA)
e
−δn
v(nk) ≥
∞
X
e−δn ean = ∞
k =n0
Wiener’s Lemma
Sept. 2009
14 / 31
Variations
Weights
Counter-Example
If v violates GRS, then there is k ∈ Zd and a > 0 such that
v(nk) ≥ ean
for n ≥ n0 .
(A non-GRS weight grows exponentially on a subgroup)
Fix δ ∈ (0, a] and set
f (t) = 1 − e−δ e2πik·t ∈ Av (Td )
Then f (t) 6= 0 for all t ∈ Td and
∞
X
1
= (1 − e−δ e2πik·t )−1 =
e−δn e2πink·t ,
f (t)
n=0
but
kf kAv =
∞
X
k =0
Karlheinz Gröchenig (EUCETIFA)
e
−δn
v(nk) ≥
∞
X
e−δn ean = ∞
k =n0
Wiener’s Lemma
Sept. 2009
14 / 31
Variations
Matrix Algebras
Variation II — Matrix Algebras
“system” A
x
b
Slowly time-varying systems — non-stationary matrices — small
variation along diagonals.
Matrix norm
X
sup |mk ,k −l | v(l)
kMkCv =
d
l∈Zd k|∈Z
{z
}
d (l)
P
P
Note: |Mx(k)| = | l∈Zd mk ,k −(k −l)xl | ≤ l∈Zd d (k − l)|x(l)|
convolution-dominated matrices
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
15 / 31
Variations
Matrix Algebras
Variation II — Matrix Algebras
“system” A
x
b
Slowly time-varying systems — non-stationary matrices — small
variation along diagonals.
Matrix norm
X
kMkCv =
sup |mk ,k −l | v(l)
d
l∈Zd k|∈Z
{z
}
d (l)
P
P
Note: |Mx(k)| = | l∈Zd mk ,k −(k −l)xl | ≤ l∈Zd d (k − l)|x(l)|
convolution-dominated matrices
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
15 / 31
Variations
Matrix Algebras
Variation II — Matrix Algebras
“system” A
x
b
Slowly time-varying systems — non-stationary matrices — small
variation along diagonals.
Matrix norm
X
kMkCv =
sup |mk ,k −l | v(l)
d
l∈Zd k|∈Z
{z
}
d (l)
P
P
Note: |Mx(k)| = | l∈Zd mk ,k −(k −l)xl | ≤ l∈Zd d (k − l)|x(l)|
convolution-dominated matrices
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
15 / 31
Variations
Matrix Algebras
Nonstationary Version of Wiener’s Lemma
Theorem (Gohberg et al., Baskakov, Kurbatov, Sjöstrand)
If M ∈ C1 and M is invertible on ℓ2 (Zd ), then M −1 ∈ C1
Theorem (Baskakov)
Assume that v satisfies the GRS condition. If M ∈ Cv and M is
invertible on ℓ2 (Zd ), then M −1 ∈ Cv .
Cv is inverse-closed in B(ℓ2 ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
16 / 31
Variations
Matrix Algebras
Nonstationary Version of Wiener’s Lemma
Theorem (Gohberg et al., Baskakov, Kurbatov, Sjöstrand)
If M ∈ C1 and M is invertible on ℓ2 (Zd ), then M −1 ∈ C1
Theorem (Baskakov)
Assume that v satisfies the GRS condition. If M ∈ Cv and M is
invertible on ℓ2 (Zd ), then M −1 ∈ Cv .
Cv is inverse-closed in B(ℓ2 ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
16 / 31
Variations
Matrix Algebras
Nonstationary Version of Wiener’s Lemma
Theorem (Gohberg et al., Baskakov, Kurbatov, Sjöstrand)
If M ∈ C1 and M is invertible on ℓ2 (Zd ), then M −1 ∈ C1
Theorem (Baskakov)
Assume that v satisfies the GRS condition. If M ∈ Cv and M is
invertible on ℓ2 (Zd ), then M −1 ∈ Cv .
Cv is inverse-closed in B(ℓ2 ).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
16 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Proof Idea
De Leeuw, Gohberg, Baskakov
Modulation Mt by (Mt c)(k) = e2πik ·t c(k) for k ∈ Zd .
(TFA)
Given matrix A, consider matrix-valued function (periodic with period 1)
f(t) = Mt AM−t
P
Mt AM−t c (k) = e2πik ·t l∈Zd akl e−2πilt cl
bf(n)kl
=
Z
[0,1]d
f(t)kl e−2πin·t dt =
Z
[0,1]d
akl e2πi(k −l)·t e−2πin·t dt
= akl δk −l−n = ak ,k −n δk −l−n
P
So bf(n) = D(n) and f(t) ≍ n∈Zd D(n)e2πin·t . We need Wiener’s
Lemma for operator-valued Fourier series (done by Bochner-Phillips
1946).
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
17 / 31
Variations
Matrix Algebras
Off-Diagonal Decay of Matrices
Other conditions:
kMkAv = sup |mkl |v(k − l)
k ,l∈Zd
kMkA1v = sup
l∈Zd
X
|mkl |v(k − l)
∀M = M ∗
k ∈Zd
Theorem (GL’06)
(a) If v −1 ∈ ℓ1 (Zd ), v −1 ∗ v −1 ≤ Cv −1 , and v satisfies the GRS
condition, then Av is inverse-closed in B(ℓ2 ) .
(b) Assume that v is submultiplicative, v(k) ≥ (1 + |k|)δ for some δ > 0,
and satisfies the GRS condition. Then A1v is inverse-closed in B(ℓ2 ).
Many open problems! Systematic constructions by A. Klotz, 2008
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
18 / 31
Variations
Absolutely Convergent Series of Time-Frequency Shifts
Intermezzo — Twisted Convolution
Fix a lattice Λ = AZ2d ⊆ R2d (for invertible 2d × 2d -matrix A)
X
(a ♮Λ x)(λ) =
aµ xλ−µ e−2πiµ1 ·(λ1 −µ1 )
µ∈Λ
• In general ♮Λ is not commutative.
(Twisted) convolution operator Ta x = a ♮Λ x
Theorem (Wiener’s lemma)
Assume that
• a ∈ ℓ1 (Λ) and
• Ta is invertible on ℓ2 (Λ),
then a is invertible on (ℓ1 (Λ), ♮Λ ) and Ta−1 = Tb for some b ∈ ℓ1 (Λ).
Follows from Wiener’s Lemma for convolution-dominated matrices.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
19 / 31
Variations
Absolutely Convergent Series of Time-Frequency Shifts
Variation III — Rotation Algebra
Time-frequency shifts associated to z = (x, ξ) ∈ R2d
π(z)f (t) = e2πiξ·t f (t − x)
x, ξ, t ∈ Rd .
Commutation Relation (CCR) w = (u, η) ∈ R2d
π(z)π(w ) = e−2πix·η π(z + w )
Fix a lattice Λ = AZ2d ⊆ R2d (for invertible 2d × 2d -matrix A)
The rotation algebra (non-commutative torus) C ∗ (Λ) is the C ∗ -Algebra
generated by {π(λ) : λ ∈ Λ}.
Interesting subalgebras
(1) “lazy man’s rotation algebra”
X
Av (Λ) = {T ∈ B(L2 (Rd )) : T =
aλ π(λ), a ∈ ℓ1v (Λ)}
P
λ∈Λ
Norm kAkAv = kak1,v = λ∈Λ |aλ | v(λ)
(2) Smooth non-commutative torus
P
A∞ (Λ) = {T ∈ B(L2 (Rd )) : T = λ∈Λ aΛ π(λ), |aλ | = O(|λ|−N )}
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
20 / 31
Variations
Absolutely Convergent Series of Time-Frequency Shifts
Wieners Lemma for the Rotation Algebra
Analogy: absolutely convergent Fourier series — absolutely
convergent series of time-frequency shifts
Theorem
Assume that
• v satisfies the GRS-condition
• A ∈ Av (Λ) and
• A is invertible on L2 (Rd ),
then A−1 ∈ Av (Λ).
“Proof”: Av (Λ) ≃ (ℓ1v (Λ), ♮Λ )
Corollary ( Connes ’80, Rieffel ’88, Janssen ’95)
P
If A = Pλ∈Λ aλ π(λ) is invertible on L2 (Rd ) and a ∈ S(Λ), then
A−1 = λ∈Λ bλ π(λ) for some b ∈ S.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
21 / 31
Variations
Time-Varying Systems and Wireless Communications
Variation IV — Continuous Time-Varying Systems and
Pseudodifferential Operators
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
22 / 31
Variations
Time-Varying Systems and Wireless Communications
Time-Varying Channels — Continuous Case
Received signal f̃ = Kσ f is a superposition of time-frequency shifts:
Z
σ
b(η, u) eπiη·u e2πiη·t f (t + u) dud η
Kσ f (t)
=
|
{z
}
2d
R
(Mη T−u f )(t)
Z
σ(x, ξ)f̂ (ξ)e2πix·ξ d ξ
=
Rd
Pseudodifferential operator with (Kohn-Nirenberg) symbol σ
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
23 / 31
Variations
Time-Varying Systems and Wireless Communications
Symbol Classes
Sjöstrand class M ∞,1 (R2d ):
Z
kσkM ∞,1 =
sup |(σ · Tz Φ)b(ζ)| d ζ < ∞
R2d z∈R2d
⇒ (σ · Tz Φ)b ∈ L1 .
Locally σ is a Fourier transform of an L1 -function!
Analogy between absolutely convergent Fourier series and M ∞,1 ?
Weighted Sjöstrand class Mv∞,1 (R2d ).
Z
sup |(σ · Tz Φ)b(ζ)| v(ζ) d ζ < ∞
kσkM ∞,1 =
v
R2d z∈R2d
, 0 ≤ b ≤ 1, is suitable symbol class for wireless
Strohmer: M ∞,1
ea|ζ|b
communications.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
24 / 31
Variations
Time-Varying Systems and Wireless Communications
Symbol Classes
Sjöstrand class M ∞,1 (R2d ):
Z
kσkM ∞,1 =
sup |(σ · Tz Φ)b(ζ)| d ζ < ∞
R2d z∈R2d
⇒ (σ · Tz Φ)b ∈ L1 .
Locally σ is a Fourier transform of an L1 -function!
Analogy between absolutely convergent Fourier series and M ∞,1 ?
Weighted Sjöstrand class Mv∞,1 (R2d ).
Z
sup |(σ · Tz Φ)b(ζ)| v(ζ) d ζ < ∞
kσkM ∞,1 =
v
R2d z∈R2d
, 0 ≤ b ≤ 1, is suitable symbol class for wireless
Strohmer: M ∞,1
ea|ζ|b
communications.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
24 / 31
Variations
Time-Varying Systems and Wireless Communications
Symbol Classes
Sjöstrand class M ∞,1 (R2d ):
Z
kσkM ∞,1 =
sup |(σ · Tz Φ)b(ζ)| d ζ < ∞
R2d z∈R2d
⇒ (σ · Tz Φ)b ∈ L1 .
Locally σ is a Fourier transform of an L1 -function!
Analogy between absolutely convergent Fourier series and M ∞,1 ?
Weighted Sjöstrand class Mv∞,1 (R2d ).
Z
sup |(σ · Tz Φ)b(ζ)| v(ζ) d ζ < ∞
kσkM ∞,1 =
v
R2d z∈R2d
, 0 ≤ b ≤ 1, is suitable symbol class for wireless
Strohmer: M ∞,1
ea|ζ|b
communications.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
24 / 31
Variations
Time-Varying Systems and Wireless Communications
Wiener’s Lemma for Mv∞,1
Theorem (Sjöstrand)
If σ ∈ M ∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ M ∞,1 .
Theorem (KG)
Assume that v is submultiplicative and GRS
limn→∞ v(nz)1/n = 1, ∀z ∈ R2d .
If σ ∈ Mv∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ Mv∞,1 .
Ingredients of Proof. Based on Wiener’s Lemma for the matrix
algebra Cv and several identities of time-frequency analysis.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
25 / 31
Variations
Time-Varying Systems and Wireless Communications
Wiener’s Lemma for Mv∞,1
Theorem (Sjöstrand)
If σ ∈ M ∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ M ∞,1 .
Theorem (KG)
Assume that v is submultiplicative and GRS
limn→∞ v(nz)1/n = 1, ∀z ∈ R2d .
If σ ∈ Mv∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ Mv∞,1 .
Ingredients of Proof. Based on Wiener’s Lemma for the matrix
algebra Cv and several identities of time-frequency analysis.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
25 / 31
Variations
Time-Varying Systems and Wireless Communications
Wiener’s Lemma for Mv∞,1
Theorem (Sjöstrand)
If σ ∈ M ∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ M ∞,1 .
Theorem (KG)
Assume that v is submultiplicative and GRS
limn→∞ v(nz)1/n = 1, ∀z ∈ R2d .
If σ ∈ Mv∞,1 (R2d ) and Kσ is invertible on L2 (Rd ), then Kσ−1 = Kτ for
some τ ∈ Mv∞,1 .
Ingredients of Proof. Based on Wiener’s Lemma for the matrix
algebra Cv and several identities of time-frequency analysis.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
25 / 31
Variations
Time-Varying Systems and Wireless Communications
Hörmander’s Class
0 if and only if ∂ α σ ∈ L∞ (R2d ), ∀α ≥ 0.
Hörmander class: σ ∈ S0,0
Observation: If vs (ζ) = (1 + |ζ|)s , then
\ ∞,1
0
S0,0
=
Mvs
s≥0
Corollary (Beals ’75)
0 and K is invertible on L2 (Rd ), then K −1 = K for some
If σ ∈ S0,0
τ
σ
σ
0 .
τ ∈ S0,0
REMARK: “Beals implies Connes”
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
26 / 31
Variations
Time-Varying Systems and Wireless Communications
Hörmander’s Class
0 if and only if ∂ α σ ∈ L∞ (R2d ), ∀α ≥ 0.
Hörmander class: σ ∈ S0,0
Observation: If vs (ζ) = (1 + |ζ|)s , then
\ ∞,1
0
S0,0
=
Mvs
s≥0
Corollary (Beals ’75)
0 and K is invertible on L2 (Rd ), then K −1 = K for some
If σ ∈ S0,0
τ
σ
σ
0 .
τ ∈ S0,0
REMARK: “Beals implies Connes”
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
26 / 31
Variations
Time-Varying Systems and Wireless Communications
Hörmander’s Class
0 if and only if ∂ α σ ∈ L∞ (R2d ), ∀α ≥ 0.
Hörmander class: σ ∈ S0,0
Observation: If vs (ζ) = (1 + |ζ|)s , then
\ ∞,1
0
S0,0
=
Mvs
s≥0
Corollary (Beals ’75)
0 and K is invertible on L2 (Rd ), then K −1 = K for some
If σ ∈ S0,0
τ
σ
σ
0 .
τ ∈ S0,0
REMARK: “Beals implies Connes”
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
26 / 31
Variations
Convolution operators on groups
Variation V — Convolution Operators on Locally Compact Groups
G . . . locally compact group with Haar measure dx
R
convolution: (f ∗ g)(x) = G f (y)g(y −1 x) dy
convolution operator Cf :
Cf h = f ∗ h
If f ∈ L1 (G), then Cf is bounded on Lp (G).
σLp (Cf ) = {λ ∈ C : Cf − λI not invertible on Lp }
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
27 / 31
Variations
Convolution operators on groups
Spectral Invariance of Convolution Operators
Lemma (Barnes)
Spectral invariance σLp (Cf ) = σL2 (Cf ) for 1 ≤ p ≤ ∞ holds, if and only
if G is amenable and symmetric.
• Amenability: existence of (translation) invariant mean on L∞ (G)
• Symmetry: spectrum of positive elements is positive
σL1 (f ∗ ∗ f ) ⊆ [0, ∞)
∀f ∈ L1 (G)
• Symmetry of A ⇔ A is inverse-closed in its enveloping
C ∗ -algebra.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
28 / 31
Variations
Convolution operators on groups
Spectral Invariance of Convolution Operators
Lemma (Barnes)
Spectral invariance σLp (Cf ) = σL2 (Cf ) for 1 ≤ p ≤ ∞ holds, if and only
if G is amenable and symmetric.
• Amenability: existence of (translation) invariant mean on L∞ (G)
• Symmetry: spectrum of positive elements is positive
σL1 (f ∗ ∗ f ) ⊆ [0, ∞)
∀f ∈ L1 (G)
• Symmetry of A ⇔ A is inverse-closed in its enveloping
C ∗ -algebra.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
28 / 31
Variations
Convolution operators on groups
Convolution Operators on Locally Compact Groups
Polynomial growth: G has polynomial growth, if for some neighborhood
U = U −1 of identity |U n | ≤ Cnd
Example: G = Rd , U = [−1, 1]d , U n = [−n, n]d , |U n | = (2n)d .
Theorem (Losert ’01)
Every compactly generated group of polynomial growth is symmetric.
Corollary
Wiener’s Lemma holds in all groups of polynomial growth, i.e.,
σLp (Cf ) = σL2 (Cf ) for all f ∈ L1 (G) and all 1 ≤ p ≤ ∞.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
29 / 31
Variations
Convolution operators on groups
Convolution Operators on Locally Compact Groups
Polynomial growth: G has polynomial growth, if for some neighborhood
U = U −1 of identity |U n | ≤ Cnd
Example: G = Rd , U = [−1, 1]d , U n = [−n, n]d , |U n | = (2n)d .
Theorem (Losert ’01)
Every compactly generated group of polynomial growth is symmetric.
Corollary
Wiener’s Lemma holds in all groups of polynomial growth, i.e.,
σLp (Cf ) = σL2 (Cf ) for all f ∈ L1 (G) and all 1 ≤ p ≤ ∞.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
29 / 31
Variations
Convolution operators on groups
Convolution Operators on Locally Compact Groups
Polynomial growth: G has polynomial growth, if for some neighborhood
U = U −1 of identity |U n | ≤ Cnd
Example: G = Rd , U = [−1, 1]d , U n = [−n, n]d , |U n | = (2n)d .
Theorem (Losert ’01)
Every compactly generated group of polynomial growth is symmetric.
Corollary
Wiener’s Lemma holds in all groups of polynomial growth, i.e.,
σLp (Cf ) = σL2 (Cf ) for all f ∈ L1 (G) and all 1 ≤ p ≤ ∞.
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
29 / 31
Variations
Convolution operators on groups
Weighted Versions
• weighted L1 with norm
kf kL1v =
Z
|f (x)|v(x) dx
G
Theorem (FGLLM)
Assume that G is compactly generated and possesses polynomial
growth and v is submultiplicative. TFAE:
(i) L1v (G) is symmetric.
Spectrum of positive operators is positive.
(ii) Spectral invariance
σL1v (Cf ) = σL2 (Cf )
∀f ∈ L1v (G)
(iii) The weight v satisfies the GRS condition (Gelfand, Raikov, Shilov)
lim v(x n )1/n = 1
n→∞
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
∀x ∈ G
Sept. 2009
30 / 31
Variations
Convolution operators on groups
Summary
Wiener’s Lemma is a deep statement about the invertilibity of operators.
Applications:
• study of input-output relations in communication systems
• time-varying channels in wireless communications
• frame theory
Tool:
• in non-commutative geometry
• analysis of pseudodifferential operators
Karlheinz Gröchenig (EUCETIFA)
Wiener’s Lemma
Sept. 2009
31 / 31