A frame of translates of a single function for Lp (R) with
2<p<∞
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
July 22, 2012 at Larsonfest, Texas A&M University
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆H
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
S(x) =
∞
X
hxi , xixi .
i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
S(x) =
∞
X
hxi , xixi .
i=1
Theorem
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
S(x) =
∞
X
hxi , xixi .
i=1
Theorem
Let (xi )∞
i=1 ⊆ H.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
S(x) =
∞
X
hxi , xixi .
i=1
Theorem
∞
Let (xi )∞
i=1 ⊆ H.The sequence (xi )i=1 ⊆ H is a frame for H if and only if
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces
Definition
A frame for a Hilbert space H is a sequence (xi )∞
i=1 ⊆ H, such that there exists
constants 0 < A ≤ B satsifying for all x ∈ H,
Akxk2 ≤
∞
X
|hxi , xi|2 ≤ Bkxk2 .
i=1
Given a sequence
(xi )∞
i=1
⊆ H the frame operator S : H → H is defined for all
x ∈ H by
S(x) =
∞
X
hxi , xixi .
i=1
Theorem
∞
Let (xi )∞
i=1 ⊆ H.The sequence (xi )i=1 ⊆ H is a frame for H if and only if the
frame operator is bounded and has bounded inverse.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
If (xi )∞
i=1 is a frame for a Hilbert space H,
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works. The frame (S −1 xi )∞
i=1 is called the
canonical dual frame of (xi )∞
i=1 .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works. The frame (S −1 xi )∞
i=1 is called the
canonical dual frame of (xi )∞
i=1 .
Definition (Schauder frame)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works. The frame (S −1 xi )∞
i=1 is called the
canonical dual frame of (xi )∞
i=1 .
Definition (Schauder frame)
Let X be an infinite dimensional Banach space.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works. The frame (S −1 xi )∞
i=1 is called the
canonical dual frame of (xi )∞
i=1 .
Definition (Schauder frame)
Let X be an infinite dimensional Banach space. A sequence
∗
(xi , fi )∞
i=1 ⊂ X × X is called a Schauder frame of X
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Frames for Hilbert spaces vs Frames for Banach spaces
Theorem
∞
If (xi )∞
i=1 is a frame for a Hilbert space H, then there exists a frame (fi )i=1 for
H such that for all x ∈ H,
x=
∞
X
hfi , xixi .
i=1
In particular, fi = S
−1
xi ∀i ∈ N works. The frame (S −1 xi )∞
i=1 is called the
canonical dual frame of (xi )∞
i=1 .
Definition (Schauder frame)
Let X be an infinite dimensional Banach space. A sequence
∗
(xi , fi )∞
i=1 ⊂ X × X is called a Schauder frame of X if for every x ∈ X ,
x=
∞
X
fi (x)xi .
i=1
The frame is called unconditional if the above series converges unconditionally
for all x ∈ X .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
Dλ (f )(x) = 2−λ/2 f (2−λ x)
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
A set (Dm Tn f )(m,n)∈Z×Z is called a wavelet frame for L2 (R) if it forms a frame
for L2 (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
A set (Dm Tn f )(m,n)∈Z×Z is called a wavelet frame for L2 (R) if it forms a frame
for L2 (R). The subspace spann∈Z {Tn f } is called the principle shift invariant
subspace generated by f .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
A set (Dm Tn f )(m,n)∈Z×Z is called a wavelet frame for L2 (R) if it forms a frame
for L2 (R). The subspace spann∈Z {Tn f } is called the principle shift invariant
subspace generated by f .
For what sets Γ, Λ ⊂ R and function f ∈ L2 (R) is (Tγ Mλ f )(γ,λ)∈Γ×Λ a Gabor
frame for L2 (R)?
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
A set (Dm Tn f )(m,n)∈Z×Z is called a wavelet frame for L2 (R) if it forms a frame
for L2 (R). The subspace spann∈Z {Tn f } is called the principle shift invariant
subspace generated by f .
For what sets Γ, Λ ⊂ R and function f ∈ L2 (R) is (Tγ Mλ f )(γ,λ)∈Γ×Λ a Gabor
frame for L2 (R)? In particular, does there exist Γ ⊂ R and f ∈ L2 (R) such that
(Tγ f )γ∈Γ is a Gabor frame?
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Gabor frames, wavelet frames, and translations of a single function
Let Γ, Λ ⊂ R. For each γ ∈ Γ, λ ∈ Λ, f ∈ L2 (R), and x ∈ R define:
Tγ (f )(x) = f (x −γ)
Mλ (f )(x) = e 2πiλx f (x)
Dλ (f )(x) = 2−λ/2 f (2−λ x)
Definition
A set (Tγ Mλ f )(γ,λ)∈Γ×Λ is called a Gabor frame for L2 (R) if it forms a frame
for L2 (R).
Definition
A set (Dm Tn f )(m,n)∈Z×Z is called a wavelet frame for L2 (R) if it forms a frame
for L2 (R). The subspace spann∈Z {Tn f } is called the principle shift invariant
subspace generated by f .
For what sets Γ, Λ ⊂ R and function f ∈ L2 (R) is (Tγ Mλ f )(γ,λ)∈Γ×Λ a Gabor
frame for L2 (R)? In particular, does there exist Γ ⊂ R and f ∈ L2 (R) such that
(Tγ f )γ∈Γ is a Gabor frame? Given, 1 ≤ p < ∞, Γ ⊂ R and f ∈ Lp (R), what
can be said about (Tγ f )γ∈Γ ?
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
If 2 < p < ∞
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
If 2 < p < ∞ then exists f ∈ Lp (R) such that spann∈Z {Tn f } = Lp (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
If 2 < p < ∞ then exists f ∈ Lp (R) such that spann∈Z {Tn f } = Lp (R).
Theorem (Olevskii ’79)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
If 2 < p < ∞ then exists f ∈ Lp (R) such that spann∈Z {Tn f } = Lp (R).
Theorem (Olevskii ’79)
For all {λn }n∈Z ⊂ R \ Z such that limn→±∞ |λn − n| = 0,
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Some known results about translations of a single function.
Theorem
If 1 ≤ p ≤ 2 then there does not exist f ∈ Lp (R) such that
spann∈Z {Tn f } = Lp (R).
Theorem (Atzmon and Olevskii ’96)
If 2 < p < ∞ then exists f ∈ Lp (R) such that spann∈Z {Tn f } = Lp (R).
Theorem (Olevskii ’79)
For all {λn }n∈Z ⊂ R \ Z such that limn→±∞ |λn − n| = 0, there exists
f ∈ L2 (R) such that span{Tλn f }n∈Z = L2 (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
For all Λ ⊂ R and f ∈ L2 (R), the set (Tλ f )γ∈Λ is not a frame for L2 (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
For all Λ ⊂ R and f ∈ L2 (R), the set (Tλ f )γ∈Λ is not a frame for L2 (R).
Theorem (Odell, Sari, Schlumprecht, Zheng ’11)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
For all Λ ⊂ R and f ∈ L2 (R), the set (Tλ f )γ∈Λ is not a frame for L2 (R).
Theorem (Odell, Sari, Schlumprecht, Zheng ’11)
Let 2 < p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
For all Λ ⊂ R and f ∈ L2 (R), the set (Tλ f )γ∈Λ is not a frame for L2 (R).
Theorem (Odell, Sari, Schlumprecht, Zheng ’11)
Let 2 < p < ∞. There exists a function f ∈ Lp (R) and Λ ⊂ N such that
(Tλ f )λ∈Λ is an unconditional basic sequence and Lp (R) embeds into
span{Tλ f }λ∈Λ
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Known results about frames and bases of translates.
Theorem (Christensen, Deng, Heil ’99)
For all Λ ⊂ R and f ∈ L2 (R), the set (Tλ f )γ∈Λ is not a frame for L2 (R).
Theorem (Odell, Sari, Schlumprecht, Zheng ’11)
Let 2 < p < ∞. There exists a function f ∈ Lp (R) and Λ ⊂ N such that
(Tλ f )λ∈Λ is an unconditional basic sequence and Lp (R) embeds into
span{Tλ f }λ∈Λ if and only if 4 < p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R),
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞. There exists g ∗ ∈ L∗p (R) and a sequence (xi )i∈N ⊂ Lp (R) such
that (xi , Ti g ∗ )i∈N is an unconditional Schauder frame for Lp (R),
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞. There exists g ∗ ∈ L∗p (R) and a sequence (xi )i∈N ⊂ Lp (R) such
that (xi , Ti g ∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
1 < p < 2.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞. There exists g ∗ ∈ L∗p (R) and a sequence (xi )i∈N ⊂ Lp (R) such
that (xi , Ti g ∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
1 < p < 2.
Theorem (F, Odell, Schlumprecht, Zsak)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞. There exists g ∗ ∈ L∗p (R) and a sequence (xi )i∈N ⊂ Lp (R) such
that (xi , Ti g ∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
1 < p < 2.
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Our results
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There exists f ∈ Lp (R) and a sequence (gi∗ )i∈N ⊂ L∗p (R) such
that (Ti f , gi∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
2 < p < ∞.
Corollary (F, Odell, Schlumprecht, Zsak)
Let 1 < p < ∞. There exists g ∗ ∈ L∗p (R) and a sequence (xi )i∈N ⊂ Lp (R) such
that (xi , Ti g ∗ )i∈N is an unconditional Schauder frame for Lp (R), if and only if
1 < p < 2.
Theorem (F, Odell, Schlumprecht, Zsak)
Let 1 ≤ p < ∞. There does not exist f ∈ Lp (R) and a sequence (λi )∞
i=1 ⊂ R
such that (Tλi f )i∈N is an unconditional basis for Lp (R).
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
The sequence (xi , fi )i∈N ⊂ X × X ∗ is called an approximate Schauder frame
if the frame operator is bounded and has bounded inverse.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
The sequence (xi , fi )i∈N ⊂ X × X ∗ is called an approximate Schauder frame
if the frame operator is bounded and has bounded inverse.
Note: If S satisfies kIdX − Sk < 1, then S is bounded and has bounded inverse.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
The sequence (xi , fi )i∈N ⊂ X × X ∗ is called an approximate Schauder frame
if the frame operator is bounded and has bounded inverse.
Note: If S satisfies kIdX − Sk < 1, then S is bounded and has bounded inverse.
Lemma (FOSZ)
If (xi , fi )i∈N is an approximate Schauder frame for a Banach space X and S is
its frame operator,
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
The sequence (xi , fi )i∈N ⊂ X × X ∗ is called an approximate Schauder frame
if the frame operator is bounded and has bounded inverse.
Note: If S satisfies kIdX − Sk < 1, then S is bounded and has bounded inverse.
Lemma (FOSZ)
If (xi , fi )i∈N is an approximate Schauder frame for a Banach space X and S is
its frame operator, then (xi , (S −1 )∗ fi )i∈N is a Schauder frame for X .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Approximate Schauder frames
Definition
Let X be a Banach space and let (xi , fi )i∈N ⊂ X × X ∗ . The frame operator
S : X → X for (xi , fi )i∈N is defined for all x ∈ X by
S(x) =
∞
X
fi (x)xi .
i=1
The sequence (xi , fi )i∈N ⊂ X × X ∗ is called an approximate Schauder frame
if the frame operator is bounded and has bounded inverse.
Note: If S satisfies kIdX − Sk < 1, then S is bounded and has bounded inverse.
Lemma (FOSZ)
If (xi , fi )i∈N is an approximate Schauder frame for a Banach space X and S is
its frame operator, then (xi , (S −1 )∗ fi )i∈N is a Schauder frame for X .
Thus, Lp (R) has a Schauder frame of translates of a single function if and only
if it has an approximate Schauder frame of translates of a single function.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0. Choose a subsequence (Ni )i∈N ∈ [N]ω
such that
(
∞
X
1−p/2 1/p
Nk
k=1
)
<
1
,
2Cu
where Cu is the unconditionality constant of (hi )i∈N .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0. Choose a subsequence (Ni )i∈N ∈ [N]ω
such that
(
∞
X
1−p/2 1/p
Nk
k=1
)
<
1
,
2Cu
where Cu is the unconditionality constant of (hi )i∈N . Choose an increasing set
(nk,i )k∈N,1≤i≤Nk ⊂ N such that,
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0. Choose a subsequence (Ni )i∈N ∈ [N]ω
such that
(
∞
X
1−p/2 1/p
Nk
k=1
)
<
1
,
2Cu
where Cu is the unconditionality constant of (hi )i∈N . Choose an increasing set
(nk,i )k∈N,1≤i≤Nk ⊂ N such that, for all (k, i) 6= (s, t),
supp(T−nk,i hk ) ∩ supp(T−ns,t hs ) = ∅, and
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0. Choose a subsequence (Ni )i∈N ∈ [N]ω
such that
(
∞
X
1−p/2 1/p
Nk
k=1
)
<
1
,
2Cu
where Cu is the unconditionality constant of (hi )i∈N . Choose an increasing set
(nk,i )k∈N,1≤i≤Nk ⊂ N such that, for all (k, i) 6= (s, t),
supp(T−nk,i hk ) ∩ supp(T−ns,t hs ) = ∅, and
for all (k, i) 6= (s, t) and (k 0 , i 0 ) 6= (s 0 , t 0 ) with (k, i, s, t) 6= (k 0 , i 0 , s 0 , t 0 ),
supp(Tns,t −nk,i hk ) ∩ supp(Tns 0 ,t 0 −nk 0 ,i 0 hk 0 ) = ∅.
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Construction
Let 2 < p < ∞ and let (hi )i∈N be an unconditional basis with
diam(supp(hi )) ≤ 1 ∀i ∈ N. Let εi & 0. Choose a subsequence (Ni )i∈N ∈ [N]ω
such that
(
∞
X
1−p/2 1/p
Nk
)
<
k=1
1
,
2Cu
where Cu is the unconditionality constant of (hi )i∈N . Choose an increasing set
(nk,i )k∈N,1≤i≤Nk ⊂ N such that, for all (k, i) 6= (s, t),
supp(T−nk,i hk ) ∩ supp(T−ns,t hs ) = ∅, and
for all (k, i) 6= (s, t) and (k 0 , i 0 ) 6= (s 0 , t 0 ) with (k, i, s, t) 6= (k 0 , i 0 , s 0 , t 0 ),
supp(Tns,t −nk,i hk ) ∩ supp(Tns 0 ,t 0 −nk 0 ,i 0 hk 0 ) = ∅.
We now set f :=
P∞ PNk
k=1
i=1
−1/2
Nk
T−nk,i hk .
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
f :=
Nk
∞ X
X
−1/2
Nk
T−nk,i hk
k=1 i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
f :=
Nk
∞ X
X
−1/2
Nk
T−nk,i hk
k=1 i=1
The following is an unconditional approximate Schauder frame for Lp (R):
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
f :=
Nk
∞ X
X
−1/2
Nk
T−nk,i hk
k=1 i=1
The following is an unconditional approximate Schauder frame for Lp (R):
−1/2 ∗
hk )k∈N,1≤i≤Nk
(Tnk,i f , Nk
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
f :=
Nk
∞ X
X
−1/2
Nk
T−nk,i hk
k=1 i=1
The following is an unconditional approximate Schauder frame for Lp (R):
−1/2 ∗
hk )k∈N,1≤i≤Nk
(Tnk,i f , Nk
−1/2 ∗
hk )k∈N,1≤i≤Nk .
Let S be the frame operator for (Tnk,i f , Nk
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
The Schauder frame of translates
f :=
Nk
∞ X
X
−1/2
Nk
T−nk,i hk
k=1 i=1
The following is an unconditional approximate Schauder frame for Lp (R):
−1/2 ∗
hk )k∈N,1≤i≤Nk
(Tnk,i f , Nk
−1/2 ∗
hk )k∈N,1≤i≤Nk .
Let S be the frame operator for (Tnk,i f , Nk
That is, for all
h ∈ Lp (R),
S(h) =
Ns
∞ X
X
Ns−1/2 hs∗ (h)Tns,t f
s=1 t=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
Tn1,t f k
as hs∗ (h1 ) = δs,1
t=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
Tn1,t f k
as hs∗ (h1 ) = δs,1
t=1
N1
X
−1/2
= kh1 − N1
t=1
Tn1,t
Nk
∞ X
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
t=1
−1/2
= kh1 − N1
N1
X
Tn1,t
Nk
∞ X
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
N1
X
t=1
Tn1,t
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
Tn1,t
t=1
−1/2
= kh1 − N1
N1
X
Nk
∞ X
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
−1/2
= 0 + N1
k
N1
X
N1
X
t=1
X
−1/2
Nk
Tn1,t
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
Tn1,t −nk,i hk k
t=1 (k,i)6=(1,t)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
Nk
∞ X
X
Tn1,t
t=1
−1/2
= kh1 − N1
N1
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
−1/2
= 0 + N1
k
N1
X
N1
X
t=1
X
−1/2
Nk
Tn1,t
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
Tn1,t −nk,i hk k
t=1 (k,i)6=(1,t)
−1/2
= N1
N1
X
(
X
−p/2 1/p
Nk
)
by disjointness of support
t=1 (k,i)6=(1,t)
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
Nk
∞ X
X
Tn1,t
t=1
−1/2
= kh1 − N1
N1
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
−1/2
= 0 + N1
k
N1
X
N1
X
t=1
X
−1/2
Nk
Tn1,t
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
Tn1,t −nk,i hk k
t=1 (k,i)6=(1,t)
−1/2
= N1
N1
X
(
X
−p/2 1/p
Nk
)
by disjointness of support
t=1 (k,i)6=(1,t)
−1/2
≤ N1
N1 ∞ Nk
X
X X −p/2 1/p
(
Nk
)
t=1 k=1 i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
Nk
∞ X
X
Tn1,t
t=1
−1/2
= kh1 − N1
N1
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
−1/2
= 0 + N1
k
N1
X
N1
X
Tn1,t
t=1
X
−1/2
Nk
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
Tn1,t −nk,i hk k
t=1 (k,i)6=(1,t)
−1/2
= N1
N1
X
(
X
−p/2 1/p
Nk
)
by disjointness of support
t=1 (k,i)6=(1,t)
−1/2
≤ N1
N1 ∞ Nk
∞
X
X X −p/2 1/p
X
1/p−1/2
1−p/2 1/p
(
Nk
)
= N1
(
Nk
)
t=1 k=1 i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
k=1
A frame of translates of a single function for Lp (R) with 2 < p < ∞
How to approximately reconstruct h1
kh1 − S(h1 )k = kh1 −
Ns
∞ X
X
Ns−1/2 hs∗ (h1 )Tns,t f k
s=1 t=1
N1
X
−1/2
= kh1 − N1
as hs∗ (h1 ) = δs,1
Tn1,t f k
t=1
N1
X
−1/2
= kh1 − N1
Nk
∞ X
X
Tn1,t
t=1
−1/2
= kh1 − N1
N1
X
−1/2
Nk
T−nk,i hk k
k=1 i=1
−1/2
Tn1,t N1
−1/2
T−n1,t h1 k + kN1
t=1
−1/2
= 0 + N1
k
N1
X
N1
X
t=1
X
−1/2
Nk
Tn1,t
X
−1/2
Nk
T−nk,i hk k
(k,i)6=(1,t)
Tn1,t −nk,i hk k
t=1 (k,i)6=(1,t)
−1/2
= N1
N1
X
(
X
−p/2 1/p
Nk
)
by disjointness of support
t=1 (k,i)6=(1,t)
−1/2
≤ N1
N1 ∞ Nk
∞
X
X X −p/2 1/p
X
1/p−1/2
1−p/2 1/p
1/p−1/2 −1 −1
(
Nk
)
= N1
(
Nk
)
< N1
2 Cu
t=1 k=1 i=1
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
k=1
A frame of translates of a single function for Lp (R) with 2 < p < ∞
Daniel Freeman, Edward Odell, Thomas Schlumprecht, Andras Zsak
A frame of translates of a single function for Lp (R) with 2 < p < ∞