Reconstruction in reproducing kernel Banach
spaces
Jens Gerlach Christensen
Tufts University
Texas A&M , July 2012
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Overview
Idea: Use smoothness to obtain sampling results.
Extends: Gröchenig, Pesenson, Feichtinger, Führ (Paley-Wiener space)
Plan for talk:
Classical sampling results
Reproducing kernel Banach spaces
Smoothness of functions and sampling
Smoothness of kernel and sampling
Sampling without continuous projections
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Band limited functions on the real line
The Fourier transform is the extension to L2 (R) of
Z
1
b
f (w ) =
f (x)e −ix·w dx.
(2π)1/2
L2π = {f ∈ L2 (R) | b
f (w ) = 0 when |w | > π}
Theorem (Gröchenig)
For an increasing sequence xn without density points and with
limn→±∞ xn = ±∞ and δ := sup(xn+1 − xn ) < 2/π we have
X xn+1 − xn−1
n
and φn (x) =
q
2
xn+1 −xn−1
sinc(π(x
2
|f (xn )|2 ∼ kf k2L2π
− xn )) form a frame for L2Ω .
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Proof of irregular sampling theorem by Gröchenig
Key 1: For x ∈ [xn − 1/π, xn + 1/π] local differences can be estimated
|f (x) − f (xn )| = |f (x) − f (x + tn )|
Z tn
=
f 0 (x + t) dt 0
(Z
0
0
≤ max
Z
|f (x + t)| dt,
−1/π
)
1/π
0
|f (x + t)| dt
0
Key 2: f 0 ∈ L2π and kf 0 k ≤ πkf k.
Key 3: L2π has reproducing kernel φ(x) = sinc(πx).
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
.
Reproducing kernel Banach spaces
G Lie group with left Haar measure dx.
B Banach function space on G .
B solid, left and right invariant.
Convergence implies convergence in measure (for example Lp (G )).
Dual B ∗ .
Assume 0 6= φ ∈ B ∩ B ∗ and
Z
φ ∗ φ(x) = φ(y )φ(y −1 x) dy = φ(x),
then
Bφ = {f ∈ B | f = f ∗ φ}
is a reproducing kernel Banach subspace of B.
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Smoothness of functions and sampling
Feichtinger and Gröchenig: Approximate f ∈ Bφ by sums
X
f (xi )ψi ,
i
P
where i ψi = 1.
Fix basis X1 , . . . , Xn for g and define
U = {e t1 X1 · · · e tn Xn | − ≤ tk ≤ }.
Choose xi and ψi such that
xi U have finite covering property
0 ≤ ψi ≤ 1xi U
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Smoothness of functions and sampling
Right differentiation in the direction X :
R(X )f (x) =
d f (xe tX )
dt t=0
Repeated differentiation:
R α f = RXα(1) RXα(2) · · · RXα(m) f
Lemma
If f ∈ B has right derivatives in B, then
X
kf −
f (xi )ψi kB ≤ C
i
X
kR α f kB ,
|α|≤dim(G )
where C → 0 as → 0.
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Smoothness of the kernel
Define T1 : Bφ → Bφ (convolution with φ continuous)
X
T1 f =
f (xi )ψi ∗ φ
i
Proposition
If convolution with φ is continuous on B and right differentiation is
continuous on Bφ , then T1 is invertible if xi are close enough.
Theorem (C.)
If f 7→ f ∗ |R α φ| continuous on B for |α| ≤ dim(G ), then T1 is invertible
when xi are close enough.
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Other reconstruction operators
Alternatively we can define
T2 f =
X
Z
λi (f )`xi φ
with λi (f ) =
f (x)ψi (x) dx
i
T3 f =
X
Z
ci f (xi )`xi φ
with ci =
ψi (x) dx
i
Theorem (C.)
If f 7→ f ∗ |R α φ| and f 7→ f ∗ |Lα φ| are continuous on B for
|α| ≤ dim(G ), then T2 and T3 are invertible when xi are close enough.
Define (Feichtinger/Gröchenig) the sequence space
X
Bd = {{λi } |
|λi |1xi U ∈ B},
i
then, since f (xi ) = hf , `xi φi, {`xi φ} is a Bd -frame if xi are close enough
and a Banach frame if convolution with φ is continuous.
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces
Bd -frames
What can be done if convolution with φ is not continuous on B?
[Casazza,Christensen,Stoeva]:
If unit vectors is Schauder basis for (Bd )∗ ,
P
then {λi } 7→ i λi `xi φ is continuous from (Bd )∗ onto (Bφ )∗ . Then point
evaluation can be rewritten
X
X
f (x) = hf , φx i = hf ,
λi (x)`xi φi =
f (xi )λi (x).
i
i
Does not work for B = L1 . Instead by idea of [Han,Nashed,Sun] this
expansion is possible if B is Köthe function space (dual given by
integration).
P
Proof uses: On Köthe space, the sum i λi 1xi U converges in norm, so
finite sequences are dense in Bd . Then the unit vectors form Schauder
basis for Bd and dual given by sequences {ci } and dual pairing is
X
λi ci .
i
Jens Gerlach Christensen
Reconstruction in reproducing kernel Banach spaces