Sampling and Interpolation on Certain Nilpotent Lie
Groups
Brad Currey
Azita Mayeli
Vignon Oussa
Department of Mathematics
and Computer Science
Saint Louis University
St. Louis, MO 63103
Email: curreybn@slu.edu
Deopartment of Mathematics
Queensboro College
City University of New York
Bayside, NY 11362
Email: amayeli@qcc.cuny.edu
Department of Mathematics
Bridgewater State University
Bridgewater, MA 02325
Email: vignon.oussa@bridgew.edu
Abstract—Rank-one left-invariant subspaces of L2 (G) that are
sampling spaces and possess the interpolation property with
respect to a class of discrete subsets of G are characterized for
certain two-step nilpotent Lie groups G.
I. I NTRODUCTION
Let G be a locally compact group, H a closed subspace
of L2 (G) consisting of continuous functions, Γ a discrete,
countably infinite subset of G, and let ρ be the restriction
mapping f 7→ f |Γ on H. The sampling problem can be posed
very generally: describe those pairs (H, Γ) for which ρ is a
bounded injective map into `2 (Γ) with bounded inverse. If
ρ is surjective then we say that (H, Γ) has the interpolation
property. Sampling on groups has been extensively studied;
see for example [6], [8], [10]. Of special relevance are several portions of the monograph [5], without which the work
presented here would not be possible.
Here we consider sampling on left-invariant subspaces. For
x ∈ G denote the left translation operator φ 7→ φ(x−1 ·)
by Lx ; a subspace H of L2 (G) is left-invariant if H is
closed and Lx (H) ⊂ H holds for all x ∈ G. Given a leftinvariant subspace H that consists of continuous functions,
and a countable discrete subset Γ of G, (H, Γ) is called a
sampling pair if there is c > 0 such that for all φ ∈ H
kφk2 =
1X
|φ(γ)|2 .
c
(I.1)
γ∈Γ
If (H, Γ) is a sampling pair, then there is S ∈ H such that
φ 7→ φ ∗ S is the orthogonal projection of L2 (G) onto H, and
φ=
1X
φ(γ)Lγ S
c
(I.2)
γ∈Γ
holds for all φ ∈ H, where the sum (I.2) converges in L2 and
uniformly [5, Theorem 2.56]. It is then immediate that the
system { √1c Lγ S : γ ∈ Γ} is a Parseval frame for H. We say
that c is the sampling constant for the sampling pair (H, Γ),
and that √1c S is the sinc-type function for (H, Γ). Now recall
the following fact.
Theorem I.1. Let (H, Γ) be a sampling pair. Then the
following are equivalent.
(i) (H, Γ) has the interpolation property
(ii) { √1c Lγ S : γ ∈ Γ} is an orthonormal basis for H.
Proof: Let A denote the analysis operator for the Parseval
frame { √1c Lγ S}γ∈Γ . If (i) holds then A is surjective. Let δγ
denote the canonical basis element in `2 (Γ); then k √1c Lγ Sk =
kA∗ δγ k = kδγ k = 1 and (ii) holds. The converse (ii) =⇒
(i) is immediate.
In what follows we describe families of sampling pairs and
characterize those that have the interpolation property, first
when G is the Heisenberg group, and then when G belongs to
a family of nilpotent Lie groups that resemble the Heisenberg
group in certain key aspects. Finally, we discuss partial results
for more general subspaces and classes of nilpotent groups.
II. T HE H EISENBERG GROUP CASE
Assume that G is the three-dimensional simply-connected
Heisenberg group, realized as R3 , with the group operation
(x1 , x2 , x3 ) · (y1 , y2 , y3 ) = (x1 + y1 , x2 + y2 , x3 + y3 + x1 y2 ).
We recall some basic facts about harmonic analysis on G. For
x ∈ G, λ ∈ R \ {0}, define the unitary operator πλ (x) on
L2 (R) by
(πλ (x)f ) (t) = e2πiλx3 e−2πiλx2 t f (t − x1 ), f ∈ L2 (R).
Then x 7→ πλ (x) is an irreducible representation of G, and
for λ 6= λ0 , the representations πλ and πλ0 are inequivalent.
The group Fourier transform is defined on L1 (G) ∩ L2 (G) by
the weak operator integral
Z
φ̂(λ) :=
φ(x)πλ (x)dx, λ ∈ R \ {0};
G
φ̂(λ) is a trace-class operator for all λ ∈ R \ {0}, and for each
\
x ∈ G, (L
x φ)(λ) = πλ (x) ◦ φ̂(λ). One has
Z
1
2
kφkL2 (G) =
kφ̂(λ)k2 |λ|dλ
(2π)2 R\{0}
where kφ̂(λ)k is the Hilbert-Schmidt norm of the operator
φ̂(λ), and the group Fourier transform extends to a unitary
isomorphism of L2 (G) with the Hilbert space of operatorvalued functions
L2 R \ {0}, HS L2 (R) , (2π)−2 |λ|dλ ;
here HS(L2 (R)) is the Hilbert space of Hilbert-Schmidt
operators on L2 (R).
Given a closed left-invariant subspace H of L2 (G) with
orthogonal projection P : L2 (G) → H, there is a unique
(up to a.e. equality) measurable field {P̂λ }λ∈Ĝ of orthogonal
projections on L2 (R), such that
[
(P
φ)(λ) = φ̂(λ)P̂λ
holds for a.e. λ. The spectrum of H is the (equivalence class
of the) set E = E(H) = {λ : rank(P̂λ ) 6= 0}. If P is given
by φ 7→ φ ∗ S where S is a convolution idempotent belonging
to H, then the corresponding measurable field is the group
Fourier transform Ŝ of S.
Now let H be a left-invariant subspace for which
rank(P̂λ ) ≤ 1 a.e.. There is a measurable field {eλ } of
functions in L2 (R), such that Pλ = eλ ⊗ eλ , a.e.λ. The map
φ 7→ φ̂(λ)eλ (t) (λ ∈ R \ {0}, t ∈ R) defines a unitary
isomorphism of H with the closed subspace L2 (E × R) of
L2 (R \ {0} × R), where E is the spectrum of H, and carries
the measure (2π)−2 |λ|dλ. Denote this isomorphism by R.
Fix ψ ∈ H and g = Rψ; then for each x ∈ G, R(Lx ψ) =
L̂x g where
L̂x g(λ, t) = e2πiλx3 e−2πiλx2 t g(λ, t − x1 ).
Now for α > 0, β > 0, let Γα,β = αZ × βZ × Z ⊂ G. It is
clear that the systems
T (ψ, α, β) = {Lγ ψ : γ ∈ Γ}
and
Tb (g, α, β) = {L̂γ g(λ, t) : γ ∈ Γ}
are equivalent. For λ 6= 0 fixed, writing g(λ, ·) = gλ , then
λ
T̂k,l,0 defines the unitary Gabor operator T̂k,l
:
λ
T̂k,l
gλ (t)
= Tk M−λl gλ (t) = e
−2πiλlt λ
g (t − k).
λ
For u ∈ L2 (R) set G(u, α, β, λ) = {T̂k,l
u : (k, l, 0) ∈ Γα,β }.
We say that g ∈ L2 (E × R) is a Gabor field over E with
respect to Γα,β if, for a.e. λ ∈ E, G(|λ|1/2 gλ , α, β, λ) is
a Parseval frame for L2 (R). If g is a Gabor field over E
with respect to Γα,β , then standard Gabor theory implies that
k|λ|1/2 gλ k2 = αβ|λ| ≤ 1.
The following provides an explicit construction of a large
family of sampling spaces.
Proposition II.1. ([3, Proposition 2.3] and [2, Proposition
2.1]) Let E ⊂ R \ {0} and g ∈ L2 (E × R).
(a) If Tb (g, α, β) is a Parseval frame for L2 (E × R), then g
is a Gabor field over E with respect to Γα,β .
(b) Suppose that E is translation congruent with a subset of
the unit interval. If g is a Gabor field over E with respect to
Γα,β , then Tb (g, α, β) is a Parseval frame for L2 (E × R).
Explicit construction of (rank-one) sampling spaces is given
by the following.
Theorem II.2. Let E be a subset of R \ {0} having finite
measure, let e ∈ L2 (E × R) such that eλ is a unit vector in
L2 (R) for a.e. λ, and let S = R−1 (e). Then the following are
equivalent.
(i) Tb ( √1c e, α, β) is a Parseval frame for L2 (E × R)
(ii) (H, Γα,β ) is a sampling pair where H is the left-invariant
subspace with convolution projection S.
If the above conditions hold, then e is a Gabor field over E,
and E is included in the interval [−1/αβ, 1/αβ].
Note that the band-limitation E ⊂ [−1/αβ, 1/αβ] arises
from the well-known condition ([7, Corollary 7.3.2]) for Gabor
systems mentioned above: k|λ|1/2 eλ k2 = αβ|λ| ≤ 1.
If α = β = 1, then the following is proved in [3, Proposition
2.5].
Corollary II.3. Let E ⊂ [−1, 1]. The the following are
equivalent.
(i) Every left invariant rank one subspace of L2 (G) with
spectrum E is a sampling space.
(ii) E is translation congruent with a subset of the unit
interval.
We have a precise criterion for the interpolation property in
this situation.
Theorem II.4. Let H be a left-invariant subspace of L2 (G)
with P̂λ rank one for a.e. λ ∈ E = E(H). Suppose that for
some lattice parameters α and β, (H, Γα,β ) is a sampling pair.
Then the sampling constant c = 1/αβ. Moreover, (H, Γα,β )
has the interpolation property if and only if
Z
1
|λ|dλ = 1/αβ
(2π)2 E
and in this case, αβ ≤ 1.
Proof: Let S be the convolution projection for H, with
R(S) = e. The sinc-type function for (H, Γα,β ) is √1c S. Then
{ √1 Tγ S}γ is a Parseval frame for H and Tb ( √1 e, α, β) so by
c
Proposition II.1,
λ ∈ E, we have
√1 e
c
c
is a Gabor field over E. Hence for a.e.
1
k|λ|1/2 √ eλ k2 = αβ|λ|,
c
and the relation c = 1/αβ follows immediately. Now H has
the interpolation property if and only if { √1c Tγ S : γ ∈ Γα,β }
is an orthonormal basis for H, if and only if k √1c Sk = 1. But
1
k √ Sk2 = αβkSk2
c
= αβkVe (S)k2
Z
= αβ
keλ k2 |λ|dλ
E
Z
= αβ
|λ|dλ.
E
This proves the first part of the theorem. Now if (H, Γα,β ) has
the interpolation property, then, since E ⊆ [−1/αβ, 1/αβ], we
have
Z
Z
1/αβ =
|λ|dλ ≤
|λ|dλ = 1/(αβ)2 .
E
λ−1
Hence for each k, the sequences {hfλ−1 , T̂k,l
eλ−1 i : l ∈ Z}
λ
and {hf (λ, ·), T̂k,l
eλ i : l ∈ Z} are Fourier coefficients for
orthogonal functions and we have
X
λ−1
λ e i = 0.
hfλ−1 , T̂k,l
eλ−1 i hfλ , T̂k,l
λ
[−1/αβ,1/αβ]
so αβ ≤ 1.
We now construct a concrete example of a sampling pair
with the interpolation property. We assume that α = β = 1;
note that in this case the interpolation property is equivalent
with E = [−1, 1]. In light of Theorems II.2 and II.4, it is
evident that in order to construct a sampling pair (H, Γ1,1 ),
it is enough to construct e ∈ L2 ([−1, 1] × R) with keλ k = 1
for all λ ∈ [−1, 1] such that Tb (e, 1, 1) is a Parseval frame
for L2 ([−1, 1] × R). The following lemma is crucial for this
purpose.
Lemma II.5. [2, Lemma 2.5] (see also [5, Lemma 6.13]) Let
e ∈ L2 ([−1, 1] × R) such that e is a Gabor field over [−1, 1]
with respect to Γα,β , and such that the orthogonality condition
X
λ−1
λ e i=0
(II.1)
hfλ−1 , T̂k,l
eλ−1 i hfλ , T̂k,l
λ
l
Hence the equation (II.1) holds.
III. A CLASS OF NILPOTENT L IE GROUPS
Let G be a simply connected Lie group with Lie algebra g of dimension n satisfying the following conditions
g = a ⊕ b ⊕ z, [a, b] ⊆ z,, where a, b, z are non-zero abelian
algebras such that
a = R-span {X1 , X2 , · · · , Xd } ,
b = R-span {Y1 , Y2 , · · · , Yd } ,
z = R-span {Z1 , Z2 , · · · , Zn−2d } ,
and where

[X1 , Y1 ]
[X2 , Y1 ]

P = det 
..

.
k,l
holds for a.e. λ ∈ (0, 1] and for all f ∈ L2 ([−1, 1] × R). Then
the system Tb (e, α, β) is a Parseval frame for L2 ([−1, 1] × R).
We can now construct e ∈ L2 ([−1, 1] × R) so that with
S = R−1 (e) and H = L2 (G)∗S, then (H, Γ1,1 ) is a sampling
pair with interpolation.
Example II.6. [2, Example 2.6] For λ ∈ (0, 1], put
eλ = 1[ 1 −1, 1 ]
λ
λ
and
eλ−1 = 1[−1,0] .
Then e defined by e(λ, t) = eλ (t) for λ ∈ (0, 1] and e(λ, t) =
1[−1,0] (t) for λ ∈ [−1, 0) is a Gabor field over [−1, 1] with
respect to Γ1,1 satisfying the conditions of Lemma II.5.
Proof: We compute that for any f ∈ L2 ([−1, 1] × R) and
for λ ∈ (0, 1],
λ−1
hf (λ − 1, ·), T̂k,l
e(λ − 1, ·)i
Z
=
f (λ − 1, t)e2πi(λ−1)lt 1[−1,0] (t − k)dt
R
Z
1
s
=
f λ − 1,
e2πils ds
1−λ
λ−1
Ikλ−1
where Ikλ−1 = [−(1 − λ)k, −(1 − λ)k + (1 − λ)]. Similarly,
Z
hf (λ, ·), ek,l,0 (λ, ·)i =
f (λ, ·)e2πiλlt 1[ 1 −1, 1 ] (t − k)dt
λ
λ
ZR 1 s
=
f (λ,
) e2πils ds
λ
λ
Ikλ
with
k,
Ikλ
[Xd , Y1 ]
···
···
···
[Xd , Y2 ] · · ·

[X1 , Yd ]
[X2 , Yd ]


..

.
[Xd , Yd ]
Λ = {λ ∈ z∗ : P (λ) 6= 0}.
For each λ ∈ Λ, there is an irreducible unitary representation
πλ of G obtained by inducing from the character χλ (exp Z) =
exp(2πiλ(Z)) of the normal subgroup corresponding to z + b.
For each φ ∈ L1 (G) ∩ L2 (G), the group Fourier transform of
φ is the trace-class operator-valued function on Λ defined by
Z
φ̂(λ) =
φ(x)πλ (x)dx
G
and
kφk2L2 (G) =
1
(2π)n−d
Z
kφ̂(λ)k2 |λ|n−2d dλ,
Λ
giving an isomorphism of L2 (G) with
L2 (Λ, HS(L2 (Rd )), (2π)n−d |P (λ)|dλ).
In this section we suppose that Γ is integral:
Γb = exp (ZY1 + · · · + ZYd ) ,
Γa = exp (ZX1 + · · · + ZXd ) ,
and
and
(Ikλ−1 + k) ∪ Ikλ = [λk, λk + 1].
(III.1)
is a non-vanishing homogeneous polynomial in the unknowns
Z1 , · · · , Zn−2d . The group Fourier transform for G is obtained
via the method of coadjoint orbits, and is very similar to that
of the Heisenberg group. Let g∗ be the linear dual space of
the vector space g; P is naturally regarded as a polynomial
function on g∗ . For ` ∈ g∗ such that P (`) 6= 0, the coadjoint
orbit of ` is ` + z⊥ = {f ∈ g∗ : z ⊂ ker(f − `)}. Thus almost
all coadjoint orbits are parametrized by
= [1 + λk − λ, 1 + λk]. It is easily seen that for each
Ikλ−1 ∩ Ikλ = ∅
[X1 , Y2 ]
[X2 , Y2 ]
..
.
Γz = exp (ZZ1 + · · · + ZZn−2d ) ,
and put
Γ = Γz Γb Γa ⊂ N.
(III.2)
Now, let Λ0 = {λ ∈ Λ : |P (λ)| ≤ 1} and C ⊂ z∗ = Rn−2d
be a bounded set such that
o
n
e2πihk,λi χC (λ) : k ∈ Zn−2d
is a Parseval frame for L2 (C, dλ) . Put E = Λ0 ∩ C.
Once again, each e ∈ L2 (E ×Rd ) such that keλ kL2 (Rd ) = 1
determines a left-invariant rank one subspace H = H(e). The
following is proved in [9].
Theorem III.1. There is a rank-one left-invariant subspace
H with spectrum E such that (H, Γ) is a sampling pair.
There is also a result for interpolation.
Theorem III.2. Suppose that E = Λ0 ∩ C is such that
Z
1
|P (λ)|dλ = 1.
(2π)n−d E
Then there is a rank-one left-invariant subspace H with
spectrum E such that (H, Γ) is a sampling pair with the
interpolation property.
IV. C ONCLUSION
The above shows that the group Fourier transform facilitates
very explicit constructions for sampling and interpolation
spaces on a class of nilpotent Lie groups, provided the spaces
are left-invariant and rank-one. Further results on sampling for
the Heisenberg group, for left-invariant subspaces that are not
necessarily rank-one, can be found in [5]. We describe a few
results and prospects, where one or more of these conditions
is relaxed.
First, suppose that G is the Heisenberg group, and that S(φ)
is a principal shift invariant space that is included in a rankone shift-invariant space. Let g = Rφ ∈ L2 (Λ × R) so that
the image of S(φ) under the group Fourier transform is
S(g, α, β) = span Tb (g, α, β) .
For each (λ, t) ∈ Λ × R put
Θgk (λ, t) :=
0
0
X
00 t − l
00 t − l
g λ−l ,
−k g λ−l ,
.
λ − l00
λ − l00
0 1
00
l ∈ β Z,l ∈Z
Then we have the following (δ0,k is the Kronecker delta.)
Theorem IV.1. [2, Theorem 2.8] Tb (g, α, β) is an orthonormal
basis for S(g, α, β) if and only if
Θgk (λ, t) = δ0,k
a.e. (λ, t).
Proof: We provide the proof for the case α = β = 1. For
each γ = (k, l, m) ∈ Γ1,1 the function
(λ, t) 7→ e2πiλm e−2πiλlt g(λ, t − k)g(λ, t)|λ|
is absolutely integrable. With Λ = R \ {0}, we can apply
periodization and Fubini’s theorem to calculate
Z Z
hT̂γ g, gi =
e2πiλm e−2πiλlt g(λ, t − k)g(λ, t)|λ|dtdλ
Λ
R
Z Z
=
e2πiλm e−2πilt g(λ, t/λ − k)g(λ, t/λ)dtdλ
Λ R
Z 1Z 1
e2πiλm e−2πilt Θgk (λ, t)dtdλ
=
0
0
Suppose that Tb (g, α, β) is an orthonormal basis for S(g, α, β).
Note that Θgk is a (1, 1)-periodic integrable function on T × T.
cg (m, l) = 0 for all integers m and l, and
If k 6= 0, then Θ
k
cg (m, l) = 0 holds for all
hence Θgk ≡ 0. If k = 0, then Θ
0
g
c (0, 0) = 1. Hence Θg ≡ 1.
(m, l) 6= 0 while Θ
0
0
On the other hand, if Θgk (λ, t) = δ0,k a.e. (λ, t), then
the above reasoning can be reversed to show that the system
Tb (g, α, β) is orthonormal.
Next, suppose that G is a simply connected nilpotent
Lie group having irreducible representations that are square
integrable modulo the center [1]. Each member of the class of
groups considered in Section III, and in particular the Heisenberg group, satisfies this condition. Let z denote the center of
its Lie algebra g. Then there is a homogeneous polynomial
function P on z∗ such that almost all irreducible unitary
representations of G are parametrized (up to equivalence) by
the set
Λ = {` ∈ z∗ : P (`) 6= 0}.
By choosing a suitable basis for the Lie algebra
of G, one can fix a global coordinate system
(x1 , x2 , . . . , xd , y1 , y2 , . . . , yd , z1 , z2 , . . . , zr )
on
G
(r = dim z) with useful properties. Principle among these
is that for each λ ∈ Λ, there is a corresponding irreducible
representation πλ acting in L2 (Rd ) by an expression of the
form
πλ (x, y, z))f (t1 , . . . , td )
= e2πihz,λi e2πiL(x,y,t,λ) f (t1 − x1 , . . . , td − xd ).
The group Fourier transform is formally exactly as before: for
φ ∈ L1 (G) ∩ L2 (G),
Z
φ̂(λ) :=
φ(x)πλ (x)dx
G
is trace-class, and
kφk2L2 (G)
1
=
(2π)n−d
Z
kφ̂(λ)k2 |P (λ)|dλ,
Λ
giving an isomorphism of L2 (G) with
L2 (Λ, HS(L2 (Rd )), (2π)n−d |P (λ)|dλ).
Given E ⊂ Λ and e ∈ L2 (E×Rd such that keλ kL2 (R) = 1 a.e.,
a rank-one left-invariant subspace H of L2 (G) is obtained just
as before, with convolution idempotent S such that Ŝ(λ) =
eλ ⊗ eλ . H is identified with L2 (E × Rd ) as before.
Fix a countable discrete subset Γ of G = Rd × Rd × Rr of
the form
The parameter set for the irreducible unitary representations
of G is the set
Γ = Γx × Γy × Zr .
Λ = {λ1 Z1∗ + λ2 Z2∗ + µY2∗ ∈ g∗ : λ1 6= 0};
Put Γ1 = Γx × Γy × {0} ⊂ Γ. For g ∈ L2 (E × Rd ), we say
that g is a Parseval frame field if {πλ (γ)gλ : γ ∈ Γ1 } is a
Parseval frame for L2 (Rd ) for a.e. λ ∈ E.
Though the group Fourier transform is conceptually no
different than that of the Heisenberg group, it differs in several
important particulars. The Heisenberg group, and the class of
groups considered in Section III, are two-step groups: all
second-order commutators in the group are trivial. By contrast,
among groups with irreducible square-integrable representations modulo the center, there are members with non-vanishing
commutators of arbitrarily high order. As a consequence, the
function L(x, y, t, λ) that appears in the expression for the
irreducible representations πλ is now polynomial of arbitrarily
high degree. In particular, the system {πλ (γ)gλ : γ ∈ Γ1 } is
no longer a proper Gabor system.
However, since the function L is independent of the central
variables z1 , . . . , zr , the proof of the following is essentially
a repetition of the proof of Proposition II.1.
for each (λ1 , λ2 , µ) ∈ Λ, there is an irreducible unitary
representation π(λ1 ,Λ2 ,µ) of G acting in L2 (R) by
π(λ1 ,Λ2 ,µ) (x, y1 , y2 , z1 , z2 )f (t)
Proposition IV.2. Let E ⊂ Λ and let H be a rank-one, leftinvariant subspace of L2 (G) with spectrum E. Let φ ∈ H
with g = Rφ ∈ L2 (E × Rd ).
(a) If {Lγ φ : γ ∈ Γ} is a Parseval frame for H, then g is a
Parseval frame field over E.
(b) Suppose that E is translation congruent with a subset of
the unit cube in Rr . If g is a Parseval frame field over E, then
{Lγ φ : γ ∈ Γ} is a Parseval frame for H.
Hence it is of interest to consider examples of such groups
for which the function L has a suitable form. Conversely, it is
also of interest to apply coorbit theory to construct sampling
spaces directly, then to study the Gabor-type systems that arise
a posteriori from the preceding result.
We remark also that for nilpotent Lie groups with irreducible representations that are square integrable modulo the
center, general shift-invariant spaces are characterized in [4],
so that further study of shift-invariant spaces in light of the
preceding would be of interest as well.
We conclude with an example of a simply-connected twostep nilpotent Lie group that does not possess irreducible
square integrable representations modulo the center. Here
we see that the expression for a typical irreducible unitary
representation of G is not a simple product of a unitary
exponential on the center and a (generalized) Gabor system.
Example IV.3. Let G = R × R2 × R2 with group product
(x, y1 , y2 , z1 , z2 ) · (x0 , y10 , y20 , z10 , z20 )
= (x + x0 , y1 + y10 , y2 + y20 , z1 + z10 + xy1 , z2 + z20 + xy2 ).
= e2πi(λ1 z1 +λ2 z2 +µy2 ) e−2πi(λ1 y1 +λ2 y2 )t f (t − x).
Let Γ be the set of integer points in G, denoted by γ =
(k, l1 , l2 , m1 , m2 ). Let Γ1 = {(k, l1 , 0, 0, 0) : k, l1 ∈ Z}
and Γ0 = {(0, 0, l2 , m1 , m2 ) : l2 , m1 , m2 ∈ Z}. If E is a
subset of Λ for which the characters {e2πi(λ1 z1 +λ2 z2 +µy2 ) :
l2 , m1 , m2 ∈ Z} are a Parseval frame, then it is natural to
define a frame field over a set E as a function g ∈ L2 (E × R)
such that for a.e. (λ1 , λ2 , µ) ∈ E,
{π(λ1 ,λ2 ,µ) (γ)g(λ1 ,λ2 ,µ) : γ ∈ Γ1 }
= {e−2πiλ1 l1 t gλ (t − k) : l1 , k ∈ Z}
is a Parseval frame for L2 (R). This is a perfectly good Gabor
system, but the expression for π(λ1 ,Λ2 ,µ) is not given by the
product of this Gabor system with a unitary exponential on the
parameter set Λ: the latter includes the additional modulation
e−2πiλ2 y2 t . It follows that the proof of Proposition IV.2 no
longer works; generalizations of Proposition IV.2 to such
groups would need to overcome this obstruction.
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