Sampling, amenability and the Kunze-Stein
phenomenon
Jens Gerlach Christensen
Gestur Ólafsson
Stephen D. Casey
Department of Mathematics
Colgate University
Email: jchristensen@colgate.edu
Department of Mathematics
Louisiana State University
Email: olafsson@math.lsu.edu
Department of Mathematics and Statistics
American University
Email: scasey@american.edu
Abstract—Inspired by recent work on the connection between
representation theory and atomic decompositions, we take a look
at convolution operators on non-unimodular amenable groups as
well as non-compact semi-simple Lie groups. We then discuss
this in context of sampling. Furthermore, we look at sampling
and optimal sampling sets for some often studied spaces.
I. I NTRODUCTION
Sampling theory tries to answer the following two questions
for an appropriate class of functions
1) How close do sample points have to be and is there an
optimal sampling density?
2) Once sampled with sufficient density, how can we reconstruct a function in a generic way?
Of course it is also of high importance to decide for which
class of functions these questions can be answered, and it is
also worth noting that the density is tied to both the geometry
of the underlying domain as well as the class of functions.
In this paper we will try to address these questions, and
our main focus is on the reconstruction process in the case
where the underlying domain is a Lie group. The first part of
the paper investigates how to reconstruct functions; a process
which typically depends on the continuity of a projection operator. In particular we investigate how this continuity can be
proven using the Kunze-Stein phenomenon for non-compact
semi-simple Lie groups. The second part of the paper deals
with optimal sampling schemes in the three basic geometries:
the sphere, the plane and hyperbolic space.
II. S AMPLING AND RECONSTRUCTION OPERATORS
A reproducing kernel Banach space is a space of continuous
functions on a domain D equipped with a norm for which
point evaluation is continuous; i.e. |f (x)| ≤ Cx kf k. In many
cases the reproducing property implies the existence of a
projection operator P from a larger Banach space B onto the
e The strategy of reconstruction
reproducing kernel space B.
of sampling then consists of choosing
P points xi ∈ D and a
partition of unity ψi of D such that i f (xi )ψi is in B and
moreover approximates f in P
the sense that there is a constant
0 ≤ C < 1 such that kf − i f (xi )ψi kB ≤ Ckf kB . Then
the projection operator can P
be applied in order to obtain the
invertible operator T f = P i f (xi )ψi .
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
In the case of wavelet theory for square integrable representations of Lie groups and for Paley-Wiener spaces on
Gelfand pairs, the projection operator P can be realized as a
convolution of functions on a group.It is therefore an important
task in sampling theory to determine the continuity of certain
convolution operators, which motivates the study to follow. We
are in particular interested in the case when the convolution
kernel does not arise from an integrable function.
III. C ONVOLUTION ON AMENABLE , NON - UNIMODULAR
GROUPS
If G is a locally compact group it can be equipped with a
left-invariant Haar measure dx and convolution of functions
on G is defined by
Z
f ∗ g(y) =
f (x)g(x−1 y) dx
G
if the integrand is integrable.
Let ∆ denote the modular function on G defined by
Z
Z
f (x) dx = ∆(y)
f (xy) dx,
G
or equivalently
Z
G
Z
f (x) dx =
G
f (x−1 )∆(x−1 ) dx.
G
The group G is called uni-modular if ∆(x) = 1 for all
x ∈ G, and in this case the measure is also right-invariant.
A linear function M : C b (G) → C is called a mean if
M (f ) ≥ 0 if f is real and f ≥ 0, and if it satisfies M (1) = 1.
The mean is left-invariant if for fy (x) = f (y −1 x) = `y f (x)
we have M fy = M f . This is equivalent to the following
statement: Given any compact set K in G then for every > 0
there is a positive function s in L1 (G) such that kskL1 = 1
and k`y (s) − skL1 < for all y ∈ K. Groups G for which
these properties hold are called amenable.
All abelian and all solvable groups are amenable, but
non-compact semi-simple Lie groups, for example SL(2, R)
and SU(n, 1), are not amenable [Re00]. Solvable means
that there is a sequence of closed subgroups G0 =
G, G1 , ..., Gn , Gn+1 = {e} such that Gk+1 is normal in Gk
and Gk /Gk+1 is abelian.
If a non-compact locally compact group G is amenable, it
follows by [Re00, Theorem8.3.10] that if f is positive and
measurable, then
g 7→ f ∗ g
is a bounded mapping on Lp (G) if and only if f ∈ L1 (G). We
start by making the note, that if G is non-unimodular, then the
order of this convolution is crucial. In particular it is possible
to have g 7→ g ∗ f continuous on Lp (G) even if f is not in
L1 (G). The precise statement of this observation is the first
contribution of this paper:
Theorem 3.1: Let G be an amenable, non-unimodular locally compact group and let f be a positive measurable
function on G. If 1 < p < ∞ then g 7→ g ∗ f is continuous
p
if and only if f ∈ L1∆(1−p)/p (G), which means
Ron L (G)(1−p)/p
f (x)∆
(x) dx < ∞.
G
Example 3.2: Let
a
b
G = (a, b) =
a > 0, b ∈ R ⊆ SL(2, R)
0 a−1
with usual matrix multiplication as composition. This group
is solvable and thus amenable, but it is also non-unimodular
with modular function
∆(a, b) = a−2
and left-invariant Haar measure
Z Z ∞
da db
f (a, b) 2 .
a
R 0
The function
f (a, b) =
1
[(a +
a−1 )2
+ b2 ]σ/2
is the modulus of a wavelet coefficient stemming from the
discrete series representations of SL(2, R). We note that it
is not in L1 (G) if 1 < σ ≤ 2. The interest in this function
comes from its relation to the Bergman projection. In particular
convolution by f has been used to provide atomic decompositions of Bergman spaces through sampling on the group G in
[FG88], [CO09]. These reference contain the details we have
skipped in this paper.
From the theorem we conclude that g 7→ g ∗ f is continuous
on all Lp (G) for which 2/σ < p < 2/(2 − σ) and 1 < σ ≤ 2.
In particular, when σ = 2, this convolution is continuous on
all Lp (G) with 1 < p < ∞ even though in this case the
convolution kernel f is not in L1 (G).
IV. T HE K UNZE -S TEIN PHENOMENON
In this section we will point out that for some amenable
subgroups of semi-simple Lie groups, the continuity of convolution operators follows from the famous Kunze-Stein phenomenon [KS60], [Cow78], [Cow08]. Let G be a semi-simple,
connected and non-compact group with Haar measure dx, then
Theorem 4.1 (Cowling): Suppose that 1 ≤ r ≤ ∞, 1/r +
1/r0 = 1 and f ∈ Lr (G). Then the mapping g 7→ g ∗ f
is bounded from Lp (G) to Lq (G) provided that one of the
following conditions hold:
1) if r = 1 and 1 ≤ p = q ≤ ∞
2) if 1 < r ≤ 2,q ≥ r, p ≤ r0 , 0 ≤ 1/p − 1/q ≤ 1/r0 and
(p, q) 6= (r, r) and (p, q) 6= (r0 , r0 ).
3) if 2 < r < ∞, q ≥ r, p ≤ r0 , 0 ≤ 1/p − 1/q ≤ 1/r0 and
(p, q) 6= (r, r0 )
4) if r = ∞,p = 1 and q = ∞.
We will only focus on the second condition with p = q, in
which case the condition simplifies to: Lp ∗ Lr ⊆ Lp if 1 <
r < 2 and r < p < r/(r − 1).
Since G is semi-simple there is a compact subgroup K,
an abelian subgroup A and a nilpotent subgroup N such that
A × N × K 3 (a, n, k) 7→ ank ∈ G is a diffeomorphism and
with proper normalizations of Haar measures
Z Z Z
Z
f (x) dx =
f (ank) da dn dk.
G
K
A
N
If f is a K-right invariant function on G, then it can be
identified with a function fe on S = AN and
Z
Z
Z Z
f (x) dx =
fe(s) ds =
f (an) da dn.
G
S
A
N
If f is K-bi-invariant and g is K-right-invariant, then the
convolution of the functions ge and fe on S = AN can be
written as a convolution of the corresponding functions on G:
ge ∗ fe = g ∗ f .
From the Kunze-Stein phenomenon we can now obtain continuity of convolution operators on the amenable (subgroup)
S = AN . This is the second contribition of this paper:
Theorem 4.2: Assume that fe is a function on S = AN
and that there is a K-bi-invariant function h on G for which
|fe(s)| ≤ h(s) for all s ∈ S. If h is in Lr (G) for some r ∈
(1, 2) then ge 7→ ge ∗ fe is continuous on Lp (S) if r < p <
r/(r − 1).
Example 4.3: For the details in this example see [CGO15].
Let G = SU(n, 1) then x ∈ G can be written as a matrix
a b
x= t
c d
where a is an n × n-matrix, b, c ∈ Cn and d ∈ C. The groups
K, A and N can be parametrized in the following manner
k
0
k ∈ U(n) ' U(n)
K = uk =
0 det(k) 



cosh(t)
0
sinh(t) 

In−1
0  t ∈ R ' R
A = at =  0


sinh(t)
0
cosh(t) 

1 − |z|2 /2 + is z T
|z|2 /2 − is
n


−z
I
z
N = nz,s =
n−1
is − |z|2 /2
zT
1 + |z|2 /2 − is o
z ∈ Cn−1 , s ∈ R
The (multivalued) function
f (x) = 1/dσ ,
corresponds to a single-valued function fe when restricted to
the simply connected subgroup S. If we define h(x) = |f (x)|,
then h is K-bi-invariant and |fe(s)| ≤ h(s). Also, the function
h is in Lr (G) for r > 2n/σ, and therefore convolution by
fe is continuous on Lp (S) when 2n/σ < p < 2n/(2n − σ)
and n < σ ≤ 2n. Note, that if n = 1 this is exactly the
statement from Example 3.2, since SL(2, R) and SU(1, 1)
are isomorphic. Also, if σ > 2n, then h is in L1 (G) and
the continuity of the convolution operator is immediate. One
interesting aspect of deriving these convolution continuities
on SU(n, 1) is their relation to the Bergman projection on the
unit ball in Cn . In particular they can be used (when applied
to weighted Lp (S)-spaces) to prove the following classical
theorem [Zh05, Thm. 2.10]
Theorem 4.4: Fix two real parameters a and b and define
the integral operator S by
Z
(1 − |w|2 )b
2 a
Sf (z) = (1 − |z| )
f (w) dv(w),
n+1+a+b
Bn |1 − hz, wi|
where dv is the volume measure on the unit ball Bn in Cn and
hz, wi = z1 w1 + · · · + zn wn . Then, for t real and 1 ≤ p < ∞,
S is bounded on Lpt (Bn ) if and only if −pa < t+1 < p(b+1).
The details of the argument can be found in the paper [C15].
V. O PTIMAL SAMPLING ON PALEY-W IENER SPACES
We first examine Rd . Let T > 0 and let f (t) be a
function such that P
supp f ⊆ [0, T ]k . The T -periodization
◦
of f is [f ] (t) =
n∈Zd f (t − nT ) . We can expand a T periodic function [f ]◦ (t) in a Fourier series. Denote the lattice
Λ = TZd , where T is the n × n matrix with T on the
main diagonal and zeroes elsewhere. The sequence of Fourier
coefficients of this periodic function on the lattice Λ = TZd
are given by
1 n
d
.
[f
]◦ [n] = d fb −
T
T
We have
X
1 X b
f (n/T )e2πin·t/T .
(PSF1)
f (t + nT ) = d
T
d
d
n∈Z
n∈Z
f (nT ) =
λ∈Λ
β∈Λ
The sampling formula follows from computations and an
application of (PSF2). We assume a single band signal. Let
Λ be a regular sampling lattice in Rd ,and let Λ⊥ be the dual
cd . Then Λ has generating vectors {τ1 , τ2 , . . . , τd },
lattice in R
and the sampling lattice can be written as Λ = {λ : λ =
z1 τ1 + z2 τ2 + . . . + zd τd } for (z1 , z2 , . . . , zd ) ∈ Zd . Let
{Ω1 , Ω2 , . . . , Ωd } be the generating vectors for the dual lattice
Λ⊥ . The dual sampling lattice can be written as Λ⊥ = {λ⊥ :
λ⊥ = z1 Ω1 + z2 Ω2 + . . . + zd Ωd } for (z1 , z2 , . . . , zd ) ∈ Zd .
The vectors {Ω1 , Ω2 , . . . , Ωd } generate a parallelepiped. We
want to use this parallelepiped to create a tiling, and therefore
we make the parallelepiped “half open, half closed” as follows.
If we shift the parallelepiped so that one vertex is at the
origin, we include all of the boundaries that contain the origin,
and exclude the other boundaries. We denote this region as a
sampling parallelepiped ΩP .
If the region ΩP is a hyper-rectangle, we get the familiar
sampling formula
f (t) =
π
X
nd sin( ω1 (t − n1 ω1 ))
n1
1
· ...·
f( , . . . , )
Vol(Λ)
ω1
ωd
π(t − n1 ω1 )
d
n∈Z
If, however, the sampling parallelepiped ΩP a general parallelepiped, we first have to compute the inverse Fourier
transform of χΩP . Let S be the generalized sinc function
S=
1
(χΩP )∨ .
Vol(Λ)
Then, the sampling formula (see [HgST]) becomes
X
f (λ)S(t − λ) .
f (t) =
λ∈Λ
Definition 5.1 (Nyquist Tiles for f ∈ PWΩP ): PWΩP =
Therefore,
X
This extends to the Schwartz class of distributions as
X
X
1
\
δλ =
δβ .
(PSF2)
Vol(Λ)
⊥
1 X b
f (n/T ) .
Td
d
cd ), supp(fb) ⊂ ΩP } ,
{f continuous : f ∈ L2 (Rd ), fb ∈ L2 (R
where {Ω1 , Ω2 , . . . , Ωd } be the generating vectors for the dual
⊥
We can write the Poisson summation formula for an arbitrary lattice Λ . Let f be a non-trivial function in PWΩP . The
Nyquist
Tile
NT(f ) for f is the sampling parallelepiped of
lattice by a change of coordinates. Let A be an invertible d×d
cd centered at the origin such that supp(fb) ⊆
d
⊥
T −1 d
matrix, Λ = A Z , and Λ = (A ) Z be the dual lattice. minimal area in R
NT(f ). A Nyquist Tiling is the set of translates {NT(f )k }k∈Zd
Then
X
X
cd .
of
Nyquist tiles which tile R
−1
f (t + λ) =
(f ◦ A)(A t + n)
Definition 5.2 (Sampling Group for f ∈ PWΩP ): Let f ∈
λ∈Λ
n∈Zd
X
PW
−1
ΩP with Nyquist Tile NT(f ). The Sampling Group G is a
=
(f ◦ A)b(n)e2πin·A (t)
cd .
symmetry group of translations such that NT(f ) tiles R
n∈Zd
Remark:
Note
that
the
sampling
group
G
of
f
∈
PW
ΩP will
X
1
T −1
2πi(AT )−1 (n)·t be isomorphic to Z ⊕ Z ⊕ . . . ⊕ Z, d-times.
b
f ((A ) (n))e
.
=
| det A|
The situation is markedly different when the geometry is
n∈Zd
not
Euclidean. We look at the sphere S2 and hyperbolic space
Note, | det A| = Vol(Λ). This last expression can be expressed
D.
more directly as
To attack sampling on hyperbolic space, let dz denote the
X
X
1
2πiβ·t
b
area
measure on the unit disc D = {z | |z| < 1, and let the
f (β)e
.
f (t + λ) =
Vol(Λ)
⊥
measure
dv be given by then the SU(1, 1)-invariant measure
λ∈Λ
β∈Λ
n∈Zd
n∈Z
on D is given by dv(z) = dz/(1 − |z|2 )2 . For functions f ∈
L1 (D, dv) the Helgason-Fourier transform is defined as
Z
b
f (λ, b) =
f (z)e(−iλ+1)hz,bi dv(z)
D
for λ > 0 and b ∈ T. Here hz, bi denotes the minimal hyperbolic distance from the origin to the horocycle through z and
a point b ∈ ∂D. The mapping f 7→ fb extends to an isometry
L2 (D, dv) → L2 (R+ ×T, (2π)−1 λ tanh(λπ/2)dλ db), i.e., the
Plancherel formula becomes
Z
dz
|f (z)|2
(1
−
|z|2 )2
D Z
1
|fb(λ, b)|2 λ tanh(λπ/2)dλ db.
=
2π R+ ×T
Here Rdb denotes the normalized measure on the circle T, such
that T db = 1, and dλ is Lebesgue measure on R. The
Helgason-Fourier inversion formula is
Z Z
1
f (z) =
fb(λ, b)e(iλ+1)hz,bi λ tanh(λπ/2) dλ db .
2π R+ T
A function f ∈ L2 (D, dv) is called band-limited if its
Helgason-Fourier transform fb is supported inside a bounded
subset [0, Ω] of R+ . The collection of band-limited functions
with band-limit inside a set [0, Ω] will be denoted PWΩ =
PWΩ (D). This definition of band-limit coincides with the
definitions given in [FP04] and [CO13] which both show that
sampling is possible for band-limited functions. The Laplacian
on D is symmetric and given by
2
∂2
∂
+
,
∆ = (1 − x2 − y 2 )−2
∂x2
∂y 2
and we note that
c (λ, b) = −(λ2 + 1)fb(λ, b).
∆f
Therefore, if f ∈ PWΩ (D), we see that the following
Bernstein inequality is satisfied
k∆n f k ≤ (1 + |Ω|2 )n/2 kf k.
We can sample via operator theory in D. The sampling
operators have previously been explored in [Gr91], [Gr92].
We cite [FP04], [FP05], and [CO13]. These papers build on
Neumann series for an operator based on sampling as well
as the Bernstein inequality. According to Pesenson [Ps00]
there is a natural number N such that for any sufficiently
small r there are points xj ∈ D for which
P B(xj , r/4) are
disjoint, B(xj , r/2) cover D and 1 ≤
j χB(xj ,r) ≤ N .
Such a collection of {xj } will be called an (r, N )-lattice.
which are supported
Let φj be smooth non-negative functions
P
in B(xj , r/2) and satisfy that
φ
=
1D and define the
j j
operator


X
T f (x) = PΩ 
f (xj )φj (x) ,
j
where PΩ is the orthogonal projection from L2 (D, dv) onto
PWΩ (D). By decreasing r (and thus choosing xj closer) one
can obtain the inequality kI − T k < 1, in which case T can
be inverted by
T −1 f =
∞
X
(I − T )k f.
k=0
For given samples we can calculate T f and the Neumann
series provides the recursion formula
fn+1 = fn + T f − T fn
and then limn→∞ fn = f with norm convergence. The rate
of convergence is determined by the estimate kfn − f k ≤
kI − T kn+1 kf k.
The paper [FP05] further provides a necessary condition
for the set {xi } to be a sampling set. They find that there
is a constant C which is determined by the geometry of
D, such that if r < C −1 (1 + |Ω|2 )k/2 )−1 for any k > 1,
then any (N, r)-lattice {xi } is a sampling set. The paper
[CO13] obtains similar results, but removes some restrictions
on the functions φj . In particular the partitions of unity
do not need to be smooth and can actually be chosen as
characteristic functions φj = χUj for a cover of disjoint
sets Uj contained in the balls B(xj , r/2). This is done by
lifting the functions to the group of isometries (which in this
case is SU(1, 1)), and by estimating local oscillations using
Sobolev norms for left-invariant vector fields on this group.
These sampling schemes are not optimal. The questions of
establishing Beurling-Landau upper and lower densities and
then constructing optimal sampling sets in terms of these is
open.
The sphere is compact, and its study requires different
tools. Fourier analysis on S2 amounts to the decomposition of
L2 (S2 ) into minimal subspaces invariant under all rotations
in SO(3). Band-limited functions on the sphere are spherical
polynomials. Sampling on the sphere is how to sample a bandlimited function, an N th degree spherical polynomial, at a
finite number of locations, such that all of the information
content of the continuous function is captured. Since the
frequency domain of a function on the sphere is discrete,
the spherical harmonic coefficients describe the continuous
function exactly. A sampling theorem thus describes how
to exactly recover the spherical harmonic coefficients of the
continuous function from its samples. Developing sampling
lattices leads to questions on how to efficiently tile the sphere,
a subject in its own right. We refer to the work of Driscoll and
Healy [DH94], Keiner, Kunis, and Potts [KKP07], McEwen
and Wiaux [MW11], and Pesenson [Ps00], [Ps04], [Ps09] for
results on the sphere. In particular, the results of Pesenson
give a potential roadmap for designing sampling schemes
using the operator theoretic approach that are essentially
optimal, at least in the case of compact manifolds. Given a
compact Riemannian manifold M , the number of sampling
points in a sampling set constructed according to the operator
theoretic sampling scheme for the space PWΩ is comparable
to Vol(M )Ωd/2 , where Vol(M ) is the volume and d is its
dimension. By Weyls formula, the dimension of the space
PWΩ is given asymptotically (when Ω −→ ∞) by the quantity
Vol(M )Ωd/2 times an absolute constant. Thus one can say
that the rate of sampling given in [Ps00] is essentially optimal
(dimension of a subspace is comparable to the number of
sampling points) at least in the case of compact manifolds.
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