Wavelet frames, Bergman spaces and Fourier transforms of
Laguerre functions
Lúıs Daniel Abreu
Abstract. The Fourier transforms of Laguerre functions play the same canon-
ical role in wavelet analysis as do the Hermite functions in Gabor analysis. We
will use them as analyzing wavelets in a similar way the Hermite functions were
recently by Gröchenig and Lyubarskii in Gabor frames with Hermite functions,
C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007). Building on Seip´s work
Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39
(1993), concerning sampling sequences on weighted Bergman spaces, we find
a sufficient density condition for constructing frames by translations and dila-
tions of the Fourier transform of the nth Laguerre function. As in Gröchenig-
Lyubarskii theorem, the density increases with n, and in the special case of
the hyperbolic lattice in the upper half plane it is given by
b log a <
2n+ α
where α is the parameter of the Laguerre function.
1. Introduction
The ”frame density problem” is one of the fundamental questions in applied
harmonic analysis: how coarse we need to sample a continuous object so that the
resulting discrete object is a frame in a certain Hilbert space? This question was
sharply solved for frames in the Bargmann-Fock [8], [24], [25] and in the Bergmann
space [28]. However, in the cases of general wavelet and Gabor transforms, very
little is known and only a few very special windows and analysing wavelets are
understood.
The short time Fourier (Gabor) transform with respect to a gaussian window
can be written in terms of the Bargmann transform, mapping isometrically the
space L2(R) onto the Bargmann-Fock space of entire functions. This is the reason
why everything is known about the geometry of sequences that generate frames by
sampling the Gabor transform with gaussian windows g(t) = e−πt
. Apart from
this example, the only cases where a description is known of the lattice sequences
that generate frames are the hyperbolic secant g(t) = (coshat)−1 [18] and the
characteristic function of an interval [19], which turned out to be a nontrivial
problem. There is also a necessary density condition for Gabor frames due to
Ramanathan and Steger [23] and it was recently observed that the frame property
is stable under small perturbations of the lattice, given the window belongs to the
Feichtinger algebra [10].
http://arxiv.org/abs/0704.1487v1
2 LUÍS DANIEL ABREU
If we consider the wavelet case, things appear to be even more mysterious. Al-
though the density and irregular grid problems have attracted some attention lately
(see [1], [31], [22], [15] and references there in), there is no known counterpart of
Ramanathan and Steeger theorem (there is, however, a version of the HAP property
for wavelet systems [16]) and the only information available so far, on particular
systems, concerns a special family of analyzing wavelets that maps the problem into
Bergmann spaces: The wavelet transform, with positive dilation parameter, with
respect to a wavelet of the form (Paul´s Wavelet in some literature)
(1.1) ψα(t) =
can be rescaled as an isometrical integral transform between spaces of analytical
functions, namely, between the Hardy space on the upper half plane H2(U) and
the Bergman space in the upper half plane A2α+1(U). It is natural to refer to
this isomorphism as the Bergman transform (the designation analytic wavelet is
also frequently used). Since the upper half plane can be mapped isomorphically
onto the unit disc by using linear fractional transformations, we can construct a
transform mapping the Hardy space onto the Bergman space in the unit disc.
The approaches used to deal with the special situations mentioned in the above
paragraphs are based in techniques which differ from case to case. It seems highly
desirable to follow a more structured approach. The natural place to look for this
structure is within the context of Hilbert spaces of analytic functions, where the
powerful methods from complex analysis may answer questions that seem hopeless
otherwise. Following this line of reasoning implies using windows that allow to carry
the problem to such spaces. In this direction a major step was taken recently by
Göchenig and Lyubarskii [12], by considering Gabor systems with Hermite func-
tions of order n as windows. They have proved that if the size of the lattice Λ is
< (n + 1)−1 then the referred Gabor system is a frame and provided an example
supporting their conjecture that the result is sharp.
In the Bargmann-Fock setting the Hermite functions play a very special role
[11, pag. 57]. They are, up to normalization constants, pre-images, under the
Bargmann transform, of the monomials {zn} and since the latter constitute an
orthogonal basis of the Bargmann-Fock space, their pre-images also constitute an
orthogonal basis of L2(R). That is, to the isomorphism
(1.2) L2(R)
B→ F 2(C)
corresponds
(1.3) hn
B→ cnzn
where B is the Bargmann transform, F 2(C) is the Bargmann-Fock space, hn are the
Hermite functions and cn some constants dependent on n. They are canonical to
time-frequency analysis in an additional sense, since they constitute the eigenfunc-
tions of the time-frequency-localization operator with Gaussian window [3]. The
Hermite functions are eigenfunctions of the Fourier transform and they can be used
[20] to describe Feichtinger´s algebra So.
Wavelet and Gabor analysis share many similarities and many of their struc-
tural aspects can be bound together in a more general theory using representations
of locally compact abelian groups [13], [9]. In looking for a Wavelet-analogue of
Gröchenig-Lyubarskii structured approach to the density problem, we must first
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 3
clarify what functions should be used instead of the Hermite functions. In analogy
to the above paragraph it is natural to consider the pre-images, under the Bergman
transform, of the monomials {zn} in the unit disc (up to an isomorphism with the
upper half plane). Since the {zn} form a basis of the weighted Bergman spaces in
the unit disc (independently of the weight), their pre-images must be an orthogonal
basis of the Hardy space H2(U) and it is reasonable to expect that such functions
will play a similar role in Wavelet analysis as do the Hermite functions in Gabor
analysis. Such functions are the Fourier transforms of the Laguerre functions and
they will be ad-hoc denoted by Sαn .
Additional structural evidence that the functions Sαn are canonical in Wavelet
analysis comes from the work of Daubechies and Paul [4], where it is shown that
they are the eigenfunctions of a differential operator that commutes with a time–
scale localization operator, once windows of the form (1.1) are chosen. This com-
pletely parallels the situation with the time-frequency-localization operators with
gaussian windows, which commute with the harmonic oscillator and therefore have
as eigenfunctions the Hermite functions [3]. It was observed by Seip [26] that these
problems have a more natural formulation when mapped into the convenient spaces
of analytical functions.
2. Description of the results
There exists an analogue of the correspondence between (1.2) and (1.3), but
involving four different functional spaces and the corresponding bases: To the se-
quence of isomorphisms between the Hilbert spaces
(2.1) L2(0,∞) F←→ H2(U) Ber
→ A2α+1(U)
Tα→ A2α+1(D),
where A2α+1(U) and A2α+1(D) are the weighted Bergmann spaces in the upper-
half plane and in the unit disc, respectively, corresponds the relations between the
basis of the respective spaces:
(2.2) lαn
F←→ Sαn
Berα→ cαnΨαn
Tα→ cαnzn.
for some constants cαn, where l
n is the nth Laguerre function of order n and pa-
rameter α, Sαn is its Fourier transform and Ψ
n is a basis of A2α+1(U) to be defined
in section 4. This correspondence was implicit in the conection between papers [4]
and [26] but since it was not stated explicitly, we devote section 4 to clarify how
exactly does it work.
The functions Sαn were computed recently in closed form [29], but the connec-
tion to the Wavelet transform seems to have been unnoticed. It is also shown in
[29] that Sαn are, up to a fractional transformation, defined in terms of a certain
system of orthogonal polynomials on the unit circle. We will show that this family
of polynomials is nothing more but the circular Jacobi orthogonal polynomials for
which there is, for computational purposes, a very convenient three term recurrence
formula. This connection will make our case to call the functions Sαn the rational
Jacobi orthogonal functions.
With a computational method available for evaluating the functions Sαn and
graphic evidence of their good localization properties (see for example the plots of
their real versions in page 44 of [2]), it is natural to investigate how to use such func-
tions as analysing wavelets and to obtain frames from the resulting discretization.
4 LUÍS DANIEL ABREU
We will obtain the following sufficient condition on the density of the parameters
of the hyperbolic lattice {(ajbk, aj)}j,k∈Z:
b log a <
2n+ α
that is, as in Gröchenig and Lyubarskii result [12], this density increases with n.
This will follow as a special case of the following more general theorem, which uses
Seip´s notion of lower Beurling density in the unit disk and constitutes the main
result in this paper:
Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping
the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform.
If D−(Γ) > n+ α
then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H
2(U).
Case α = 0 gives the sufficient part of theorem 7.1 in [28]. The proof of theorem
1 will be dramatically simplified with the observation of the following remarkable
ocurrence: There exist constants {C} and sequences of numbers {ak}, both de-
pending on α and n, such that
(2.3) Sαn (
) = C
akψk+α
This will allow to obtain a formula which expresses the wavelet transform with
window Sαn as a linear combination of simple but non-analytic functions. As a
result we will have to deal with a situation that is reminiscent of the non-analiticity
problem that Gröchenig and Lyubarskii faced with formula (15) of [12] and which
they solved by using the so called Wexler-Raz identities [11]. Since there is no
known analogue of the Wexler-Raz identities in wavelet analysis, we have to deal
with this in a more direct way. The key idea is to map the problem into the unit
circle and use a deep sampling formula proved by Seip in [28]. Althought this
approach lacks the simplicity of [12] it gives the result for more general sequences,
while the method in [12] works only for the lattice case due to the restritions
imposed by the use of the Wexler-Raz identities.
We organize our ideas as follows. The next section contains the main definitions
and facts concerning wavelet transforms, Bergman spaces and Laguerre functions.
Section four contains the evaluation of the pre-images required to build up the
correspondences (2.1) and (2.2). The fifth section contains our main results on
wavelet frames with Fourier transforms of Laguerre functions. We conclude the
paper collecting some further properties of the functions Sαn and clarifying their
classification within known families of orthogonal polynomials.
3. Tools
3.1. The Bergman transform. Now we present a sinthesis of ideas that
appeared in the section 3.2 of [13] and in [4] (see also [2, pag. 31]) Here they will
be exposed in such a way that the role of the Bergman spaces is emphasized.
Consider the dilation and translation operators
Dsf(x) = |s|−
2 f(s−1x)
Txf(t) = f(t− x)
and define
ψx,s(t) = TxDsψ(t) = |s|−
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 5
The wavelet transform of a function f , with respect to the wavelet ψ is
Wψf(x, s) = 〈f, TxDsψ(t)〉L2(R) =
f(t)ψx,s(t)dt.
A function ψ ∈ L2(R) is said to be admissible if
|Fψ(s)|2 ds
where K is a constant. If ψ is admissible, then for all f ∈ L2(R) we have
(3.1)
s−2 |Wψf(x, s)|2 dxds = K ‖f‖2
We will restrict ourselves to parameters s > 0 and functions f ∈ H2(U), where
H2(U) is the Hardy space in the upper half plane U = {z = x+ is : s > 0}
H2(U) = {f : f is analytic in U and sup
0<s<∞
|f(x+ is)|2 dx <∞}.
Let F denote the Fourier transform
(Ff)(t) = 1√
e−itxf(x)dx
By the Paley-Wiener theorem, H2(U) is constituted by the functions whose Fourier
transform is supported in (0,+∞) and belongs to L2(0,∞).
Now take the special window ψα(t) defined in (1.1). Since
(3.2) Fψ−x,sα (t) = 1[0,∞]sα+
2 tαei(x+is)t
(3.3) Wψαf(−x, s) = s
tαeizt(Ff)(t)dt
where the function defined by the integral is analytic in z = x + si. The identity
(3.1) gives
(3.4)
∣Wψαf(−x, s)
s−2dxds = ‖f‖H2(R)
This motivates the definition of the Bergman transform, or the analytic wavelet
transform:
(3.5) Berα f(z) =
tαeizt(Ff)(t)dt = s−α− 12Wψαf(−x, s),
where z = x+ is (see for instance [15], where the authors use this Bergman trans-
form with the same normalization in the case α = 1). Introducing the scale of
weighted Bergman spaces
Aα(U) = {f analytic in U such that
|f(z)|2 sα−2dxds <∞},
it is clear from (3.4) and (3.5) that Berα f(z) ∈ A2α+1(U). We have therefore an
isometric transformation
Berα : H2(R)→ A2α+1(U)
6 LUÍS DANIEL ABREU
The weighted Bergman spaces in the unit disc are denoted by Aα(D) and defined
Aα(D) =
f analytic in D such that
|f(z)|2 (1− |z|)α−2dxdy <∞
For detailed expositions of the theory of Bergman spaces we point out the mono-
graphs [32], [6], [14]
3.2. Frames and sampling sequences. A sequence of functions {ej} is said
to be a frame in a Hilbert space H if there exist constants A and B such that
(3.6) A ‖f‖2 ≤
|〈f, ej〉|2 ≤ B ‖f‖2 .
In spaces of analytic functions a related concept to frames is the one of a
sampling sequence. A set Γ = {zj} is said to be a sampling sequence for the
Bergman space Aα(U) if there exist positive constants A and B such that
(3.7) A
|f(z)|2 yα−2dxds ≤
|f(zj)|2 yαj ≤ B
|f(z)|2 yα−2dxds.
The corresponding definition of a sampling sequence for the Bergman space on
the unit disc is
|f(z)|2 (1− |z|2)α−2dz ≤
|f(zj)|2 (1− |z|2)α
|f(z)|2 (1− |z|2)α−2dz.
Define the pseudohyperbolic metric on the unit disk by
̺(z, ζ) =
z − ζ
1− ζz
Following [28], a sequence Γ = {zj} ⊂ D is separated if infn6=j
zj−zn
zj−zn
> 0 and its
lower density D− is given by
D−(Γ) = lim
inf inf
̺(zj ,z)<r
(1− ̺(zj , z))
log 1
The next Theorem is from [28]:
Theorem A A separated sequence Γ ⊂ D is a sampling sequence for Aα(D)
iff D−(Γ) > α
It is a generalization of an earlier lattice result [27]:
Theorem B Let Γ(a, b) = {zjk}j,k∈Z, where zjk = aj(bk + i). Γ(a, b) is a
sampling sequence for Aα(U) if and only if b log a <
3.3. Fourier transforms of Laguerre functions. The Laguerre polynomi-
als will play a central role in our discussion. One way to define them is by means
of the Rodrigues formula
(3.8) Lαn(x) =
exx−α
e−xxα+n
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 7
and this gives, in power series form
Lαn(x) =
(α + 1)n
(−n)k
(α+ 1)k
For information on specific systems of orthogonal polynomials, we suggest [17]. The
Laguerre functions are defined as
lαn(x) = 1[0,∞](x)e
−x/2xα/2Lαn(x)
and they are known to constitute an orthogonal basis for the space L2(0,∞). By the
Paley-Wiener theorem, the Fourier transform is an isomorphism between H2(U)
and L2(0,∞). Therefore, the Fourier transform of the functions lαn form an orthog-
onal basis for the space H2(U). From now on we will set
(3.9) FSαn (t) = lαn(t)
The functions Sαn (t) can be evaluated explictly using (3.8). This was already done
by Shen in [29], where the author was interested in describing the orthogonal poly-
nomials which arise from an application of the Fourier transform on the Laguerre
polynomials. A description of this method of generating new families of orthogonal
polynomials with an interesting historical account is in the paper [21] where a link
is provided between Jacobi and Meixner-Polaczek polynomials. Following [29], we
(3.10) Sαn (t) =
(1 + α)n
(−n)k(α2 + 1)k
k!(α+ 1)k
4. Bergman transform of Sαn (t)
In this section the correspondences (2.1) and (2.2) will be established via a
direct calculation. The complex linear fractional transformations will play an im-
portant role, in a style that is reminiscent of the way they are used in the discrete
series representation of SL(2,R) over the Bergman space [7, chapter IX].
We first define a set of functions that, as we shall see later, constitute a basis
of Aα(U). For every n ≥ 0 and α > −1 let
Ψαn(z) =
iz + 1
iz − 1
)−α−1
Now define a map Tα such that for every function f ∈ Aα(U) the action of Tα
(4.1) Tαf(w) = f
w + 1
w − 1
The range space of Tα is a weighted Bergman space in the unit disc.
Lemma 4.1 The map
Tα : Aα(U)→ Aα(D)
is an unitary isometry between Hilbert spaces.
Proof. Argue as in the proof of lemma 1 in [7, pag.185]. �
Proposition 4.1 For n = 0, 1, ..., the following relations hold:
(4.2) Berα(S2αn ) = c
n , with c
n = (−1)α+1(2α+ n)!/n!
8 LUÍS DANIEL ABREU
and, for |z| < 1,
(4.3) Tα(Ψ
n) = z
In other words, the function S2αn is the pre-image, under Tα ◦
, of zn ∈
Aα(D).
Proof. Using the definition of the Bergman transform (3.5) and (3.9) we have
Berα S2αn (z) =
tαeizt(FS2αn (t))(t)dt
e(iz−
)tt2αL2αn (t)dt
e−xx2α+n
(−iz + 1
= cαnΨ
n (z),
where in the last two identities we are applying Rodrigues formula (3.8) for the
Laguerre polynomials and writing the integral in terms of the Laplace transform L,
whose well known properties establish the last identity. Now, the linear fractional
transformation
iz + 1
iz − 1
is an analytic isomorphism between the upper half plane and the unit circle. Since
the inverse of this transformation is given by
w + 1
w − 1
a short calculation with the definition of Tα gives(4.3). �
Remark 1. {Ψαn(z)} is a basis of Aα(U) and the map Tα is an unitary isomorphism
between Aα(U) and Aα(D). Indeed, since {zn} is an orthogonal basis of the space
Aα(D) [7, pag. 186] and it is contained in the range of Tα, Tα is onto and therefore
an unitary isomorphism.The functions {Ψαn(z)} form a basis for the space Aα(U)
since they are the pre-images of the basis {zn}.
As a consequence we obtain a new proof of the (known) isomorphic property
of the Bergman transform.
Corollary 1. The transform Berα : H2(R)→ A2α+1(U) is an isometric isomor-
phism.
Proof. The isometry is a consequence of the isometric property of the wavelet
transform, so we need only to prove that the Ber
2 is onto. But in view of the
preceding remark and Theorem 5.1, the range of Ber
2 contains a basis of Aα(U).
Therefore, Ber
2 is onto. �
Remark 2. Similar calculations as we have seen here also play a role in [5], in the
context of Laplace transformations and group representations and in [4], to obtain
an explicit formula for the eigenvalues of the time-scale localization operator.
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 9
5. Wavelet frames with Fourier transforms of Laguerre functions
5.1. Preparation. We wish to construct wavelet frames with analysing wavelets
Sαn , by using the common discretization of the continuous wavelet transform via
the hyperbolic lattice. Discretizing the scale parameter s in the Wavelet transform
by a sequence aj and the parameter x by ajbm gives
(5.1) Wψf(a
jbm, aj) = 〈f, TajbmDajψ〉 .
We want to know conditions in a and b under which {TajbkDajψ} is a frame, for a
given wavelet ψ. More generally we can consider the non lattice problem replacing
(ajbm, aj) by more sequences (ajm, bjm).
Remark 3. Observe that it follows immeadiately from Theorem A and Theorem B
that if Γ ⊂ D is the sequence obtained from mapping the sequence {zk,j = aj(bk +
i)} ⊂ U into the unit disk via a Cayley transform, then D−(Γ) = 2π
b log a
(to move
between the notation of [28] and [27] set α = 2n+ 1).
Remark 4. Observe that {TajbmDajψα}j,k∈Z is a frame of H2(U) iff Γ(a, b) =
{z = ajbm+ aji}j,k∈Z is a set of sampling for A2α+1(U). Indeed, since functions
in A2α+1(U) can be identified with Ber
α transforms of H2(U) functions, it follows
from (3.7) and (??) that Γ(a, b) is a set of sampling for A2α+1(U) if and only if
∣Wψαf(x, s)
2 dxds
∣Wψαf(a
jbm, aj)
2 ≤ B
∣Wψαf(x, s)
2 dxds
Using (3.4) this is equivalent to
A ‖f‖2H2(U) ≤
|〈f, TajbmDajψα〉|
2 ≤ B ‖f‖2H2(U)
which says that {TajbmDajψα}j,k is a frame in H2(U).
We now collect the required preliminary lemmas for the proof of the main
result. The first was quoted in the introduction and is just a simple modification
of the representation (3.10). The second appeared in [27] (see [6, pag. 160] for a
proof) and the third one is a quite deep sampling theorem which is formula (30) in
[28] with s = 0 and ǫ = 0 (see also [6, pag. 216] for a direct derivation with this
choice of parameters).
Lemma 5.2 The functions Sαn can written as the linear combination (2.3) of
analyzing wavelets ψk+α
(t) defined by
Sαn (
) = C
akψk+α
where the constants C and the coeficients ak given as
C = Cα,n =
(1 + α)n
ak = a
k = (2i)
(−n)k(α2 + 1)k
k!(α+ 1)k
Lemma 5.3 For 1 < s < t we have
(5.2)
(1− |zj |2)s
∣1− ζzj
≤ C(1− |ζ|2)s−t
10 LUÍS DANIEL ABREU
for any separated sequence {zj}.
Lemma 5.4 Let D({wj}) > β. Then every f ∈ A2β+1(D) satisfies
(5.3) (1− |w|2)βf(w) =
(1 − |wj |2)βf(wj)hj(w),
with the estimate for hj(w)
(5.4) |hj(w)| ≤ C
(1− |w|2)(1 − |wj |2)
|1− wjw|2
5.2. Main result. To prove our main result we will adhere to the following
plan: we begin by deriving from Lemma 5.2 a formula expressing the wavelet trans-
form with analysing wavelet Sαn in terms of the wavelet transform with analysing
wavelet ψk+α
. The proof of the right frame inequality is then straightforward (as
usual in frame inequalities). To prove the left hand inequality we first rewrite the
formula in terms of complex functions defined on the upper half plane and then map
it into the unit disk by means of a Cayley transform. Manipulating the resulting
formula and using lemma 5.3 and lemma 5.4 gives an estimate in the unit disk. We
proceed backwards to the half plane to end up with the required estimate in terms
of wavelet frames.
Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping
the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform.
If D−(Γ) > n+ α
then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H
2(U).
Proof. The definition of the wavelet transform gives, using (2.3),
)f(−x, s) =
f, T−xDsS
f, T−xDsψk+α
akWψk+α
f(−x, s).
The right frame inequality is now easy: observe that, since {Tam,jDbm,jψk+α
is a frame, then for every k = 1, ..., n there exists a Bk such that
f, Tam,jDbm,jψk+α
≤ Bk ‖f‖2H2(U) .
Let Bn = max1≤k≤nBk. Then,
f, Tam,jDbm,jS
f, Tam,jDbm,jψk+α
|ak|2
f, Tam,jDbm,jψk+α
≤ B ‖f‖2 ,
where B = BnC
k=0 |ak|
and we are done.
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 11
Now the left frame inequality. Define the function F (z) as
(5.5) F (z) =WSα
)f(−x, s) = C
2Berk+
2 f(z)
where z = x+is belongs to the upper half plane U. We construct a related function
in the unit disc, by setting w = z−i
G(w) = C
ak(1− |w|2)k+
2Gk+α
where
G(w) = F (i
1 + w
(w) =
2k+α+1
Berk+
2 f(i
1 + w
) ∈ A2k+α+1(D).
Therefore, if D({wj}) > n+ α2 we can use (5.3) to write, for every k ≤ n,
(1 − |w|2)k+
2 Gk+α
(w) =
(1 − |wj |2)k+
2 Gk+α
(wj)hj(w).
As a result,
(1 − |w|2)− 12G(w) = Cα,n
ak(1− |w|2)k+
2 Gk+α
(1− |wj |2)k+
2 Gk+α
(wj)hj(w)
hj(w)C
ak(1− |wj |2)k+
2 Gk+α
hj(w)(1 − |wj |2)−
2G(wj).
Now use estimate (5.4), then Cauchy-Schwarz and finally (5.2) in the following way.
(1− |w|2)−1 |G(w)|2 ≤
(1 − |w|2)
|1− wjw|2
(1− |wj |2)
2G(wj)
(1 − |w|2)2 |G(wj)|2
(1− |wj |2)1
|1− wjw|4
(1 − |w|2)−1 |G(wj)|2 .
Thus the factor (1− |w|2)−1 cancels and we have
|G(w)|2 ≤ C
|G(wj)|2 .
12 LUÍS DANIEL ABREU
This last inequality gives at once
|G(w)|2 (1− |w|2)−2dw ≤ C
|G(wj)|2
or, in the upper half plane,
|F (z)|2 s−2dxds ≤
|F (zj)|2 .
In the wavelet notation this is
)f(−x, s)
s−2dxds ≤
jbm, aj)
Taking into account that Sαn (
) is admissible, we can apply (3.1) and (5.1) to write
the above inequality as
(5.6) A ‖f‖2 ≤
f, Tam,jDbm,jS
This is the left frame inequality. The sequence {Tam,jDbm,jSαn ( t2 )}j,m is thus a
frame. �
In view of remark 5.1, the lattice result is a special case of our main theorem.
Corollary 2. If b log a < 4π
, then {TajbmDajSαn ( t2 )}j,m is a frame of H
2(U).
Remark 5. Theorem 5.1 parallels theorem 3.1 in [12], which states that, in the
lattice case of the time-frequency plane, if the density of the lattice Λ is > n + 1
(or if the size of Λ is < (n + 1)−1) then the Gabor system {e2πiλ2tHn(t − λ1) :
λ = (λ1, λ2) ∈ Λ}, where Hn stands for the Hermite function of order n, is a
frame for L2(R). In particular, if one is dealing with the Von Neumann lattice
with parameters a and b, the condition is ab < (n+1)−1. Corollary 5.2 makes this
analogy even more explicit.
Remark 6. It is interesting to notice that, since Berα f(z) =
tαeizt(Ff)(t)dt,
we clearly have
ikBerk+
2 f(z) =
2 f(z).
This allows to rewrite (5.5) as
)f(−x, s) = C
2 f(z)
which is, in some sense, reminiscent of Proposition 3.2 in [12].
Remark 7. Observe that combining the Paley-Wiener with the Plancherel theorem,
we have ‖f‖H2(U) = ‖f‖L2(0,∞) and the results of Theorem 1 and 2 say also that
we have a frame for L2(0,∞).
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 13
6. Further properties of the functions Sαn (t)
The notation
F (a, b; c;x) =
(a)k(b)k
k!(c)k
for the hypergeometric function is used in this section. Rewriting Sαn in this notation
gives
Sαn (t) = C
F (−n, α
+ 1;α+ 1;
(observe that the infinite sum becomes a polynomial of order n, since (−n)k = 0 if
k > 0). Composing the functions Sαn (t) with the fractional linear transformation
(6.1) z =
2t− i
2t+ i
the result is
(6.2) Sαn (t) = Γ
(1 − z)α2 +1gαn(z)
where
gαn(z) =
(α/2)n
F (−n, α
+ 1;−n− α/2 + 1; z)
is a polynomial in z of degree n. This was pointed out in [29]. It was also shown
that these polynomials satisfy the orthogonality
(6.3)
|z|=1
gαn(z)g
n(z) |1− z|
= 0 if m 6= n
and therefore are orthogonal on the unit circle with respect to the weight
(6.4) w(z) = sinα
This fact implies many properties, since there exists a very rich theory for orthogo-
nal polynomials on the unit circle (see [30] and references therein and also chapter
8 of [17]). For example, the general theory assures that all the zeros of gαn(z) lay
within the unit disc.
Remark 8. Setting a = α
in Example 8.2.5 at [17], and using the identity 1−eiθ =
4 sin2 θ
to write the measure (8.2.21) as (6.4) we recognize that the polynomials
gαn(z) are, up to a normalization, a family of orthogonal polynomials on the unit
circle known as the circular Jacobi orthogonal polynomials.
Remark 9. From (6.3) the functions z−
2 (1−z)α2 gαn(z) are orthogonal on the circle
and form a basis of the Hardy space on the unit disc
H2(D) = {f : f is analytic in D and sup
∣f(reit)
dt <∞}
it is also clear that Sαn (t) are orthogonal on the real line (the boundary of the upper
half place). Since gαn(z) are the circular Jacobi orthogonal polynomials, the basis
functions z−
2 (1 − z)α2 gαn(z) are the circular Jacobi orthogonal functions and it is
therefore natural to call Sαn (t) the rational Jacobi orthogonal functions.
14 LUÍS DANIEL ABREU
Remark 10. From the general theory [17, (8.2.10)] follows that, if κn is the leading
coefficient of the polynomial, then the sequence of polynomials {gαn(z)} satisfies a
three term recurrence relation
n(0)g
n+1(z) + κn−1g
n+1(0)zg
n−1(z) = [κng
n+1(0) + κn+1g
n(0)z]g
where κn is the leading coefficient of the polynomial. From the explicit representa-
tion of the polynomials gαn(z) it is easily seen that
+ 1)n
; φn(0) =
+ 1)n−1
This three term recurrence relation provides a effective method for computational
purposes: To evaluate the functions Sαn (t) it is sufficient to combine this recurrence
relation with formulas (6.1) and (6.2).
References
[1] A. Aldroubi, C. Cabrelli, U. M., Molter Wavelets on irregular grids with arbitrary dilation
matrices and frame atoms for L2(Rd), Appl. Comput. Harmon. Anal. 17, no. 2, 119–140
(2004).
[2] I. Daubechies, ”Ten lectures on wavelets”, CBMS-NSF Regional conference series in applied
mathematics (1992).
[3] I. Daubechies, Time-frequency localization operators: A geometric phase space approach
IEEE Trans. Inform. Theory, 34, pp. 605-612. (1988).
[4] I. Daubechies, T. Paul, Time-frequency localization operators: A geometric phase space ap-
proach II. The use of dilations and translations. Inverse Prob., pp 661-680. (1988).
[5] M. Davidson, G. Ólafsson, G. Zhang, Laguerre polynomials, restriction principle, and holo-
morphic representations of SL(2;R). Acta Appl. Math. 71 , no. 3, 261–277 (2002).
[6] P. Duren, A. Schuster, ”Bergman spaces”, Mathematical Surveys and Monographs, 100.
American Mathematical Society, Providence, RI, (2004).
[7] S. Lang, ”SL(2,R)”, Springer (1985).
[8] Y. Lyubarskii, Frames in the Bargmann space of entire functions, Entire and subharmonic
functions, 167-180, Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI (1992)
[9] H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations
and their atomic decompositions, I, J. Funct. Anal. 86 (2), 307-340 (1989).
[10] H. G. Feichtinger, N. Kaiblinger, Varying the time-frequency lattice of Gabor frames. Trans.
Amer. Math. Soc. 356 (2004), no. 5, 2001–2023
[11] K. Gröchenig, ”Foundations of time-frequency analysis”, Birkhäuser, Boston, (2001)
[12] K. Gröchenig, Y. Lyubarskii, Gabor frames with Hermite functions, C. R. Acad. Sci. Paris,
Ser. I 344 157-162 (2007).
[13] A. Grossman, J. Morlet, T. Paul, Transforms associated to square integrable group represen-
tations II: examples, Ann. Inst. H. Poincaré Phys. Théor., 45, 293-309, (1986).
[14] H. Hedenmalm, B. Korenblum, K. Zhu, ”Theory of Bergman spaces.” Graduate Texts in
Mathematics, 199. Springer-Verlag, New York, (2000).
[15] C. Heil, G. Kutyniok, Density of wavelet frames, J. Geom. Anal. 13, no. 3, 480-493. (2003).
[16] C. Heil, G. Kutyniok, The Homogeneous Approximation Property for wavelet frames , J.
Approx. Theory., online first.
[17] M. E. H. Ismail, ”Classical and quantum orthogonal polynomials in one variable”, Encyclo-
pedia of Mathematics and its Applications No. 98. (2005).
[18] A. J. E. M. Janssen; T. Strohmer, Hyperbolic secants yield Gabor frames. Appl. Comput.
Harmon. Anal. 12 , no. 2, 259–267 (2002).
[19] A. J. E. M. Janssen, Zak transforms with few zeros and the tie, in ”Advances in Gabor
Analysis” (H.G. Feichtinger, T.Strohmer, eds.), Boston, 2003, pp. 31-70.
[20] A. J. E. M. Janssen, Hermite function description of Feichtinger´s Space S0, J. Fourier Anal.
Appl., 11 (5) 577-588 (2005).
[21] H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124,
no. 3, 887-898 (1996).
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 15
[22] G. Kutyniok, Affine density, frame bounds, and the admissibility condition for wavelet
frames. Constr. Approx., 25, no. 3, 239-253 , (2006)
[23] J. Ramanathan, T. Steger, Incompleteness of sparse coherent states. Appl. Comput. Harmon.
Anal. 2 , no. 2, 148–153 (1995).
[24] K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space I, J.
Reine Angew. Math. 429, 91-106 (1992).
[25] K. Seip, R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock
space II, J. Reine Angew. Math. 429 (1992), 107-113.
[26] K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces,
SIAM J. Math. Anal. 22, No. 3 pp. 856-876 (1991).
[27] K. Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer.
Math. Soc. 117, no. 1, 213-220 (1993).
[28] K. Seip, Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993).
[29] L. C. Shen, Orthogonal polynomials on the unit circle associated with the Laguerre polyno-
mials, Proc. Amer. Math. Soc. 129, no. 3, 873-879 (2001).
[30] B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42 (2005), 431-460.
[31] W. Sun, X. Zhou, Density and stability of wavelet frames, Appl. Comput. Harmon. Anal. 15,
no. 2, 117–133 (2003).
[32] K. H. Zhu, ”Operator theory in function spaces”. Monographs and Textbooks in Pure and
Applied Mathematics, 139. Marcel Dekker, Inc., New York, 1990.
	1. Introduction
	2. Description of the results
	3. Tools
	4. Bergman transform of Sn0=x"010B(t)
	5. Wavelet frames with Fourier transforms of Laguerre functions
	6. Further properties of the functions Sn0=x"010B(t)
	References

  The Fourier transforms of Laguerre functions play the same canonical role in
wavelet analysis as do the Hermite functions in Gabor analysis. We will use
them as analyzing wavelets in a similar way the Hermite functions were recently
by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C.
R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K.
Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113,
21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we
find a sufficient density condition for constructing frames by translations and
dilations of the Fourier transform of the nth Laguerre function. As in
Groechenig-Lyubarskii theorem, the density increases with n, and in the special
case of the hyperbolic lattice in the upper half plane it is given by b\log
a<\frac{4\pi}{2n+\alpha}, where alpha is the parameter of the Laguerre
function.

Introduction
The ”frame density problem” is one of the fundamental questions in applied
harmonic analysis: how coarse we need to sample a continuous object so that the
resulting discrete object is a frame in a certain Hilbert space? This question was
sharply solved for frames in the Bargmann-Fock [8], [24], [25] and in the Bergmann
space [28]. However, in the cases of general wavelet and Gabor transforms, very
little is known and only a few very special windows and analysing wavelets are
understood.
The short time Fourier (Gabor) transform with respect to a gaussian window
can be written in terms of the Bargmann transform, mapping isometrically the
space L2(R) onto the Bargmann-Fock space of entire functions. This is the reason
why everything is known about the geometry of sequences that generate frames by
sampling the Gabor transform with gaussian windows g(t) = e−πt
. Apart from
this example, the only cases where a description is known of the lattice sequences
that generate frames are the hyperbolic secant g(t) = (coshat)−1 [18] and the
characteristic function of an interval [19], which turned out to be a nontrivial
problem. There is also a necessary density condition for Gabor frames due to
Ramanathan and Steger [23] and it was recently observed that the frame property
is stable under small perturbations of the lattice, given the window belongs to the
Feichtinger algebra [10].
http://arxiv.org/abs/0704.1487v1
2 LUÍS DANIEL ABREU
If we consider the wavelet case, things appear to be even more mysterious. Al-
though the density and irregular grid problems have attracted some attention lately
(see [1], [31], [22], [15] and references there in), there is no known counterpart of
Ramanathan and Steeger theorem (there is, however, a version of the HAP property
for wavelet systems [16]) and the only information available so far, on particular
systems, concerns a special family of analyzing wavelets that maps the problem into
Bergmann spaces: The wavelet transform, with positive dilation parameter, with
respect to a wavelet of the form (Paul´s Wavelet in some literature)
(1.1) ψα(t) =
can be rescaled as an isometrical integral transform between spaces of analytical
functions, namely, between the Hardy space on the upper half plane H2(U) and
the Bergman space in the upper half plane A2α+1(U). It is natural to refer to
this isomorphism as the Bergman transform (the designation analytic wavelet is
also frequently used). Since the upper half plane can be mapped isomorphically
onto the unit disc by using linear fractional transformations, we can construct a
transform mapping the Hardy space onto the Bergman space in the unit disc.
The approaches used to deal with the special situations mentioned in the above
paragraphs are based in techniques which differ from case to case. It seems highly
desirable to follow a more structured approach. The natural place to look for this
structure is within the context of Hilbert spaces of analytic functions, where the
powerful methods from complex analysis may answer questions that seem hopeless
otherwise. Following this line of reasoning implies using windows that allow to carry
the problem to such spaces. In this direction a major step was taken recently by
Göchenig and Lyubarskii [12], by considering Gabor systems with Hermite func-
tions of order n as windows. They have proved that if the size of the lattice Λ is
< (n + 1)−1 then the referred Gabor system is a frame and provided an example
supporting their conjecture that the result is sharp.
In the Bargmann-Fock setting the Hermite functions play a very special role
[11, pag. 57]. They are, up to normalization constants, pre-images, under the
Bargmann transform, of the monomials {zn} and since the latter constitute an
orthogonal basis of the Bargmann-Fock space, their pre-images also constitute an
orthogonal basis of L2(R). That is, to the isomorphism
(1.2) L2(R)
B→ F 2(C)
corresponds
(1.3) hn
B→ cnzn
where B is the Bargmann transform, F 2(C) is the Bargmann-Fock space, hn are the
Hermite functions and cn some constants dependent on n. They are canonical to
time-frequency analysis in an additional sense, since they constitute the eigenfunc-
tions of the time-frequency-localization operator with Gaussian window [3]. The
Hermite functions are eigenfunctions of the Fourier transform and they can be used
[20] to describe Feichtinger´s algebra So.
Wavelet and Gabor analysis share many similarities and many of their struc-
tural aspects can be bound together in a more general theory using representations
of locally compact abelian groups [13], [9]. In looking for a Wavelet-analogue of
Gröchenig-Lyubarskii structured approach to the density problem, we must first
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 3
clarify what functions should be used instead of the Hermite functions. In analogy
to the above paragraph it is natural to consider the pre-images, under the Bergman
transform, of the monomials {zn} in the unit disc (up to an isomorphism with the
upper half plane). Since the {zn} form a basis of the weighted Bergman spaces in
the unit disc (independently of the weight), their pre-images must be an orthogonal
basis of the Hardy space H2(U) and it is reasonable to expect that such functions
will play a similar role in Wavelet analysis as do the Hermite functions in Gabor
analysis. Such functions are the Fourier transforms of the Laguerre functions and
they will be ad-hoc denoted by Sαn .
Additional structural evidence that the functions Sαn are canonical in Wavelet
analysis comes from the work of Daubechies and Paul [4], where it is shown that
they are the eigenfunctions of a differential operator that commutes with a time–
scale localization operator, once windows of the form (1.1) are chosen. This com-
pletely parallels the situation with the time-frequency-localization operators with
gaussian windows, which commute with the harmonic oscillator and therefore have
as eigenfunctions the Hermite functions [3]. It was observed by Seip [26] that these
problems have a more natural formulation when mapped into the convenient spaces
of analytical functions.
2. Description of the results
There exists an analogue of the correspondence between (1.2) and (1.3), but
involving four different functional spaces and the corresponding bases: To the se-
quence of isomorphisms between the Hilbert spaces
(2.1) L2(0,∞) F←→ H2(U) Ber
→ A2α+1(U)
Tα→ A2α+1(D),
where A2α+1(U) and A2α+1(D) are the weighted Bergmann spaces in the upper-
half plane and in the unit disc, respectively, corresponds the relations between the
basis of the respective spaces:
(2.2) lαn
F←→ Sαn
Berα→ cαnΨαn
Tα→ cαnzn.
for some constants cαn, where l
n is the nth Laguerre function of order n and pa-
rameter α, Sαn is its Fourier transform and Ψ
n is a basis of A2α+1(U) to be defined
in section 4. This correspondence was implicit in the conection between papers [4]
and [26] but since it was not stated explicitly, we devote section 4 to clarify how
exactly does it work.
The functions Sαn were computed recently in closed form [29], but the connec-
tion to the Wavelet transform seems to have been unnoticed. It is also shown in
[29] that Sαn are, up to a fractional transformation, defined in terms of a certain
system of orthogonal polynomials on the unit circle. We will show that this family
of polynomials is nothing more but the circular Jacobi orthogonal polynomials for
which there is, for computational purposes, a very convenient three term recurrence
formula. This connection will make our case to call the functions Sαn the rational
Jacobi orthogonal functions.
With a computational method available for evaluating the functions Sαn and
graphic evidence of their good localization properties (see for example the plots of
their real versions in page 44 of [2]), it is natural to investigate how to use such func-
tions as analysing wavelets and to obtain frames from the resulting discretization.
4 LUÍS DANIEL ABREU
We will obtain the following sufficient condition on the density of the parameters
of the hyperbolic lattice {(ajbk, aj)}j,k∈Z:
b log a <
2n+ α
that is, as in Gröchenig and Lyubarskii result [12], this density increases with n.
This will follow as a special case of the following more general theorem, which uses
Seip´s notion of lower Beurling density in the unit disk and constitutes the main
result in this paper:
Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping
the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform.
If D−(Γ) > n+ α
then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H
2(U).
Case α = 0 gives the sufficient part of theorem 7.1 in [28]. The proof of theorem
1 will be dramatically simplified with the observation of the following remarkable
ocurrence: There exist constants {C} and sequences of numbers {ak}, both de-
pending on α and n, such that
(2.3) Sαn (
) = C
akψk+α
This will allow to obtain a formula which expresses the wavelet transform with
window Sαn as a linear combination of simple but non-analytic functions. As a
result we will have to deal with a situation that is reminiscent of the non-analiticity
problem that Gröchenig and Lyubarskii faced with formula (15) of [12] and which
they solved by using the so called Wexler-Raz identities [11]. Since there is no
known analogue of the Wexler-Raz identities in wavelet analysis, we have to deal
with this in a more direct way. The key idea is to map the problem into the unit
circle and use a deep sampling formula proved by Seip in [28]. Althought this
approach lacks the simplicity of [12] it gives the result for more general sequences,
while the method in [12] works only for the lattice case due to the restritions
imposed by the use of the Wexler-Raz identities.
We organize our ideas as follows. The next section contains the main definitions
and facts concerning wavelet transforms, Bergman spaces and Laguerre functions.
Section four contains the evaluation of the pre-images required to build up the
correspondences (2.1) and (2.2). The fifth section contains our main results on
wavelet frames with Fourier transforms of Laguerre functions. We conclude the
paper collecting some further properties of the functions Sαn and clarifying their
classification within known families of orthogonal polynomials.
3. Tools
3.1. The Bergman transform. Now we present a sinthesis of ideas that
appeared in the section 3.2 of [13] and in [4] (see also [2, pag. 31]) Here they will
be exposed in such a way that the role of the Bergman spaces is emphasized.
Consider the dilation and translation operators
Dsf(x) = |s|−
2 f(s−1x)
Txf(t) = f(t− x)
and define
ψx,s(t) = TxDsψ(t) = |s|−
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 5
The wavelet transform of a function f , with respect to the wavelet ψ is
Wψf(x, s) = 〈f, TxDsψ(t)〉L2(R) =
f(t)ψx,s(t)dt.
A function ψ ∈ L2(R) is said to be admissible if
|Fψ(s)|2 ds
where K is a constant. If ψ is admissible, then for all f ∈ L2(R) we have
(3.1)
s−2 |Wψf(x, s)|2 dxds = K ‖f‖2
We will restrict ourselves to parameters s > 0 and functions f ∈ H2(U), where
H2(U) is the Hardy space in the upper half plane U = {z = x+ is : s > 0}
H2(U) = {f : f is analytic in U and sup
0<s<∞
|f(x+ is)|2 dx <∞}.
Let F denote the Fourier transform
(Ff)(t) = 1√
e−itxf(x)dx
By the Paley-Wiener theorem, H2(U) is constituted by the functions whose Fourier
transform is supported in (0,+∞) and belongs to L2(0,∞).
Now take the special window ψα(t) defined in (1.1). Since
(3.2) Fψ−x,sα (t) = 1[0,∞]sα+
2 tαei(x+is)t
(3.3) Wψαf(−x, s) = s
tαeizt(Ff)(t)dt
where the function defined by the integral is analytic in z = x + si. The identity
(3.1) gives
(3.4)
∣Wψαf(−x, s)
s−2dxds = ‖f‖H2(R)
This motivates the definition of the Bergman transform, or the analytic wavelet
transform:
(3.5) Berα f(z) =
tαeizt(Ff)(t)dt = s−α− 12Wψαf(−x, s),
where z = x+ is (see for instance [15], where the authors use this Bergman trans-
form with the same normalization in the case α = 1). Introducing the scale of
weighted Bergman spaces
Aα(U) = {f analytic in U such that
|f(z)|2 sα−2dxds <∞},
it is clear from (3.4) and (3.5) that Berα f(z) ∈ A2α+1(U). We have therefore an
isometric transformation
Berα : H2(R)→ A2α+1(U)
6 LUÍS DANIEL ABREU
The weighted Bergman spaces in the unit disc are denoted by Aα(D) and defined
Aα(D) =
f analytic in D such that
|f(z)|2 (1− |z|)α−2dxdy <∞
For detailed expositions of the theory of Bergman spaces we point out the mono-
graphs [32], [6], [14]
3.2. Frames and sampling sequences. A sequence of functions {ej} is said
to be a frame in a Hilbert space H if there exist constants A and B such that
(3.6) A ‖f‖2 ≤
|〈f, ej〉|2 ≤ B ‖f‖2 .
In spaces of analytic functions a related concept to frames is the one of a
sampling sequence. A set Γ = {zj} is said to be a sampling sequence for the
Bergman space Aα(U) if there exist positive constants A and B such that
(3.7) A
|f(z)|2 yα−2dxds ≤
|f(zj)|2 yαj ≤ B
|f(z)|2 yα−2dxds.
The corresponding definition of a sampling sequence for the Bergman space on
the unit disc is
|f(z)|2 (1− |z|2)α−2dz ≤
|f(zj)|2 (1− |z|2)α
|f(z)|2 (1− |z|2)α−2dz.
Define the pseudohyperbolic metric on the unit disk by
̺(z, ζ) =
z − ζ
1− ζz
Following [28], a sequence Γ = {zj} ⊂ D is separated if infn6=j
zj−zn
zj−zn
> 0 and its
lower density D− is given by
D−(Γ) = lim
inf inf
̺(zj ,z)<r
(1− ̺(zj , z))
log 1
The next Theorem is from [28]:
Theorem A A separated sequence Γ ⊂ D is a sampling sequence for Aα(D)
iff D−(Γ) > α
It is a generalization of an earlier lattice result [27]:
Theorem B Let Γ(a, b) = {zjk}j,k∈Z, where zjk = aj(bk + i). Γ(a, b) is a
sampling sequence for Aα(U) if and only if b log a <
3.3. Fourier transforms of Laguerre functions. The Laguerre polynomi-
als will play a central role in our discussion. One way to define them is by means
of the Rodrigues formula
(3.8) Lαn(x) =
exx−α
e−xxα+n
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 7
and this gives, in power series form
Lαn(x) =
(α + 1)n
(−n)k
(α+ 1)k
For information on specific systems of orthogonal polynomials, we suggest [17]. The
Laguerre functions are defined as
lαn(x) = 1[0,∞](x)e
−x/2xα/2Lαn(x)
and they are known to constitute an orthogonal basis for the space L2(0,∞). By the
Paley-Wiener theorem, the Fourier transform is an isomorphism between H2(U)
and L2(0,∞). Therefore, the Fourier transform of the functions lαn form an orthog-
onal basis for the space H2(U). From now on we will set
(3.9) FSαn (t) = lαn(t)
The functions Sαn (t) can be evaluated explictly using (3.8). This was already done
by Shen in [29], where the author was interested in describing the orthogonal poly-
nomials which arise from an application of the Fourier transform on the Laguerre
polynomials. A description of this method of generating new families of orthogonal
polynomials with an interesting historical account is in the paper [21] where a link
is provided between Jacobi and Meixner-Polaczek polynomials. Following [29], we
(3.10) Sαn (t) =
(1 + α)n
(−n)k(α2 + 1)k
k!(α+ 1)k
4. Bergman transform of Sαn (t)
In this section the correspondences (2.1) and (2.2) will be established via a
direct calculation. The complex linear fractional transformations will play an im-
portant role, in a style that is reminiscent of the way they are used in the discrete
series representation of SL(2,R) over the Bergman space [7, chapter IX].
We first define a set of functions that, as we shall see later, constitute a basis
of Aα(U). For every n ≥ 0 and α > −1 let
Ψαn(z) =
iz + 1
iz − 1
)−α−1
Now define a map Tα such that for every function f ∈ Aα(U) the action of Tα
(4.1) Tαf(w) = f
w + 1
w − 1
The range space of Tα is a weighted Bergman space in the unit disc.
Lemma 4.1 The map
Tα : Aα(U)→ Aα(D)
is an unitary isometry between Hilbert spaces.
Proof. Argue as in the proof of lemma 1 in [7, pag.185]. �
Proposition 4.1 For n = 0, 1, ..., the following relations hold:
(4.2) Berα(S2αn ) = c
n , with c
n = (−1)α+1(2α+ n)!/n!
8 LUÍS DANIEL ABREU
and, for |z| < 1,
(4.3) Tα(Ψ
n) = z
In other words, the function S2αn is the pre-image, under Tα ◦
, of zn ∈
Aα(D).
Proof. Using the definition of the Bergman transform (3.5) and (3.9) we have
Berα S2αn (z) =
tαeizt(FS2αn (t))(t)dt
e(iz−
)tt2αL2αn (t)dt
e−xx2α+n
(−iz + 1
= cαnΨ
n (z),
where in the last two identities we are applying Rodrigues formula (3.8) for the
Laguerre polynomials and writing the integral in terms of the Laplace transform L,
whose well known properties establish the last identity. Now, the linear fractional
transformation
iz + 1
iz − 1
is an analytic isomorphism between the upper half plane and the unit circle. Since
the inverse of this transformation is given by
w + 1
w − 1
a short calculation with the definition of Tα gives(4.3). �
Remark 1. {Ψαn(z)} is a basis of Aα(U) and the map Tα is an unitary isomorphism
between Aα(U) and Aα(D). Indeed, since {zn} is an orthogonal basis of the space
Aα(D) [7, pag. 186] and it is contained in the range of Tα, Tα is onto and therefore
an unitary isomorphism.The functions {Ψαn(z)} form a basis for the space Aα(U)
since they are the pre-images of the basis {zn}.
As a consequence we obtain a new proof of the (known) isomorphic property
of the Bergman transform.
Corollary 1. The transform Berα : H2(R)→ A2α+1(U) is an isometric isomor-
phism.
Proof. The isometry is a consequence of the isometric property of the wavelet
transform, so we need only to prove that the Ber
2 is onto. But in view of the
preceding remark and Theorem 5.1, the range of Ber
2 contains a basis of Aα(U).
Therefore, Ber
2 is onto. �
Remark 2. Similar calculations as we have seen here also play a role in [5], in the
context of Laplace transformations and group representations and in [4], to obtain
an explicit formula for the eigenvalues of the time-scale localization operator.
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 9
5. Wavelet frames with Fourier transforms of Laguerre functions
5.1. Preparation. We wish to construct wavelet frames with analysing wavelets
Sαn , by using the common discretization of the continuous wavelet transform via
the hyperbolic lattice. Discretizing the scale parameter s in the Wavelet transform
by a sequence aj and the parameter x by ajbm gives
(5.1) Wψf(a
jbm, aj) = 〈f, TajbmDajψ〉 .
We want to know conditions in a and b under which {TajbkDajψ} is a frame, for a
given wavelet ψ. More generally we can consider the non lattice problem replacing
(ajbm, aj) by more sequences (ajm, bjm).
Remark 3. Observe that it follows immeadiately from Theorem A and Theorem B
that if Γ ⊂ D is the sequence obtained from mapping the sequence {zk,j = aj(bk +
i)} ⊂ U into the unit disk via a Cayley transform, then D−(Γ) = 2π
b log a
(to move
between the notation of [28] and [27] set α = 2n+ 1).
Remark 4. Observe that {TajbmDajψα}j,k∈Z is a frame of H2(U) iff Γ(a, b) =
{z = ajbm+ aji}j,k∈Z is a set of sampling for A2α+1(U). Indeed, since functions
in A2α+1(U) can be identified with Ber
α transforms of H2(U) functions, it follows
from (3.7) and (??) that Γ(a, b) is a set of sampling for A2α+1(U) if and only if
∣Wψαf(x, s)
2 dxds
∣Wψαf(a
jbm, aj)
2 ≤ B
∣Wψαf(x, s)
2 dxds
Using (3.4) this is equivalent to
A ‖f‖2H2(U) ≤
|〈f, TajbmDajψα〉|
2 ≤ B ‖f‖2H2(U)
which says that {TajbmDajψα}j,k is a frame in H2(U).
We now collect the required preliminary lemmas for the proof of the main
result. The first was quoted in the introduction and is just a simple modification
of the representation (3.10). The second appeared in [27] (see [6, pag. 160] for a
proof) and the third one is a quite deep sampling theorem which is formula (30) in
[28] with s = 0 and ǫ = 0 (see also [6, pag. 216] for a direct derivation with this
choice of parameters).
Lemma 5.2 The functions Sαn can written as the linear combination (2.3) of
analyzing wavelets ψk+α
(t) defined by
Sαn (
) = C
akψk+α
where the constants C and the coeficients ak given as
C = Cα,n =
(1 + α)n
ak = a
k = (2i)
(−n)k(α2 + 1)k
k!(α+ 1)k
Lemma 5.3 For 1 < s < t we have
(5.2)
(1− |zj |2)s
∣1− ζzj
≤ C(1− |ζ|2)s−t
10 LUÍS DANIEL ABREU
for any separated sequence {zj}.
Lemma 5.4 Let D({wj}) > β. Then every f ∈ A2β+1(D) satisfies
(5.3) (1− |w|2)βf(w) =
(1 − |wj |2)βf(wj)hj(w),
with the estimate for hj(w)
(5.4) |hj(w)| ≤ C
(1− |w|2)(1 − |wj |2)
|1− wjw|2
5.2. Main result. To prove our main result we will adhere to the following
plan: we begin by deriving from Lemma 5.2 a formula expressing the wavelet trans-
form with analysing wavelet Sαn in terms of the wavelet transform with analysing
wavelet ψk+α
. The proof of the right frame inequality is then straightforward (as
usual in frame inequalities). To prove the left hand inequality we first rewrite the
formula in terms of complex functions defined on the upper half plane and then map
it into the unit disk by means of a Cayley transform. Manipulating the resulting
formula and using lemma 5.3 and lemma 5.4 gives an estimate in the unit disk. We
proceed backwards to the half plane to end up with the required estimate in terms
of wavelet frames.
Theorem 5.1 Let Γ ⊂ D denote a separated sequence obtained from mapping
the sequence {zm,j = am,j + ibm,j} ⊂ U into the unit disk via a Cayley transform.
If D−(Γ) > n+ α
then {Tam,jDbm,jSαn ( t2 )}m,j is a frame of H
2(U).
Proof. The definition of the wavelet transform gives, using (2.3),
)f(−x, s) =
f, T−xDsS
f, T−xDsψk+α
akWψk+α
f(−x, s).
The right frame inequality is now easy: observe that, since {Tam,jDbm,jψk+α
is a frame, then for every k = 1, ..., n there exists a Bk such that
f, Tam,jDbm,jψk+α
≤ Bk ‖f‖2H2(U) .
Let Bn = max1≤k≤nBk. Then,
f, Tam,jDbm,jS
f, Tam,jDbm,jψk+α
|ak|2
f, Tam,jDbm,jψk+α
≤ B ‖f‖2 ,
where B = BnC
k=0 |ak|
and we are done.
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 11
Now the left frame inequality. Define the function F (z) as
(5.5) F (z) =WSα
)f(−x, s) = C
2Berk+
2 f(z)
where z = x+is belongs to the upper half plane U. We construct a related function
in the unit disc, by setting w = z−i
G(w) = C
ak(1− |w|2)k+
2Gk+α
where
G(w) = F (i
1 + w
(w) =
2k+α+1
Berk+
2 f(i
1 + w
) ∈ A2k+α+1(D).
Therefore, if D({wj}) > n+ α2 we can use (5.3) to write, for every k ≤ n,
(1 − |w|2)k+
2 Gk+α
(w) =
(1 − |wj |2)k+
2 Gk+α
(wj)hj(w).
As a result,
(1 − |w|2)− 12G(w) = Cα,n
ak(1− |w|2)k+
2 Gk+α
(1− |wj |2)k+
2 Gk+α
(wj)hj(w)
hj(w)C
ak(1− |wj |2)k+
2 Gk+α
hj(w)(1 − |wj |2)−
2G(wj).
Now use estimate (5.4), then Cauchy-Schwarz and finally (5.2) in the following way.
(1− |w|2)−1 |G(w)|2 ≤
(1 − |w|2)
|1− wjw|2
(1− |wj |2)
2G(wj)
(1 − |w|2)2 |G(wj)|2
(1− |wj |2)1
|1− wjw|4
(1 − |w|2)−1 |G(wj)|2 .
Thus the factor (1− |w|2)−1 cancels and we have
|G(w)|2 ≤ C
|G(wj)|2 .
12 LUÍS DANIEL ABREU
This last inequality gives at once
|G(w)|2 (1− |w|2)−2dw ≤ C
|G(wj)|2
or, in the upper half plane,
|F (z)|2 s−2dxds ≤
|F (zj)|2 .
In the wavelet notation this is
)f(−x, s)
s−2dxds ≤
jbm, aj)
Taking into account that Sαn (
) is admissible, we can apply (3.1) and (5.1) to write
the above inequality as
(5.6) A ‖f‖2 ≤
f, Tam,jDbm,jS
This is the left frame inequality. The sequence {Tam,jDbm,jSαn ( t2 )}j,m is thus a
frame. �
In view of remark 5.1, the lattice result is a special case of our main theorem.
Corollary 2. If b log a < 4π
, then {TajbmDajSαn ( t2 )}j,m is a frame of H
2(U).
Remark 5. Theorem 5.1 parallels theorem 3.1 in [12], which states that, in the
lattice case of the time-frequency plane, if the density of the lattice Λ is > n + 1
(or if the size of Λ is < (n + 1)−1) then the Gabor system {e2πiλ2tHn(t − λ1) :
λ = (λ1, λ2) ∈ Λ}, where Hn stands for the Hermite function of order n, is a
frame for L2(R). In particular, if one is dealing with the Von Neumann lattice
with parameters a and b, the condition is ab < (n+1)−1. Corollary 5.2 makes this
analogy even more explicit.
Remark 6. It is interesting to notice that, since Berα f(z) =
tαeizt(Ff)(t)dt,
we clearly have
ikBerk+
2 f(z) =
2 f(z).
This allows to rewrite (5.5) as
)f(−x, s) = C
2 f(z)
which is, in some sense, reminiscent of Proposition 3.2 in [12].
Remark 7. Observe that combining the Paley-Wiener with the Plancherel theorem,
we have ‖f‖H2(U) = ‖f‖L2(0,∞) and the results of Theorem 1 and 2 say also that
we have a frame for L2(0,∞).
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 13
6. Further properties of the functions Sαn (t)
The notation
F (a, b; c;x) =
(a)k(b)k
k!(c)k
for the hypergeometric function is used in this section. Rewriting Sαn in this notation
gives
Sαn (t) = C
F (−n, α
+ 1;α+ 1;
(observe that the infinite sum becomes a polynomial of order n, since (−n)k = 0 if
k > 0). Composing the functions Sαn (t) with the fractional linear transformation
(6.1) z =
2t− i
2t+ i
the result is
(6.2) Sαn (t) = Γ
(1 − z)α2 +1gαn(z)
where
gαn(z) =
(α/2)n
F (−n, α
+ 1;−n− α/2 + 1; z)
is a polynomial in z of degree n. This was pointed out in [29]. It was also shown
that these polynomials satisfy the orthogonality
(6.3)
|z|=1
gαn(z)g
n(z) |1− z|
= 0 if m 6= n
and therefore are orthogonal on the unit circle with respect to the weight
(6.4) w(z) = sinα
This fact implies many properties, since there exists a very rich theory for orthogo-
nal polynomials on the unit circle (see [30] and references therein and also chapter
8 of [17]). For example, the general theory assures that all the zeros of gαn(z) lay
within the unit disc.
Remark 8. Setting a = α
in Example 8.2.5 at [17], and using the identity 1−eiθ =
4 sin2 θ
to write the measure (8.2.21) as (6.4) we recognize that the polynomials
gαn(z) are, up to a normalization, a family of orthogonal polynomials on the unit
circle known as the circular Jacobi orthogonal polynomials.
Remark 9. From (6.3) the functions z−
2 (1−z)α2 gαn(z) are orthogonal on the circle
and form a basis of the Hardy space on the unit disc
H2(D) = {f : f is analytic in D and sup
∣f(reit)
dt <∞}
it is also clear that Sαn (t) are orthogonal on the real line (the boundary of the upper
half place). Since gαn(z) are the circular Jacobi orthogonal polynomials, the basis
functions z−
2 (1 − z)α2 gαn(z) are the circular Jacobi orthogonal functions and it is
therefore natural to call Sαn (t) the rational Jacobi orthogonal functions.
14 LUÍS DANIEL ABREU
Remark 10. From the general theory [17, (8.2.10)] follows that, if κn is the leading
coefficient of the polynomial, then the sequence of polynomials {gαn(z)} satisfies a
three term recurrence relation
n(0)g
n+1(z) + κn−1g
n+1(0)zg
n−1(z) = [κng
n+1(0) + κn+1g
n(0)z]g
where κn is the leading coefficient of the polynomial. From the explicit representa-
tion of the polynomials gαn(z) it is easily seen that
+ 1)n
; φn(0) =
+ 1)n−1
This three term recurrence relation provides a effective method for computational
purposes: To evaluate the functions Sαn (t) it is sufficient to combine this recurrence
relation with formulas (6.1) and (6.2).
References
[1] A. Aldroubi, C. Cabrelli, U. M., Molter Wavelets on irregular grids with arbitrary dilation
matrices and frame atoms for L2(Rd), Appl. Comput. Harmon. Anal. 17, no. 2, 119–140
(2004).
[2] I. Daubechies, ”Ten lectures on wavelets”, CBMS-NSF Regional conference series in applied
mathematics (1992).
[3] I. Daubechies, Time-frequency localization operators: A geometric phase space approach
IEEE Trans. Inform. Theory, 34, pp. 605-612. (1988).
[4] I. Daubechies, T. Paul, Time-frequency localization operators: A geometric phase space ap-
proach II. The use of dilations and translations. Inverse Prob., pp 661-680. (1988).
[5] M. Davidson, G. Ólafsson, G. Zhang, Laguerre polynomials, restriction principle, and holo-
morphic representations of SL(2;R). Acta Appl. Math. 71 , no. 3, 261–277 (2002).
[6] P. Duren, A. Schuster, ”Bergman spaces”, Mathematical Surveys and Monographs, 100.
American Mathematical Society, Providence, RI, (2004).
[7] S. Lang, ”SL(2,R)”, Springer (1985).
[8] Y. Lyubarskii, Frames in the Bargmann space of entire functions, Entire and subharmonic
functions, 167-180, Adv. Soviet Math., 11, Amer. Math. Soc., Providence, RI (1992)
[9] H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations
and their atomic decompositions, I, J. Funct. Anal. 86 (2), 307-340 (1989).
[10] H. G. Feichtinger, N. Kaiblinger, Varying the time-frequency lattice of Gabor frames. Trans.
Amer. Math. Soc. 356 (2004), no. 5, 2001–2023
[11] K. Gröchenig, ”Foundations of time-frequency analysis”, Birkhäuser, Boston, (2001)
[12] K. Gröchenig, Y. Lyubarskii, Gabor frames with Hermite functions, C. R. Acad. Sci. Paris,
Ser. I 344 157-162 (2007).
[13] A. Grossman, J. Morlet, T. Paul, Transforms associated to square integrable group represen-
tations II: examples, Ann. Inst. H. Poincaré Phys. Théor., 45, 293-309, (1986).
[14] H. Hedenmalm, B. Korenblum, K. Zhu, ”Theory of Bergman spaces.” Graduate Texts in
Mathematics, 199. Springer-Verlag, New York, (2000).
[15] C. Heil, G. Kutyniok, Density of wavelet frames, J. Geom. Anal. 13, no. 3, 480-493. (2003).
[16] C. Heil, G. Kutyniok, The Homogeneous Approximation Property for wavelet frames , J.
Approx. Theory., online first.
[17] M. E. H. Ismail, ”Classical and quantum orthogonal polynomials in one variable”, Encyclo-
pedia of Mathematics and its Applications No. 98. (2005).
[18] A. J. E. M. Janssen; T. Strohmer, Hyperbolic secants yield Gabor frames. Appl. Comput.
Harmon. Anal. 12 , no. 2, 259–267 (2002).
[19] A. J. E. M. Janssen, Zak transforms with few zeros and the tie, in ”Advances in Gabor
Analysis” (H.G. Feichtinger, T.Strohmer, eds.), Boston, 2003, pp. 31-70.
[20] A. J. E. M. Janssen, Hermite function description of Feichtinger´s Space S0, J. Fourier Anal.
Appl., 11 (5) 577-588 (2005).
[21] H. T. Koelink, On Jacobi and continuous Hahn polynomials, Proc. Amer. Math. Soc. 124,
no. 3, 887-898 (1996).
WAVELET FRAMES WITH FOURIER TRANSFORMS OF LAGUERRE FUNCTIONS 15
[22] G. Kutyniok, Affine density, frame bounds, and the admissibility condition for wavelet
frames. Constr. Approx., 25, no. 3, 239-253 , (2006)
[23] J. Ramanathan, T. Steger, Incompleteness of sparse coherent states. Appl. Comput. Harmon.
Anal. 2 , no. 2, 148–153 (1995).
[24] K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space I, J.
Reine Angew. Math. 429, 91-106 (1992).
[25] K. Seip, R. Wallstén, Density theorems for sampling and interpolation in the Bargmann-Fock
space II, J. Reine Angew. Math. 429 (1992), 107-113.
[26] K. Seip, Reproducing formulas and double orthogonality in Bargmann and Bergman spaces,
SIAM J. Math. Anal. 22, No. 3 pp. 856-876 (1991).
[27] K. Seip, Regular sets of sampling and interpolation for weighted Bergman spaces, Proc. Amer.
Math. Soc. 117, no. 1, 213-220 (1993).
[28] K. Seip, Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993).
[29] L. C. Shen, Orthogonal polynomials on the unit circle associated with the Laguerre polyno-
mials, Proc. Amer. Math. Soc. 129, no. 3, 873-879 (2001).
[30] B. Simon, OPUC on one foot, Bull. Amer. Math. Soc. 42 (2005), 431-460.
[31] W. Sun, X. Zhou, Density and stability of wavelet frames, Appl. Comput. Harmon. Anal. 15,
no. 2, 117–133 (2003).
[32] K. H. Zhu, ”Operator theory in function spaces”. Monographs and Textbooks in Pure and
Applied Mathematics, 139. Marcel Dekker, Inc., New York, 1990.
	1. Introduction
	2. Description of the results
	3. Tools
	4. Bergman transform of Sn0=x"010B(t)
	5. Wavelet frames with Fourier transforms of Laguerre functions
	6. Further properties of the functions Sn0=x"010B(t)
	References
