We define q-analogues of Fourier-Bessel series, by means of complete q-
orthogonal systems constructed with the third Jackson q-Bessel function. Sufficient
conditions for pointwise convergence of these series are obtained, in terms of a
general convergence principle valid for other Fourier series on grids defined over
numerable sets. The results are illustrated with specific examples of developments
in q-Fourier-Bessel series

Abreu, L. D.

Alvarez-Nodarse, R.

Cardoso, J. L.

Estudo Geral

Pointwise convergent expansions in q-Fourier-Bessel series

Pre´-Publicac¸o˜es do Departamento de Matema´tica
Universidade de Coimbra
Preprint Number 06–25
POINTWISE CONVERGENT EXPANSIONS IN
q-FOURIER-BESSEL SERIES
L. D. ABREU, R. ALVAREZ-NODARSE AND J. L. CARDOSO
Abstract: We define q-analogues of Fourier-Bessel series, by means of complete q-
orthogonal systems constructed with the third Jackson q-Bessel function. Sufficient
conditions for pointwise convergence of these series are obtained, in terms of a
general convergence principle valid for other Fourier series on grids defined over
numerable sets. The results are illustrated with specific examples of developments
in q-Fourier-Bessel series.
Keywords: q-Fourier series, third Jackson q-Bessel function.
AMS Subject Classification (2000): 42C10, 33D15.
1. Introduction
The theory of Fourier-Bessel series, based on the orthogonality relation∫ 1
0
tJν(jνmt)Jν(jνnt)dt = 0, if m 6= n ,
where Jν stands for the Bessel functions of order ν and jνn is their n-
th zero, was developed by the classical analysis school of the early twenty
century [19, XVIII], in a close parallelism to the classical theory of Fourier
series and maintained its status as an active research topic till our days (see
e.g. [12] and references therein).
Among the generalizations of the Bessel function, considerable attention
has been given to the third Jackson (or Hahn-Exton) q-Bessel function J
(3)
ν (z; q)
defined by
J (3)ν (z; q) = z
ν (q
ν+1; q)∞
(q; q)∞
∞∑
k=0
(−1)k
q
k(k+1)
2
(qν+1; q)k(q; q)k
z2k,
where ν > −1 is a real parameter. When q → 1− we recover the Bessel
function from J
(3)
ν (z; q) , after a normalization. It is a well known fact (see
Received June 2, 2006.
1
2 L. D. ABREU, R. ALVAREZ-NODARSE AND J. L. CARDOSO
e.g. [16, Prop. 3.5]) that these functions satisfy the orthogonality relation∫ 1
0
x J (3)ν
(
jnνqx; q
2
)
J (3)ν
(
jmνqx; q
2
)
dqx = 0, if m 6= n , (1.1)
with j1ν < j2ν < j3ν < . . . where jnν := jnν(q
2) are the zeros of Jν(z; q
2)
arranged in ascending order of magnitude and dqx stands for the measure
of the Jackson q-integral. The notation J
(k)
ν , k = 1, 2, 3 , is due to Ismail [14]
and is used to distinguish the three q-analogues of the Bessel function defined
by Jackson. However, since the only q-Bessel function to appear on the text
is J
(3)
ν , we will drop the superscript for shortness of the notation and simply
write
Jν(z; q
2) := J (3)ν (z; q
2).
These functions appear as the matrix elements of irreducible unitary rep-
resentations of the quantized universal enveloping algebra of the quantum
group of plane motions [15]. They are, within some boundaries, the only
functions q-orthogonal with respect to their own zeros [2] in the sense of
(1.1).
It is the purpose of this note to develop convergence results for q-Fourier-
Bessel series, based on the above orthogonality relation (1.1) and on results
about the completeness of these system [4]. Associated with such a q-Fourier-
Bessel theory there is a q-analogue of the Hankel transform with an inver-
sion formula, introduced in [17], whose kernel is the third Jackson q-Bessel
function. The relation between these q-Fourier-Bessel expansions and the
q-Hankel transform was exploited in [1] to obtain a sampling theorem. How-
ever, since such sampling theorems sit in a reproducing kernel Hilbert space,
convergence issues are of a different nature of those discussed here.
The q-Fourier theory to be present in this paper are different from the
theory of Fourier expansions in terms of q-exponential functions [9, 18], since
the latter is based on a continuous orthogonality property and the former
is based on a discrete one. As we shall see, as a consequence of this fact,
pointwise convergence issues are much simpler, and they follow from general
properties of complete orthogonal systems in Hilbert spaces associated to
general discrete orthogonality measures.
We will organize our ideas in four sections. The second is devoted to the
required background on q-analysis, the third gives the convergence results,
first in a general set-up and then in the specialization to the q-Fourier-Bessel
q-FOURIER-BESSEL SERIES 3
and other examples. We finish the paper with a section containing two
examples of expansions in q-Fourier-Bessel series.
2. Definitions and preliminary results
Following the standard notations in [13], consider 0 < q < 1 and define the
q-shifted factorial for a nonnegative positive integer n
(a; q)0 = 1, (a; q)n = (1− q) (1− aq) · · ·
(
1− aqn−1
)
, n = 1, 2, . . .
and the infinite case as
(a; q)∞ = lim
n→∞
(a; q)n.
The q−Jackson integral in the interval (0, a) is defined by∫ a
0
f (t) dqt = (1− q)
∞∑
k=0
f
(
aqk
)
aqk. (2.1)
Using this definition we may consider an inner product by setting
〈f, g〉 =
∫ 1
0
f (t) g(t)dqt .
The resulting Hilbert space is commonly denoted by L2q(0, 1). The space
L2q(0, 1) is a separable Hilbert space [5].
The third Jackson q-Bessel function has a countable infinite number of
real and simple zeros jk,ν , k = 0, 1, 2, . . ., as it was shown in [16]. Further
properties of the zeros of the third Jackson q-Bessel function were considered
in [3].
We will need the following theorem from [4]:
Theorem A The orthonormal sequence (u
(ν)
k )k≥0 defined by
u
(ν)
k (x) =
x
1
2Jν(jkνqx; q
2)∥∥∥x 12Jν(jkνqx; q2)∥∥∥
L2q(0,1)
is complete in L2q(0, 1).
This means that, whenever a function f is in L2q(0, 1) , if
∫ 1
0 f(x)u
(ν)
k (x)dqx =
0 , k = 1, 2, 3, . . . , then f
(
qk
)
= 0 , k = 0, 1, 2, . . . .
4 L. D. ABREU, R. ALVAREZ-NODARSE AND J. L. CARDOSO
3. q-Fourier-Bessel Series
Using the orthogonality relation (1.1), we may consider the q−Fourier-Bessel
series, Sνq [f ], associated with a function f ,
Sνq [f ](x) =
∞∑
k=1
ak (f) Jν(qjkνx; q
2), (3.1)
with the coefficients ak given by
ak (f) =
1
µk
∫ 1
0
tf(t)Jν(qjkνt; q
2)dqt, (3.2)
where [16, Prop. 3.5],
µk =
∥∥∥x 12Jν(jkνqx; q2)∥∥∥2
L2q(0,1)
=
∫ 1
0
[
x J (3)ν
(
jnνqx; q
2
) ]2
dqx
=
q − 1
2
qν−1Jν+1(qjkν ; q
2)J ′ν(jkν; q
2).
(3.3)
Throughout the paper we will use the set Vq defined by
Vq = {q
n : n = 0, 1, 2, . . .} , (3.4)
which coincides with the support points of the q-integral (2.1) in [0, 1].
3.1. Pointwise convergence: A general set-up. With a view to study
convergence of the series (3.1) when x ∈ Vq, we first establish a general
result concerning the pointwise convergence of these series. The setting to
be used in this section is a very general one, designed to cover not only the
convergence of q-Fourier-Bessel series but also other Fourier systems based
on discrete orthogonality relations, as in [8], [10] and [6].
Let N = {an |n ∈ N} be a numerable space and let µ a positive measure
on N such that µn = µ({an}) > 0. We will denote by L2µ, the space of all
functions f : N 7→ C, such that
‖f‖22 =
∞∑
n=1
|f(an)|
2µn < +∞.
In such a space, the scalar product 〈f, g〉 of two functions is defined by
〈f, g〉 =
∞∑
n=1
f(an)g(an)µn.
q-FOURIER-BESSEL SERIES 5
An orthonormal basis in L2µ is the sequence of functions (en)n such that
en(ak) =
{
µ
−1/2
n , k = n,
0, k 6= n.
(3.5)
To check that the above system is a complete orthonormal system in L2µ,
notice that the function gN , N ∈ N, defined as
gN = f −
N∑
n=1
〈f, en〉en, f ∈ L
2
µ
is such that gN(ak) = 0 for all k ≤ N and gN(ak) = f(ak) for all k > N .
Therefore,
‖gN‖
2
2 =
∞∑
n=N+1
|f(an)|
2µn → 0, as N →∞.
Thus, for an arbitrary f ∈ L2µ , we have
f =
∞∑
n=1
〈f, en〉en,
with convergence in norm ‖ · ‖22 . This is also true for any other complete
orthonormal system (un)n, i.e., for an arbitrary f ∈ L2µ one has
f =
∞∑
n=1
〈f, un〉un,
with convergence in norm ‖·‖22. It remains only to check when the convergence
of the above series is pointwise. The answer to this question is in the following
lemma.
Lemma 1. Let (un)n a complete orthonormal system of functions in L
2
µ.
Then for any arbitrary f ∈ L2µ
f(ak) = lim
N→∞
N∑
n=1
〈f, un〉un(k), ∀ak ∈ N .
Proof : Let ak be an arbitrary element ofN . Then, the function dk := µ
−1/2
k ek,
where ek is the function given in (3.5), satisfies the property
〈f, dk〉 = 〈f, µ
−1/2
k ek〉 = f(ak)µ
−1/2
k ek(ak)µk = f(ak).
6 L. D. ABREU, R. ALVAREZ-NODARSE AND J. L. CARDOSO
In particular, 〈un, dk〉 = un(ak). Then,
f(ak) = 〈f, dk〉 =
〈
lim
N→∞
N∑
n=1
〈f, un〉un, dk
〉
= lim
N→∞
N∑
n=1
〈f, un〉〈un, dk〉,
and, therefore,
f(ak) = lim
N→∞
N∑
n=1
〈f, un〉un(ak).
3.2. Application to q-Fourier-Bessel series. Let be N = Vq and in Vq
define the measure µ associated to the Jackson q-integral (2.1). Let L2q(0, 1)
be the corresponding L2µ space. Since the set of functions
uνn(x) :=
x
1
2Jν(jnνqx; q
2)∥∥∥x 12Jν(jnνqx; q2)∥∥∥,
is a complete orthogonal system in L2q(0, 1), then, for an arbitrary f ∈
L2q(0, 1), i.e., f such that ∫ 1
0
|f(x)|2dqx < +∞,
we have the equality
f(qk) = lim
N→∞
N∑
n=1
〈f, uνn〉u
ν
n(ak), ∀k = 0, 1, 2, . . . ,
where
ak (f) =
1
µk
∫ 1
0
tf(t)Jν(qjkνt; q
2)dqt ,
being µk the norm of the q-Bessel functions given in (3.3). This summarizes
in the following theorem.
Theorem 1. If f ∈ L2q(0, 1), then the q-Fourier-Bessel series (3.1) converges
to the function f at every point x = qk−1, k = 1, 2, 3, . . . of the set Vq.
Remark 1. Let us mention that in the case of the standard trigonometric
series the equivalent result of Lemma 1 (L2 convergence implies pointwise
convergence) is not true. In fact this problem leads to the celebrated Carleson
Theorem (see e.g. [7]). The main difference between these two cases is that,
q-FOURIER-BESSEL SERIES 7
contrary to the case of the discrete space L2µ (see the function dk used in the
proof of Lemma 1), for the L2[0,2pi] space for an arbitrary a ∈ [0, 2pi] there not
exists a function fa such that 〈f, fa〉 = f(a).
Remark 2. In the paper [10] were established some convergence theorems
of the q-Fourier series associated with the q-trigonometric orthogonal system{
1, Cq
(
q
1
2ωkx
)
, Sq (qωkx)
}
, where the q-cosines Cq and q-sinus Sq are given
in terms of the third q-Bessel functions by
Cq(z) = q
−3/8(q
2; q2)∞
(q; q2)∞
z1/2J
(3)
−1/2
(
q−3/4z; q2
)
,
Sq(z) = q
1/8(q
2; q2)∞
(q; q2)∞
z1/2J
(3)
1/2
(
q−1/4z; q2
)
,
being ωk is the positive zeros of the function Sq. Notice that, since this
orthogonal system is also a complete system in L2q(0, 1), then we have that
the q-trigonometric Fourier series converge to f at every point of Vq, i.e.,
f(x) =
a0
2
+
∞∑
k=1
[
akCq
(
q
1
2ωkx
)
+ bkSq (qωkx)
]
, x = qm, m = 0, 1, 2, . . . ,
with a0 =
∫ 1
−1 f(t)dqt and, for k = 1, 2, 3, . . . ,
ak =
1
µk
∫ 1
−1
f(t)Cq
(
q
1
2ωkt
)
dqt, bk =
q
1
2
µk
∫ 1
−1
f(t)Sq (qωkt) dqt,
where,
µk = (1− q)Cq(q
1/2ωk)S
′
q(ωk) .
Then, for any f ∈ L2q(0, 1), the q-trigonometric Fourier series defined above
(see [10]) converges to the function f at every point x = qk−1, k = 1, 2, 3, . . .
of the set Vq (3.4). Thus, the corresponding open problem posed in the con-
cluding remarks section of [10] is completely solved.
Remark 3. In [6] a rigorous theory of q-Sturm-Liouville systems was devel-
oped. In particular it was shown that the set of all normalized eigenfunctions
forms an orthonormal basis for L2q(0, a). Therefore Lemma 1 can be used to
show that the Fourier expansions in terms of the eigenfunctions of q-Sturm-
Liouville systems are pointwise convergent.
8 L. D. ABREU, R. ALVAREZ-NODARSE AND J. L. CARDOSO
4. Examples of q-Fourier-Bessel expansions
Consider
g(x; q) = xν
(x2q2; q2)∞
(x2q2µ−2ν; q2)∞
.
Using the q-binomial theorem [13, (1.3.2)] we have
g(x; q) =
xν
(1− x2)
[
∞∑
n=0
(q2µ−2ν; q2)n
(q2; q2)n
x2n
]−1
.
Since
lim
q→1−
∞∑
n=0
(q2µ−2ν; q2)n
(q2; q2)n
x2n =
∞∑
n=0
(2µ− 2ν)n
n!
x2n = (1− x2)−2µ+2ν,
it becomes clear that g(x; q) is a q-analogue of g(x) = xν(1− x2)2µ−2ν−1 .
Let consider the case when ν > 0, µ > 1/2. Then g(x; q) ∈ L2q(0, 1) and by
Theorem 1 the q-Fourier-Bessel series is pointwise convergent in Vq. Setting
x = qjkν in the formula [1, Eq. (4.11)] (where a misprint is corrected; see
also [11, (54)] for a similar formula with a different normalization)
(q2; q2)∞
(q2u−2ν; q2)∞
xν−uJ (3)u (x; q
2) =
1
1− q
∫ 1
0
tν+1
(t2q2; q2)∞
(t2q2u−2ν; q2)∞
J (3)ν (xt; q
2)dqt,
follows that∫ 1
0
t g(t; q)Jν(qjkνt; q
2)dqt = (1− q)(qjkν)
ν−µ (q
2; q2)∞
(q2µ−2ν; q2)∞
Jµ(qjkν; q
2) . (4.1)
Therefore, from (3.2)-(3.3), (4.1) follows that
xν (x2q2; q2)∞
(x2q2µ−2ν; q2)∞
= −
2 q1−µ
(
q2; q2
)
∞(
q2(µ−ν); q2
)
∞
∞∑
k=1
(jkν)
ν−µJµ
(
jqkν ; q
2
)
Jν+1
(
jkνq; q2
)
J ′ν(jkν ; q
2)
Jν
(
jqkνx; q
2
)
,
for all x = qn , n = 0, 1, 2, . . . .
Notice that putting µ = ν + 1 in the above formula it gives the following
expansion of f(x) = xν :
xν = −2 q−ν
∞∑
k=1
Jν(jkνqx; q
2)
jkνJ ′ν(jkν; q
2)
for all x = qn , n = 0, 1, 2, . . . .
q-FOURIER-BESSEL SERIES 9
The convergence of the expansion of xν in the classical Fourier-Bessel series
is studied in [19, 18.22] using contour integral methods.
Acknowledgements. The authors are very grateful to Prof. Juan Arias for
stimulating discussions and for pointing out the Lemma 1. This work was
partially supported by CMUC from University of Coimbra (LDA, JLC) and
by the Ministerio de Ciencias y Tecnolog´ıa of Spain under the grant BFM-
2003-6335-C03-01 and the Junta de Andaluc´ıa under grant FQM-262 (RAN).
LDA was also supported by a post-doctoral grant from Fundac¸a˜o Cieˆncia e
Tecnologia from Portugal during the completion of this work.
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L. D. Abreu
Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universidade de
Coimbra, Apdo. 3008, 3001-454 Coimbra, PORTUGAL
E-mail address : abreu@mat.uc.pt
R. Alvarez-Nodarse
Departamento de Ana´lisis Matema´tico, Facultad de Matema´ticas, Universidad de Sevilla,
Apdo. 1160, 41080 Sevilla, SPAIN
E-mail address : ran@us.es
J. L. Cardoso
Departamento de Matema´tica, Universidade de Tra´s-os-Montes e Alto Douro, Apdo.
1013, 5001-801 Vila Real, PORTUGAL
E-mail address : jluis@utad.pt


We define q-analogues of Fourier-Bessel series, by means of complete q-orthogonal systems constructed with the third Jackson q-Bessel function. Sufficient conditions for pointwise convergence of these series are obtained, in terms of a general convergence principle valid for other Fourier series on grids defined over numerable sets. The results are illustrated with specific examples of developments in q-Fourier-Bessel series

L. D. Abreu

R. Alvarez-nodarse

J. L. Cardoso

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POINTWISE CONVERGENT EXPANSIONS IN q-FOURIER-BESSEL SERIES

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Pointwise convergent expansions in q-Fourier-Bessel series

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