On Basic Fourier-Bessel Expansions

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu}$ and the"shifted"zeros, $qj_{k\nu}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the"shifted"ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the"shifted"zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.


Introduction
When dealing with basic Fourier-Bessel expansions, due to convergence issues, it is crucial to know the asymptotic behavior of the third Jackson q-Bessel function when computed in its own shifted zeros. For this purpose, in the sequel of Rahman, as pointed out by Koelink and Swarttouw [22, p. 696], "the intermingling of (ordinary) analysis and q-analysis may be fruitful".
In the literature, the function J (3) ν (z; q) ≡ J ν (z; q), where ν and q are parameters satisfying ν > −1 and 0 < q < 1, is usually identified by the third Jackson q-Bessel function or by the Hahn-Exton q-Bessel function: Using the basic hypergeometric representation [19, p. 4] for r φ s , it is very well known that (1.1) can be written as J ν (z; q) := z ν q ν+1 ; q ∞ (q; q) ∞ 1 φ 1 0; q ν+1 ; q, qz 2 . (1.2) In [2] it was shown, under some restrictions, that these functions are the only ones that satisfy a q-analogue of the Hardy result [20] about functions orthogonal with respect to their own zeros.
Abreu and Bustoz showed in [3] that the sequence {u k } k , where then f q k = 0, k = 0, 1, 2, . . .. Regarding criteria for completeness in L q [0, 1] see [1], with an interesting application to a qversion of the uncertainty principle via the q-Hankel transform and a completeness property of the third Jackson q-Bessel function.
Basic Fourier expansions were studied in [9,10], with respect to quadratic grids and to linear grids, respectively. For an overview over basic Fourier expansions see [26]. In [11,12] were presented results regarding convergence issues concerning basic Fourier expansions involving the basic sine and cosine functions considered by Suslov [25], which are equivalent to the ones introduced by Exton [18]. With properties connected to this or related functions we refer to [28]. For new definitions of q-exponential and q-trigonometric functions see [15]. Since we are using and proving some asymptotic results, we highlight [24] on this subject, where a complete asymptotic expansion for the q-Pochhammer symbol (or, the infinite q-shifted factorial (z; q) ∞ ) was exhibited. We also point out [29] with an appendix where, among others, asymptotic results for the theta function, for the 1 φ 1 (0; ω; q, z) function and for its derivative were presented. This last two ones, with an useful separation of the terms that increase from the terms that decrease, were crucial to establish our results of the third Jackson q-Bessel function.
Other publications also show estimates or inequalities involving the third Jackson q-Bessel function: for instance, equation (3.2.14) of [27] when ν = n, equation (2.4) of Proposition 2.1 from [23], Lemma 1 from [8] and [16] for the particular case of third Jackson q-Bessel function of order zero.
In our judgement the main results of this work are Theorems 3.5, 3.6, 3.9 and 4.1. We also emphasize Theorem 3.1 since it was decisive for the proof of Theorem 3.6 and mostly because of its important Corollaries 3.3 and 3.4. We believe that all the results stated in Section 3 and Theorem 4.1 of Section 4 are original.
To know the asymptotic behaviors given by Theorems 3.5 and 3.9, are decisive to study convergence properties of the basic Fourier-Bessel expansions. This is an important issue and, in our opinion, it is the most relevant contribution of the present work.
Many questions concerning basic Fourier or basic Fourier-Bessel expansions can be raised: analogues of Dirichlet's kernel, Riemann-Lebesgue theorem, Dini's condition or summability (Fejér's theorem) and many other topics are open problems since some of the nice properties used in the corresponding proofs are no longer valid in the context of basic expansions. Regarding this, we approach these difficulties and push a little further towards a q-analogue of the Riemann-Lebesgue theorem.
The paper is organized as follows: in Section 2 we collect the main definitions and preliminary results that were taken from other publications; in Section 3 we present some asymptotic behavior of the third Jackson q-Bessel function and of its derivative when computed at certain points. We also study the asymptotic behavior of the zeros j mν and their relations with the "shifted" zeros qj mν or jmν q , for large values of m = 1, 2, 3, . . . and explore its consequences to obtain other results; we finish with Section 4 where an analogue of the Riemann-Lebesgue theorem concerning basic Fourier-Bessel expansion is proved.

Definitions and preliminary results
Fixing 0 < q < 1 and following the standard notations of [5,19], the q-shifted factorial for a finite positive integer n is defined by (a; q) n = (1 − q)(1 − aq) · · · 1 − aq n−1 and the zero and infinite cases as The third Jackson q-Bessel function has a countable infinite number of real and simple zeros, as it was shown in [22]. In [4, Theorem 2.3] it was proved the following theorem: On this subject see [7] and [29]. The latter one improved the accuracy of the asymptotic expression for the zeros of the basic hypergeometric function 1 φ 1 0; ω; q, z , which appears in the definition (1.2) of the Hahn-Exton q-Bessel function. See also [21] where Hayman obtained an expression for the asymptotic behavior of the zeros of a certain class of entire functions, which perhaps may be extended to the third Jackson q-Bessel function.
Using Taylor expansion it can be shown that, as k → ∞, Formally, the q-Fourier Bessel series associated with a function f , by the orthogonal relation (1.3), is defined by or, which we rather prefer, and η k,ν by where the last equality can be derived from [22,Proposition 3.5] or [13,Proposition 5].
The asymptotic behavior of the q-integral that appears in the Fourier coefficient (2.3), as well as the asymptotic behavior (as k → ∞) of the factors J ν qj kν ; q 2 and J ν j kν ; q 2 appearing in η k,ν , are crucial for further developments related with convergence issues of the Fourier-Bessel expansion (2.2). To study those behaviors will be the main purpose in this work.
represents the integer part of x ∈ R. With the notation (2.5), (2.6) and assuming that 0 ≤ ω < 1, there exist functions B(ω, z) and C(ω, z) such that Under the same assumptions of the previous theorem, as z → +∞.
3 Asymptotic properties of the function J ν z; q 2 and its derivative Theorem 2 of [13, p. 12], establishes a superior bound for the asymptotic behavior of J ν qj kν ; q 2 as k → ∞. We notice that this bound can be enlarged for the cases Its prove is essentially coincident with the corresponding one of the [13] so we omit it. However, at least when µ = ν or µ = ν + 1, the above estimate for J µ qj kν ; q does not seem accurate. As a matter of fact, by Theorem 2.1 and (2.1), as k → ∞, the product qj kν is "closed" to the positive zero j k−1,ν of the function J ν z; q 2 and, as a consequence, we expect J ν qj kν ; q 2 to approach zero. So we look for a better bound when µ = ν or µ = ν + 1.
Because of the basic hypergeometric representation (1.2), in order to keep the results more general and applicable to other situations, at the final of this section we present a subsection with the corresponding main results for the function 1 φ 1 (0; ω; q, z).
3.1 Asymptotic properties of J ν z; q 2 During this subsection and to avoid any confusion, most of the times we prefer to use ∂Jν (z;q 2 ) ∂z rather than J ν z; q 2 . Proof . By the definition of the Hahn-Exton q-Bessel function (1.1) we may write Computing its derivative one gets Now, by Theorems 2.2 and 2.3, with the notation ω = q 2(ν+1) and (2.5), (2.6) with q shifted to q 2 , as z → +∞, we have Taking into account that, shifting q to q 2 and putting z = q −m+θm for m ∈ N, β q 2 z 2 = (−m + 1 + θ m )π, K q 2 z 2 = m − 1 2 − θ m , identity (3.1) gives, as m → ∞, ∂J ν z; q 2 ∂z  Also, when the sequence {mθ m } m converges to a strictly positive real number, or when it is a bounded but not convergent sequence, then it is also possible to state conditions in order to obtain conclusions.
We notice that if {θ * m } m is any sequence which satisfies 0 < θ * m < 1 for all m = 1, 2, 3, . . . and lim m→∞ mθ * m = 0 then, by (3.2), we conclude that part (i) of the Theorem 3.1 remains true for every other sequence {γ m } m such that 0 < γ m ≤ θ * m . This implies the next result. In particular, because of (2.1), it follows immediately the following corollary. We end this subsection with the following theorem.
This guarantees that, for large values of k, which proves the theorem since, trivially by Theorem 2.1, qj k,ν = q 1−k+ (ν) The following corollaries are immediate consequences of the previous theorem.
Corollary 3.7. For large values of k, the sequence (ν) k k that appears in Theorem 2.1 is strictly decreasing, i.e., there exists a positive integer k 0 such that Proof . The previous theorem guarantees that, for large values of k, qj kν > j k−1,ν , which is k−1 for large values of k.
We now prove the following theorem.
Theorem 3.9. For large values of k, Proof . On one hand, being j kν , for k = 1, 2, 3, . . ., the positive zeros of the Hahn-Exton q-Bessel function, we have On the other hand, by [8, equation (12), p. 1205], This last result was first presented in [23] and it can also be obtained in an equivalent form using directly Theorem 2.2.
Notice that, by Theorem 2.1, j k−1,ν = q −k+1+ (ν) k−1 where, as a consequence of (2.1), lim k−1 = 0. Thus, by Corollary 3.3, J ν z; q 2 is strictly monotone in each interval ]j k−1,ν , q −k+1 [, for large values of k. Now, since by Theorem 3.6, qj k,ν ∈ ]j k−1,ν , q −k+1 [, then, using (3.8) and (3.9), the theorem follows. Then, Theorem 2.3 enables one to write cos (π(−m + τ m )) + 8π ln (q) ∞ k=1q k 1 −q k e −2i(π(−m+τm)) 2 sin 2 (π(−m + τ m )) cos (π(−m + τ m )) as m → +∞, or equivalently, We notice that if {τ m } m is any sequence satisfying the condition (i) of the previous theorem then, by (3.10), the same conclusion of part (i) remains true for any other sequence {γ m } m such that 0 ≤ γ m ≤ τ m . This implies the next result.   Following the framework of [14], we rewrite the system {u m } m mentioned in the introduction as where, by (2.4), The sequence {u m } m defines a system of functions which is orthonormal with respect to the inner product defined in the L 2 q [0, 1] space by (1.5) and with the norm · L 2 q [0,1] induced by it. In this context, we are able to state the following analogue of the Riemann-Lebesgue theorem, based on a indirect proof within the scope of the inner product spaces: Alternatively, with a direct approach, we can extend the set of functions which satisfy the above property and state the following q-analogue of the Riemann-Lebesgue theorem.