On the Zeros of the Big q-Bessel Functions and Applications
Department of Mathematics, College of Sciences, King Saudi University, P. O Box 2455, Riyadh 11451, Saudi Arabia
Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, Zarzouna 7021, Tunisia
Department of Mathematics, Faculty of Science, King Saud University, Riyadh 11451, Saudi Arabia
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 237; https://doi.org/10.3390/math8020237
Received: 2 December 2019 / Revised: 29 January 2020 / Accepted: 31 January 2020 / Published: 13 February 2020
This paper deals with the study of the zeros of the big q-Bessel functions. In particular, we prove new orthogonality relations for functions which are similar to the one for the classical Bessel functions. Also we give some applications related to the sampling theory.
The classical Bessel functions which are defined by satisfy the orthogonality relationswhere are the zeros of .
Moreover, a function can be represented as the Fourier-Bessel serieswhereIn the literature there are many basic extensions of the Bessel functions . The oldest one was introduced by Jackson in 1903–1905 and rewritten in modern notation by Ismail . Other q-analogues can be obtained as formal limits of the three q-analogues of Jacobi polynomials; i.e., of little q-Jacobi polynomials, big q-Jacobi polynomials and Askey-Wilson polynomials. For this reason we propose to speak about little q-Bessel functions, big q-Bessel functions and AW type q-Bessel functions for the corresponding limit cases.
Recently, Koelink and Swarttouwn established orthogonality relations for the little q-Bessel (see [3,4]). Other orthogonality relations for Askey-Wilson functions were founded by Bustoz and Suslov (see, ). In this paper we discuss a new orthogonality relations for the big q-Bessel functions In this work we show that all zeros of the big q-Bessel function are real and simple. Further, using a similar technique as Bergweiler-Hayman  we derive an explicit asymptotic formula for these zeros, which is denoted :In signal processing it is known that the space of band-limited signals is characterized as the set of all functions of whose Fourier transforms have supports contained in , see [8,9]. The classical sampling theorem of Whittaker-Kotelnikov-Shannon (WKS), states that band-limited functions can be recovered from their values at the integers. In this work we provide a q-version of the sampling theorem of Whittaker-Kotelnikov-Shannon, and q-type band-limited signals which are defined in terms of Jackson’s q-integral. The sampling points are the zeros of .
The paper is organized as follows: in Section 2, we define the big q-Bessel function, we give some recurrence relations and we prove that the big q-Bessel function is an eigenfunction of a q-difference equation of second order. Section 3, is devoted to study of the zeros of the big q-Bessel functions. In Section 4, we show that the set of functions is a complete orthogonal system in . Finally, in the last section, we give a q-version of the sampling theorem in the points .
2. The Big q-Bessel Functions
For the convenience of the reader, we provide a summary of the notations and definitions used in this paper.
Let and ; the q-shifted factorials are defined by :We also denoteThe basic hypergeometric series is defined by The q-integral of a continuous function f on is defined byWe introduce q-integration by parts. This will involve backward and forward q-derivativesand provided exists.
If f and g are continuous on thenThe big q-Bessel functions are defined by For , the functions are analytic in in their variables x and and satisfy
The big q-Bessel functions satisfy the following recurrence difference relationswhere
A simple computation shows thatHence,Then, we obtain after making the change in the second member of (15).On the other hand, from the following relationThen we get□
The trigonometric functions and are related to the Bessel function bySimilarly, there are two q-trigonometric functions associated to the big q-Bessel function givenandIn particular, we have
The big q-Bessel function is solution of the q-difference equation:where is defined in (14).
The big q-Bessel functions satisfy the recurrence relations
3. On the Zeros of the Big q-Bessel Functions
By using a similar method, as in , we prove in this section that the big q-Bessel function has infinite simple zeros on the real line, and by an explicit evaluation of a q-integral, we establish new orthogonality relations for this function.
Let and For every , we have
Let and . The zeros of the function are real.
Suppose is a zero of We haveFor Equation (3) yieldsNow if and only if or , then in all other cases we haveUsing the definition of the q-integral we getthen,and defines an analytic function on . Hence, Now if , with , then we haveFor this expression cannot be zero, which proves the corollary. □
To obtain an expression for the q-integral in Equation (3) with we use l’Hopital’s rule. The result isThis formula is reduced tofor a real zero of
The non-zero real zeros of with are simple zeros.
Let be a non-zero real zero of with . The integralis strictly positive. If it were zero, this would imply that the big q-Bessel function is identically zero as in the proof of Corollary 1. Hence, (21) implies thatwhich proves the lemma. □
For and , the big q-Bessel function has infinitely many zeros.
We havewithBy , Theorem 1.2.5, it suffices to show that .
Since , we havethere exists such that and for we haveThe Jacobi ’s triple identity (see, ) leads toSet for and . ClearlyWe haveandadditionally,HencewhereThis implies. □
For we order the positive zeros of asNext, we derive an explicit asymptotic formula for the zeros of the big q-Bessel function , using the same technique of Bergweiler-Hayman  and Annaby-Mansour [13,14]. We start with two preliminary results.
If and are the positive zeros of then we have for the sufficiently large n,
The proof is similar to the proof of Theorem 2.1 in  and is omitted. □
Additionally, we can prove the following
If and are the positive zeros of , then we have for sufficiently large n,
In order to investigate the asymptotic of the functions , we define the suitable sets of annuli in terms of the zeros such that for every , intersects with only on the boundary. The sets of annuli are constructed such that the annulus contains only the zeros of . Then we study the behavior of in when n is large enough. LetThen is a decreasing nonnegative sequence. Moreover, from (22) we havefor sufficiently large n. That is, . ThereforeWe define the positive sequences and :andwhere . We can easily verify thatPutdividing the region into annuli with common boundaries. Now we introduce the asymptotic of in the set of annuli , when m is large enough.
Assume that and , , is the annulus defined in (29). Then we have the asymptotic relationuniformly when m is sufficiently large.
The proof is similar to the proof of Theorem 3.2 in  and is omitted. □
Let be fixed, and let and . The asymptoticholds true with
Using the transformation, see ,we haveThis representation shows that is also an entire function in x and . Using thatwe obtainUsing also thatwe getwhere□
4. Orthogonality Relation and Completeness
The Proposition 3 and Relation (3) are useful to state the orthogonality relations for the big q-Bessel functions.
Let and be the positive zeros of the big q-Bessel function Then
We consider the inner product giving byLetand
If and then is entire of order 0.
We first show that is entire of order 0. From the definition of the q-integral, we haveThe series (36) converges uniformly in any disk . Hence is complete and we haveSince we have that
Both the numerator and the denominator of are entire functions of order 0. If we write a factor of and as canonical products, each factor of that divides out with a factor of by hypothesis is thus entirety of order 0. □
For the quotient is bounded on the imaginary axis.
We will make use of the simple inequalitiesandWe get for real,Thus, we have□
For the system is complete in .
Lemma 3 implies that is bounded. Since is entire of order 0, we can apply one of the versions of the Phragmén-Lindelöf theorem, see  and Lemma 3 and conclude that is bounded in the entire -plane. Next, by Liouville’s theorem we conclude that is constant. Say that . We will prove that Indeed, we haveandIt follows thatDividing to common factors, we haveand letting giveshence,We can now conclude thatorWe complete the proof with a simple argument that gives
IfthenLetting gives . Then, dividing by and again letting gives Continuing this process, we have which completes the proof. □
Using the orthogonality Relation (35), we consider the big q-Fourier-Bessel series, , associated with a function f,with the coefficients given bywhere
5. Sampling Theorem
The classical Kramer sampling is as follows [8,16]. Let be a function, continuous in such that, as a function of x, for every real number , where I is an interval of the real line. Assume that there exists a sequence of real numbers , with n belonging to an indexing set contained in such that is a complete orthogonal sequence of functions of Then for any F of the formwhere we havewithThe series (39) converges uniformly wherever is bounded.
Now we give a q-sampling theorem for the q-integral transform of the form
Let f be a function in . Then the q-integral transformhas the point-wise convergent sampling expansionThe series (41) converges uniformly over any compact subset of .
Set and is the k-th positive zero of and is a complete orthogonal sequence of function in Then we getBut is analytic on so is bounded on any compact subset of , and hence is bounded. Substituting from (3) with we obtained (42) and the theorem follows. □
Define a function f on :ThenThus, applying Theorem 7 givesWe define the Paley-Wiener space related to the big q-Bessel function bywhere the finite big q-Hankel transform is defined byBy quite similar arguments to those in the proof of , Theorem 1, we see that the space equipped with the inner productis a Hilbert space, and the finite big q-Hankel transform (44) becomes a Hilbert space isometry between and . Therefore, from (, Theorem A) we deduce that the big q-Bessel function has an associated reproducing kernel.
Formal analysis, F.B., H.B.M. and M.G.; Methodology, F.B., H.B.M. and M.G.; Writing—original draft, F.B., H.B.M. and M.G.; Writing—review and editing, F.B., H.B.M. and M.G. All authors have read and agreed to the published version of the manuscript.
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this research group (RG-1437-020).
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saudi University for funding this research group (RG-1437-020). We thank M.E.H. Ismail for his valuable comments during the work in this paper.
Conflicts of Interest
The authors declare no conflict of interest.
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