Uniform convergence of basic Fourier-Bessel series on a q-linear grid

The Ramanujan Journal https://doi.org/10.1007/s11139-018-0070-3 Uniform convergence of basic Fourier–Bessel series on a q-linear grid L. D. Abreu1·R. Álvarez-Nodarse2·J. L. Cardoso3 Received: 19 October 2017 / Accepted: 16 July 2018 © The Author(s) 2018 Abstract We study Fourier–Bessel series on a q-linear grid, defined as expansions in complete q-orthogonal systems constructed with the third Jackson q-Bessel function, and obtain sufficient conditions for uniform convergence. The convergence results are illustrated with specific examples of expansions in q-Fourier–Bessel series. Keywords Hahn–Exton q-Bessel function ·Third Jackson q-Bessel function · q-Fourier series ·Basic Fourier expansions ·Uniform convergence ·q-Linear grid Mathematics Subject Classification 42C10 ·33D15 1 Introduction Based on the orthogonality relation 1 0 Jν(jνmt)Jν(jνnt)dt=0, The research of L. D. Abreu was supported by Austrian Science Foundation (FWF), through START-project FLAME (“Frames and Linear Operators for Acoustical Modeling and Parameter Estimation”, FWF Y 551-N13) and FWF project ‘Operators and Time-Frequency Analysis’ FWF P 31225-N32. The research of R. Álvarez-Nodarse was partially supported by MTM2015-65888-C4-1-P (Ministerio de Economía y Competitividad), FQM-262, FQM-7276 (Junta de Andalucía) and Feder Funds (European Union). The research of J. L. Cardoso was supported by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia - under the Project UID-MAT-00013/2013. BL. D. Abreu [email protected].ac.at J. L. Cardoso [email protected] 1Acoustics Research Institute, Austrian Academy of Sciences, Vienna, Austria 2Dep. de Análisis Matemático, Universidad de Sevilla, Sevilla, Spain 3Dep. de Matemática, Universidade de Trás-os-Montes e Alto Douro (UTAD), Vila Real, Portugal 123 L. D. Abreu et al. if m= n, where Jνstands for the Bessel functions of order νand jνnis their nth positive zero, a theory of Fourier–Bessel series was developed (see e.g. [48, §XVIII]), in a close parallelism to the classical theory of Fourier series. Hardy [33] proved that, within some boundaries, the Bessel functions are the most general functions satisfying such an orthogonality “with respect to their own zeros”, giving no space for generalizations of the theory of Fourier–Bessel series in the scope of Lebesgue measure. However, for a certain q-analogue of the Bessel function, such an extension is possible, when considering the proper measure. In the following we will use the standard notation for the q-calculus (more precisely, we will follow [31] for basic hypergeometric series, [14]forq-calculus and [39]forq-Bessel functions). The third Jackson q-Bessel function or, for some authors, the Hahn–Exton q-Bessel function, is defined as J(3) ν(z;q)≡Jν(z;q):= zν(qν+1;q)∞ (q;q)∞ ∞  k=0 (−1)kqk(k+1) 2 (qν+1;q)k(q;q)k z2k, where ν>−1 is a real parameter. The notation J(k) ν,k=1,2,3, from [35]isusedto distinguish the three q-analogues of the Bessel function defined by Jackson. We will often drop the superscript for notational convenience and simply write J(3) ν(z;q)≡ Jν(z;q). When q→1−one recovers the Bessel function after proper normalization. It is a well known fact [30,42] that Jν(z;q)satisfies the orthogonality relation 1 0 xJ ν(jnνqx;q2)Jν(jmνqx;q2)dqx=ηn,νδn,m, ηn,ν =q−1 2qν−1Jν+1(qj nν;q2)J ν(jnν;q2), (1.1) where jnν(q2)≡jnνare the positive zeros of Jν(z;q2)arranged in ascending order of magnitude, j1ν<j2ν<j3ν<···, and dqxstands for the measure of the Jackson q-integral. In the papers [22–24], a theory of Fourier series on a q-linear grid was developed, using a q-analogue of the exponential function and the corresponding q-trigonometric functions introduced by Exton [30]. This was motivated by Bustoz–Suslov orthogonality and completeness results of q-quadratic Fourier series [21]. Later, a simple argument has been found to prove such orthogonality and completeness results [37], leading, together with the solution of [38] developed in [7,8] , to a very general theory of expansions relying on expansion formulas of the plane wave type. The plane wave expansion in Gegenbauer polynomials was extended to the q-quadratic case [40] and, more recently, to the q-linear case [9] (combined with the orthogonality in [43, Theorem 4.2] this provides an alternative way to derive the results in [1,5,42]), using a general method to derive plane-wave type formulas [8]. Since several special function identities can be obtained from Fourier expansions and the associated sampling theorems (see the section by Butzer and Hauss in [34], last section of [1] and the expansions in the last section of this paper for some examples), this may be seen, 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid modulo some poetic licence, as a small contribution towards Gian-Carlo Rota’s 5th problem [46] of finding a unified approach that will give the identities satisfied by both hypergeometric and q-hypergeometric functions. This paper investigates the most delicate convergence issues of the basic Fourier– Bessel series on a q-linear grid, based on the orthogonality relation (1.1), on mean convergence results [5,6], and on the localization of the zeros jnν[4]. We will first prove that pointwise convergence associated with orthogonal discrete systems follows directly from the mean convergence. Our main contribution will be a result providing sufficient conditions for uniform convergence. Since it was proved [2], under the same general conditions imposed by Hardy, that the above orthogonality relation characterizes the functions J(3) ν(z;q2), this is the most general Fourier theory based on functions q-orthogonal with respect to their own zeros. Moreover, the third Jackson q-Bessel function provides a q-analogue of the Hankel transform with an inversion formula [44], leading to a full theory of expansions parallel to Fourier theory, including sampling and Paley–Wiener type theorems [1,3]. As a further evidence of its remarkable structure, we note in passing that the function J(3) ν(z;q2)also shows up naturally in the study of the quantum group of plane motions [41]. It should be emphasized that Ismail stimulated a considerable research activity by conjecturingpropertiesofthezerosofq-Besselfunctions,confirmedin[4,32].First,as documented in [20], the asymptotic expansion for the zeros of q-difference equations has been conjectured in a letter from Ismail to Hayman. Then, in a preprint that circulatedintheearly2000’s[36], Ismailconjecturedproperties ofthepositivezerosof q-Bessel functions. Several results followed, among which we can single out [20,32], the bounds for the zeros of the third Jackson q-Bessel function [4], the asymptotic results of [13] and the recent improvement in the corresponding accuracy [47, Prop. A.3]. Since q-series provide a wealth of examples of nontrivial functions of order zero, all these results are contributions to the intriguing and relatively overlooked topic of functions of order zero started in Littlewood’s PhD thesis, published in [45]. Since it is well known, by a result of Boas [19, Theorem 5.1], that functions of order zero cannot belong to Lp(R)(p≥1)without vanishing identically, there is no possibility of expanding them using classical Fourier analysis. Actually it is more a rule than an exception that the methods and concepts used to study functions of positive order do not suffice to study functions of order zero (this is particular notorious if the notion of type is used, which explains why classical tools succeed in [2,6] and in the study of radii of starlikeness of functions of order zero [10,17]). Such obstructions lead to the search of alternative methods and provide strong motivation for a theory of expansions in q-analogues of classical orthogonal basis functions. In this paper we will investigate series expansions Sν q[f]of functions f∈L2 q[0,1]of the form Sν q[f](x)=∞  k=1 aν k(f)x1 2Jν(qj kνx;q2).(1.2) Since the measure of L2 q[0,1]is discrete, pointwise convergence is a direct consequence of the completeness results of [5,6]. We will make some comments about this in Sect. 5of the paper. In the following we will use the notation V+ q= 123 L. D. Abreu et al. {qn:n=0,1,2,... }for the support points of the q-integral (2.3)in[0,1], and introduce the concept of a q-linear Hölder function [24, p. 103]. Definition 1.1 If two constants Mand λexist such that fqn−1−fqn≤Mqλn,n=0,1,2,..., then the function fis said to be q-linear Hölder of order λ(in V+ q∪{q−1}). If the inequality only holds for n=1,2,3,... then we say that fis q-linear Hölder of order λin V+ q. Our main result is the following sufficient conditions for uniform convergence of (1.2). Theorem 1.2 If the function f is q-linear Hölder of order λ>1in V + q, such that t−3 2f(t)∈L2 q[0,1]and the limit limx→0+f(x)=f(0+)is finite, the correspondent basic Fourier–Bessel series S(ν) q[f](x)converges uniformly to f on V + qwhenever ν>0. The following is a brief outline of the paper. In the next section, we collect the main definitions and preliminary results. The third section is devoted to the evaluation of a few finite sums. The fourth section contains a brief introduction to q-Fourier–Bessel series and the fifth section discusses pointwise convergence for systems associated with discrete orthogonality relations. We prove our main result in Sect. 6, relying on fine estimates for the coefficients of basic Fourier–Bessel series. In the last section of the paper, two examples of basic Fourier–Bessel expansions are provided. 2 Definitions and preliminary results Following the standard notations of [31], consider 0 <q<1, the q-shifted factorial for a finite positive integer nis defined as (a;q)n=(1−q)( 1−aq)···1−aqn−1,(a;q)−n=1 aq−n;qn and the zero and infinite cases as (a;q)0=1,(a;q)∞=lim n→∞(a;q)n. The symmetric q-difference operator acting on a suitable function fis defined by δqf(x)=f(q1/2x)−f(q−1/2x), (2.1) 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid hence, the symmetric q-derivative becomes δqf(x) δqx=⎧ ⎪ ⎨ ⎪ ⎩ f(q1 2x)−f(q−1 2x) (q1 2−q−1 2)x if x= 0, f(0)if x=0 and f(0)exists. (2.2) The q-integral in the interval (a,b)is defined by b a f(t)dqt=b 0 f(t)dqt−a 0 f(t)dqt(2.3) where a 0 f(t)dqt=(1−q)∞  k=0 faqkaqk. This is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points qk, with the jump at the point qkbeing q.Ifwe call this step function μq(t), then dμq(t)=dqt. One can define an inner product by setting f,g=1 0 f(t)g(t)dqt. The resulting Hilbert space is commonly denoted by L2 q(0,1). The space L2 q(0,1)is a separable Hilbert space [11]. For the properties of the more general spaces Lp q(a,b) and Lp q,ω(a,b), with p≥1, see [27]. We will also need the following formula of qintegration by parts [25, Lemma 2, p. 327], valid for a,b∈R, assuming the involved limits exist: b a gq±1 2xδqf(x) δqxdqx:= −b a fq∓1 2xδqg(x) δqxdqx +q1 2(fg)(bq−1 2)−(fg)(aq−1 2) −lim n→+∞(fg)(bq 1 2+n)−lim n→+∞(fg)(aq 1 2+n). (2.4) The third Jackson q-Bessel function has a countable infinite number of real and simple zeros [42]. In [4, Theorem 2.3] it was proved that, when q2ν+2<(1−q2)2, the positive zeros jkνof the function Jν(z;q2)satisfy jkν=q−k+(ν) k(q2)(2.5) with 0< (ν) k(q2)<α (ν) k(q2), (2.6) 123 L. D. Abreu et al. where α(ν) k(q2)=log 1−q2(k+ν)/(1−q2k) 2logq. (2.7) Using Taylor expansion it is plain that, as k→∞, α(ν) k(q2)=Oq2k. (2.8) The restriction q2ν+2<(1−q2)2can be dropped if kis chosen large enough [4, Remark 2.5, p. 4247] because (2.5)–(2.7) remain valid for every k≥k0whenever q2(k0+ν) ≤1−q21−q2k0. Hence, the following theorem holds. Theorem A For every q ∈]0,1[,k0∈N exists such that, if k ≥k0then jkν=q−k+(ν) k(q2), where 0< (ν) k(q2)<α (ν) k(q2)and α(ν) k(q2)is given by (2.7). In the remaining of the paper we will simplify the notation by setting ε(ν) k= ε(ν) kq2. The definition of basic Fourier–Bessel series on a q-linear grid depends on the following mean convergence result [5,6]. Theorem B The orthonormal sequence {uk}k≥1defined by u(ν) k(x)= x1 2Jν(jkνqx;q2)   x1 2Jν(jkνqx;q2)   is complete in L2 q(0,1). More precisely, whenever a function fis in L2 q(0,1)and 1 0f(x)u(ν) k(x)dqx= 0,k=1,2,3,..., then fqk=0,k=0,1,2,.... Thus, the orthogonal complement of the space generated by {u(ν) k}k≥1in L2 q(0,1)is {0}and any f∈L2 q(0,1)can be expanded in terms of the sequence u(ν) k(x). The following estimate [26] will also be key in the proof of the main results. Theorem C For large values of k, Jνqj kν;q2≤−q2,−q2(ν+1);q2∞ (q2;q2)∞ q(k+ν)(k−1). 3 Identities for finite sums in q-calculus In this section we gather some new and old identities for finite sums. First observe that one can rewrite the obvious identity (a;q)maqm;qk=(a;q)kaqk;qm, 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid which holds for every aand for every integers mand k,as aqm;qk (a;q)k=aqk;qm (a;q)m (3.1) valid for any a= q−j,j=0,1,2,...,and mand knonnegative integers. Proposition 3.1 For each complex a, the identity n  k=0 qka;qk (q;q)k=aq;qn (q;q)n holds for all n =0,1,2,.... Proof We argue by induction. The proposition is obvious when n=0. For the induction step, one can write the sum for n+1as n+1  k=0 qka;qk (q;q)k= n  k=0 qka;qk (q;q)k+qn+1a;qn+1 (q;q)n+1 =aq;qn (q;q)n+qn+1a;qn+1 (q;q)n+1. From the induction hypothesis, n+1  k=0 qka;qk (q;q)k=aq;qn (q;q)n1+qn+11−a 1−qn+1 =aq;qn (q;q)n 1−aqn+1 1−qn+1=aq;qn+1 (q;q)n+1.  Proposition 3.2 For each a and λin the complex plane, the identity n  k=0 q2ka q;qk (q;q)kq1+n−k+λ;q1=(q;q)1(aq;q)n (q;q)n+qλ;q1q1+n(a;q)n (q;q)n holds for all n =0,1,2,.... Proof We will perform the proof in two steps. First, using induction on n, we prove the case λ=0: n  k=0 q2ka q;qk (q;q)kq1+n−k;q1=(q;q)1(aq;q)n (q;q)n (3.2) 123 L. D. Abreu et al. and then reduce the general case to this particular one. (i) Step 1:λ=0. For n=0, the identity is trivial. For the induction step one decomposes the sum as n  k=0 q2ka q;qk (q;q)kq1+n−k;q1 = n−1  k=0 q2ka q;qk (q;q)kq1+n−k;q1+q2na q;qn (q;q)n (q;q)1 = n−1  k=0 q2ka q;qk (q;q)kqn−k;q)1+(q;q)1qn n−1  k=0 qka q;qk (q;q)k+q2na q;qn (q;q)n (q;q)1, using q1+n−k;q1=1−q1+n−k=1−qn−k+qn−k(1−q)=qn−k;q)1+ qn−k(q;q)1in the last identity. Combining the induction hypothesis with Proposition 3.1 yields n  k=0 q2ka q;qk (q;q)kq1+n−k;q1=(q;q)1(aq;q)n−1 (q;q)n−1+(q;q)1qn(a;q)n−1 (q;q)n−1 +q2na q;qn (q;q)n (q;q)1 =aq;qn−2 (q;q)n (q;q)11−aqn−11−aqn =aq;qn (q;q)n (q;q)1, leading to (3.2). (ii) Step 2:λ∈C. First notice that q1+n−k+λ;q1=1−q1+n−k+λ=1−q1+n−k+q1+n−k1−qλ and then use Proposition 3.1 and the case λ=0(3.2) of the first step.  Lemma 3.3 For each complex a and each non-negative integer i, the identity i  k=0 q2ka q;qk (q;q)k⎛ ⎝ n  j=0 qjaqi;qj q1+i;qjq1+i+j−k;q1 (q1+i+j;q)1⎞ ⎠=(aq;q)n+i (q;q)n+iq1+n;q1 q1+n+i;q1 holds for all n =0,1,2,.... 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid Proof We use induction over n. The case n=0 is precisely identity (3.2) and writing i  k=0 q2ka q;qk (q;q)k⎛ ⎝ n+1  j=0 qjaqi;qj q1+i;qjq1+i+j−k;q1 (q1+i+j;q)1⎞ ⎠ = i  k=0 q2ka q;qk (q;q)k⎛ ⎝ n  j=0 qjaqi;qj q1+i;qjq1+i+j−k;q1 (q1+i+j;q)1⎞ ⎠ +qn+1aqi;qn+1 q1+i;qn+2 i  k=0 q2ka q;qk (q;q)kqn+2+i−k;q1 the result follows from algebraic manipulations, after combining the induction hypothesis with Proposition 3.2. Remark 3.4 Using identity (3.1) Proposition 3.1 can be written as i  k=0 qkq1+k;qj−1 (q;q)j−1=q1+i;qj (q;q)j . Now we will consider the sum a(n,ν) 0:= q−nν n  i=0q2νi, where nis a nonnegative integer and νis a fixed parameter, which will show up in Sect. 6. The notation [x]to denote the greatest integer which does not exceed xwill be adopted. Lemma 3.5 For a given sequence {γλ}we have, for m =0,1,2,..., m  λ=0 a(λ,ν) 0a(m−λ,ν) 0γλ=m 2  θ=0 a(m−2θ,ν) 0m−θ  λ=θ γλ. Proof Using induction on mit can be proved that a(λ,ν) 0a(m−λ,ν) 0= λ  θ=0 a(m−2θ,ν) 0if 0 ≤λ≤m 2(3.3) holds for all m=0,1,2,.... As a consequence, a(λ,ν) 0a(m−λ,ν) 0= m−λ  θ=0 a(m−2θ,ν) 0if m 2+1≤λ≤m.(3.4) 123 L. D. Abreu et al. −q−νq2na(n−1,ν) j−1+q2(n−1)a(1,ν) 0a(n−2,ν) j−1+q2(n−2)a(2,ν) 0a(n−3,ν) j−1 =a(3,ν) 0a(n−3,ν) j−a(2,ν) 0a(n−4,ν) j −q−νq2na(n−1,ν) j−1+q2(n−1)a(1,ν) 0a(n−2,ν) j−1+q2(n−2)a(2,ν) 0a(n−3,ν) j−1. Iterating n−jtimes the same argument provides the identity a(n,ν) j=a(n−j,ν) 0a(j,ν) j−a(n−j−1,ν) 0a(j−1,ν) j−q−ν n−1−j  λ=0 q2(n−λ)a(λ,ν) 0a(n−1−λ,ν) j−1. Since a(j−1,ν) j=0 and a(j,ν) j=−q−ν+2ja(j−1,ν) j−1, a(n,ν) j=−q−ν n−j  λ=0 q2(n−λ)a(λ,ν) 0a(n−1−λ,ν) j−1, and, therefore, ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a(n,ν) j=−q2−ν"n−j λ=0q2(n−1−λ)a(λ,ν) 0a(n−1−λ,ν) j−1,j≤n; a(0,ν) 0=1;a(i,ν) −1=0 whenever i≥0; a(n,ν) j=0 whenever j>n. (6.6) The following result uses (6.6) to compute the quantities a(n,ν) jin explicit form. Proposition 6.2 An explicit expression for the polynomial coefficients a(n,ν) jis given by a(n,ν) j=(−1)jqj(j+1−ν)n−j 2  i=0 a(n−j−2i,ν) 0q2iq2j;q2i q2;q2i q21+j;q2n−j−2i q2;q2n−j−2i ×q21+n−2i;q2i q2n−j−2i+2;q2i =(−1)jqj(j+1−ν)n−j 2  i=0 a(n−j−2i,ν) 0q2iq2j;q2i q2;q2i q21+j;q2n−j−i q2;q2n−j−i ×q21+n−j−2i;q21 q21+n−j−i;q21 , with 0≤j≤n, n =0,1,2,.... 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid Proof The proof, once again carried out by induction, is a bit long and technical. We will simply present a sketch. The case n=0 is obvious. We point out that Proposition 6.2 is true for every nwhen j=0 . Let us now assume that a(k,ν) l=(−1)lql(l+1−ν)k−l 2  i=0 a(k−l−2i,ν) 0q2iq2l;q2i q2;q2iq21+l;q2k−l−2i q2;q2k−l−2i ×q21+k−2i;q2i q2k−l−2i+2;q2i holds true for k=0,1,2,...,n−1 and 0 <l≤k. Then, for 0 <j≤n,itfollows from (6.6) that a(n,ν) j=−q2−ν n−j  λ=0 q2(n−1−λ)a(λ,ν) 0(−1)j−1q(j−1)( j−ν) n−j−λ 2  i=0 a(n−j−λ−2i,ν) 0cλ,i =(−1)jqj(j+1−ν) n−j  λ=0 ⎛ ⎜ ⎜ ⎝ n−j−λ 2  i=0q2n−j−λa(λ,ν) 0a(n−j−λ−2i,ν) 0cλ,i⎞ ⎟ ⎟ ⎠ , with cλ,i=q2iq2j−1;q2i (q2;q2)iq2j;q2n−j−λ−2i (q2;q2)n−j−λ−2iq2n−λ−2i;q2i (q2)n−j−λ−2i+2;q2i . Hence a(n,ν) j=(−1)jqj(j+1−ν) n−j 2  i=0⎛ ⎝ n−j−2i  λ=0q2n−j−λa(λ,ν) 0a(n−j−λ−2i,ν) 0cλ,i⎞ ⎠. Setting γλ,i=q2n−j−λ−icλ,iin the last identity and using Lemma 3.5 yields a(n,ν) j=(−1)jqj(j+1−ν) n−j 2  λ=0q2i⎛ ⎜ ⎜ ⎝ n−j 2−i  θ=0 a(n−j−2i−2θ,ν) 0⎛ ⎝ n−j−2i−θ  λ=0 γλ,i⎞ ⎠⎞ ⎟ ⎟ ⎠ , which can be rewritten as a(n,ν) j=(−1)jqj(j+1−ν) n−j 2  i=0 a(n−j−2i,ν) 0q2iq2j;q2i q2;q2i 123 L. D. Abreu et al. ×⎛ ⎜ ⎝ i  θ=0q22θq2j−1;q2θ q2;q2θ ×⎛ ⎜ ⎝ n−j−2i  λ=0q2λq2j;q2i+λ q2;q2i+λq21+i+λ−θ;q21 q21+i+λ;q21 ⎞ ⎟ ⎠⎞ ⎟ ⎠. Finally, we use Lemma 3.3 and the proposition follows.  Remark 6.3 Notice that Proposition 6.2 also holds true for every nonnegative integers nand jsince, when j>n, then, by (6.3), both members become identically zero. Lemma 6.4 For n =0,1,2,...and ν>0fixed, the sequence %Jνq1+njkν;q2&k∈N is uniformly bounded (with respect to n). Proof One must show that there exists C, independent of kand n, such that Jνq1+njkν;q2≤C for every kand n. Using Theorem A we may write, for klarge enough, Jνq1+njkν;q2=Jνq1+n−k+(ν) k;q2. This suggests dividing the proof in two cases, according to the behavior of n−k. (i) When k−n>0 is sufficiently large then, by Corollary 3 of [26], jk−n−1,ν =q1+n−k+(ν) k−n−1<q1+n−k+(ν) k=q1+njkν<q1+n−k. Therefore, if n∈Nthen q1+njkν∈jk−n−1,ν ,q1+n−k, whenever k−n>0is sufficiently large. Now, by definition, Jνjk−n−1,ν;q2=0,(6.7) and, by virtue of (12) in [18, p. 1205], for large (positive) values of k−n, Jνq1+n−k;q2≤−q2,−q2(ν+1);q2∞ q2;q2∞ q(k−n+ν)(k−n−1).(6.8) Moreover, it follows from Theorem A that jk−n−1,ν,q1+n−k⊂q1+n−k+α(ν) k−n−1,q1+n−k. By Corollary 2 of [26], the function Jνx;q2is monotone in the interval jk−n−1,ν,q1+n−k. Hence, by (6.7) and (6.8), Jνq1+njkν;q2is bounded whenever k−nis sufficiently large (positive). 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid (ii) Now, let us consider k−nbounded above. Then n−kis bounded below. Thus, if ν>0, Jνq1+njkν;q2=Jνq1+n−k+(ν) k;q2is bounded.  6.2 Behavior of J qjk;q2 The asymptotic behavior of J νqj kν;q2when k→∞was recently investigated [26, Lemma 1], combining the asymptotic properties of the infinite q-shifted factorial (z;q)∞from [29] with the ideas developed in [47]. We provide here a more direct proof, based only on the definition of the Hahn–Exton q-Bessel function and its derivative. Lemma 6.5 For large values of k, J ν(jkν;q2)=Aν(q)q−k+ν 2−1−(ν) k2 Sk, where Aν(q)=2q2(ν+1);q2∞ q2;q2∞ q(ν−1)(ν−3) 4and lim infk→∞ |Sk|>0. Proof We will present only the main steps of the proof. Computing the derivative of the function Jν(z;q2)and setting z=jkν,gives J ν(jkν;q2)=2q2(ν+1);q2∞ q2;q2∞ ∞  n=0 (−1)nnqn(n+1) (q2(ν+1),q2;q2)njkν2n+ν−1. By Theorem A, jkν=q−k+(ν) k, and the above identity becomes J ν(jkν;q2)=Aν(q)q−k+ν 2−1−(ν) k2 Sk, (6.9) where Aν(q)=2q2(ν+1);q2∞ q2;q2∞ q(ν−1)(ν−3) 4and Sk=∞ " n=0 (−1)nnqn−k+1/2+(ν) k2 (q2(ν+1),q2;q2)n. Considering m=n−k, straightforward manipulations lead to (−1)kSk=∞  m=−k (−1)mmqm+1/2+(ν) k2 (q2(ν+1),q2;q2)m+k . Thus, if pis a positive integer, (−1)kSk=−(2p+2)  m=−k F(ν) m,k(q)+ 2p  m=−(2p+1) F(ν) m,k(q)+∞  m=2p+1 F(ν) m,k(q), 123 L. D. Abreu et al. with F(ν) m,k(q)=(−1)mmqm+1/2+(ν) k2 (q2(ν+1),q2;q2)m+k . Hence, |Sk|≥ 2p  m=−(2p+1) F(ν) m,k(q)−−(2p+2)  m=−kF(ν) m,k(q)−∞  m=2p+1F(ν) m,k(q). Since this estimate is independent of p, one can resort to (2.6)–(2.8) to show that the last two sums on the right side of the previous inequality tend to zero when k→∞. Moreover, as k→∞, 2p " m=−(2p+1) F(ν) m,k(q)=1 q2(ν+1),q2;q2∞ 2p " m=−(2p+1) (−1)mmqm+1/22 +o(1) =q1/4 q2(ν+1),q2;q2∞ 2p " m=0 (−1)m(2m+1)qm(m+1)+o(1). Therefore, lim inf k→∞ |Sk|≥q1/4 q2(ν+1),q2;q2∞ ∞  i=0 (−1)i(2i+1)qi(i+1) .(6.10) Identity (10.4.9) of Corollary 10.4.2 due to Jacobi [15, p. 500] guarantees that ∞  i=0 (−1)i(2i+1)qi(i+1)=∞ ' i=11−q2i3>0.(6.11) The lemma now follows from (6.9), (6.10) and (6.11).  Noticethattheabove lemmaimplies that J ν(jkν;q2)=Oq−k(k+ν−2)ask→∞. 6.3 Sufficient conditions Recall the notation V+ q={qn:n=0,1,2,... }for the support points of the qintegral (2.3)in[0,1]and the concept of a q-linear Hölder function in V+ qdefined in theintroduction.In[25] thefollowingupperboundforbasicFourier–Besselcoefficient (4.2) has been obtained. However, the uniform convergence of basic Fourier–Bessel expansions requires the slightly more restrictive conditions of Theorem 1.2 . 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid Theorem 6.6 If the function f is q-linear Hölder of order λ>0in V + q∪{q−1},with t−1 2f(t)∈L2 q(0,1)and the limit limx→0+f(x)=f(0+)is finite, then 1 0 tf(t)Jν(qj kνt;q2)dqt ≤(1−q)qν−1 jkνfq−1Jν+1(qj kν;q2) +(1−q)qν−3 2 jkν η 1 2 k,ν ⎛ ⎝ qν+1 2M1 (1−q)1 21−q2α1 2+qν 2−q−ν 2 q1 2−q−1 2(M2⎞ ⎠, where M1and M2are independent of k and ηk,ν is given by (4.3). The proof of the main result depends on a refinement of the above estimates. We start proving the following identity. Lemma 6.7 Let ν>0. If f is a function such that limx→0+f(x)<+∞, then 1 0 tf(t)Jν(qj kνt;q2)dqt =(1−q)qν−2fq−1Jν(qj kν;q2) j2 kν −qν−2 j2 kνqν 2−q−ν 2qν 21 0 Jν(qj kνt;q2)f(qt) tdqt −q−ν 21 0 Jν(qj kνt;q2)f(t) tdqt −qν 2qν 21 0 Jν(qj kνt;q2)f(qt)−f(t) tdqt −q−ν 21 0 Jν(qj kνt;q2)f(t)−f(t/q) tdqt, provided all q-integrals converge. Proof We will use the symmetric operator δqnotation (2.1). From (3.7) in [25, Proposition 4, p. 329], δqxνJνx;q2 δqx=q−ν 2 1−qxνJν−1q−1 2x;q2, and, using q-integration by parts (2.4) together with the assumption limx→0+f(x)= f(0+)<+∞, 1 0 tf(t)Jν(qj kνt;q2)dqt 123 L. D. Abreu et al. =(1−q)qν−1fq−1Jν+1(qj kν;q2) jkν−(1−q)qν−3 2 jkν ×)−qν 2−q−ν 2 q1 2−q−1 21 0 Jν+1(qj kνt;q2)f(t)dqt+qν 21 0 Jν+1(qj kνt;q2)f2(t)dqt*, (6.12) where f2(t):= tδqfq−1 2t δqt,(6.13) and the operator δqis given in (2.1). We will now rewrite the integrals on the right hand side of (6.12). Using (3.8) of [25, Proposition 4, p. 329], (2.4) and limx→0+f(x)= f(0+), the first integral on the right hand side of (6.12) becomes 1 0 Jν+1(qj kνt;q2)f(t)dqt=(1−q)qν−3 2 jkν1 0 Jν(qj kνt;q2) t−νδqtνfq1 2t δqtdqt. Evaluating the right hand side using (2.2 ), 1 0 Jν+1(qj kνt;q2)f(t)dqt=−qν−2 2 jkνqν 21 0 Jν(qj kνt;q2)f(qt) tdqt −q−ν 21 0 Jν(qj kνt;q2)f(t) tdqt.(6.14) The evaluation of the second q-integral on the right side of (6.12) is similar and provides 1 0 Jν+1(qj kνt;q2)f2(t)dqt =qν−1 2 (1−q)jkν ×qν 21 0 Jν(qj kνt;q2)f(qt)−f(t) tdqt −q−ν 21 0 Jν(qj kνt;q2) f(t)−ft q tdqt*.(6.15) Finally, the lemma follows by substituting (6.14) and (6.15)into(6.12) and using the identity (4.4).  Remark 6.8 To assure the convergence of all q-integrals involved in the previous lemma, one is required to assume the same sufficient conditions of Theorem 1.2. We are now in conditions to prove our main result. The beef of the proof consists of finding an n-independent (convergent) upper bound for the q-Fourier–Bessel series. 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid Proof of Theorem 1.2 As observed before, the assumptions of Theorem 1.2 allow the use of Lemma 6.7, yielding 1 0 tf(t)Jν(qj kνt;q2)dqt ≤ (1−q)qν−2f1 qJν(qj kν;q2) j2 kν +qν−2 j2 kνqν 2−q−ν 2qν 21 0 Jν(qj kνt;q2)f(qt) tdqt +q−ν 21 0 Jν(qj kνt;q2)f(t) tdqt +qν 2qν 21 0 Jν(qj kνt;q2)f(qt)−f(t) tdqt +q−ν 21 0 Jν(qj kνt;q2) f(t)−ft q tdqt*. (6.16) Employing the q-Hölder type inequality [27, Th. 3.4, p. 346] with p=2, the four q-integrals appearing on the right side of (6.16) can be estimated as follows: 1 0 Jν(qj kνt;q2)f(qt) tdqt≤1 0 tJ2 ν(qj kνt;q2)dqt 1 2 1 0 f2(qt) t3dqt 1 2 ≤η 1 2 kν1 0 f2(qt) t3dqt 1 2, (6.17) 1 0 Jν(qj kνt;q2)f(t) tdqt≤η 1 2 kν1 0 f2(t) t3dqt 1 2 ,(6.18) 1 0 Jν(qj kνt;q2)f(qt)−f(t) tdqt≤η 1 2 kν1 0 (f(qt)−f(t))2 t3dqt1 2 , (6.19) 1 0 Jν(qj kνt;q2) f(t)−ft q tdqt≤η 1 2 kν⎛ ⎜ ⎝1 0f(t)−ft q2 t3dqt⎞ ⎟ ⎠ 1 2 . (6.20) Now, by virtue of the assumption t−3 2f(t)∈L2 q[0,1], one can write the q-integral as an infinite convergent sum 1 0 f2(qt) t3dqt 123 L. D. Abreu et al. =(1−q)∞  n=0 qnf2(qn+1) q3n =(1−q)q2∞  n=0f(qn+1) qn+12 =S<+∞, (6.21) and 1 0 f2(t) t3dqt=(1−q)∞  n=0f(qn) qn2 =T<+∞. (6.22) Moreover, since fis q-linear Hölder of order λ>1inV+ q∪%q−1&, there exist constants Uand Wsuch that 1 0 (f(qt)−f(t))2 t3dqt=(1−q)∞  n=0fqn+1−fqn2 q2n ≤(1−q)U2∞  n=0 q2αn q2n=(1−q)U2 1−q2(α−1),(6.23) and 1 0f(t)−ft q2 t3dqt=(1−q)∞  n=0fqn−fqn−12 q2n ≤(1−q)W2∞  n=0 q2αn q2n=(1−q)W2 1−q2(α−1).(6.24) The constants S≡Sq(f),T≡Tq(f),U≡Uq(f)and W≡Wq(f)in (6.21), (6.21), (6.23) and (6.24) are independent of k. Notice also that the extra condition involving the point q−1can be removed (or neglected) since it only affects the choice of the constant W. Introducing inequalities (6.21), (6.22), (6.23) and (6.24) into inequalities (6.17), (6.18), (6.19) and (6.20), respectively, and using (6.16), gives: 1 0 tf(t)Jν(qj kνt;q2)dqt≤ (1−q)qν−2f1 qJν(qj kν;q2) j2 kν +qν−2η 1 2 k j2 kνqν 2−q−ν 2qν 2√S+q−ν 2√T +qν 2⎛ ⎝qν 2+(1−q)U2 1−q2(α−1)+q−ν 2+(1−q)W2 1−q2(α−1)⎞ ⎠⎤ ⎦, 123 Uniform convergence of basic Fourier–Bessel series on a q-linear grid or, equivalently, 1 0 tf(t)Jνqj kνt;q2dqt≤1 j2 kν.C1Jνqj kν;q2+C2ηk,ν1 2, where C1and C2depend on f,νand q, but not on k. Thus, the absolute value of the kth term in (6.1) can be bounded as follows: a(ν) k(f)Jνqn+1jkν;q2 =1 ηk,ν 1 0 tf(t)Jνqj kνt;q2dqtJνqn+1jkν;q2 ≤⎧ ⎨ ⎩ C1Jνqj kν;q2 j2 kνηk,ν +C2 j2 kνη 1 2 k,ν ⎫ ⎬ ⎭Jνq1+njkν;q2 ≤⎧ ⎨ ⎩ qC1 jkνJ νjkν;q2+q1 2C2 j 3 2 kνJ νjkν;q2 1 2Jνqj kν;q2 1 2⎫ ⎬ ⎭Jνq1+njkν;q2, (6.25) using (4.3). To deal with the first term in (6.25), one uses Theorem A and Lemmas 6.4 and 6.5 to assure the existence of a constant M>0, independent of kand n, such that Jνq1+njkν;q2 jkνJ νjkν;q2≤Mqk. By Proposition 6.1, the second term in (6.25) reads Jνq1+njkν;q2 j 3 2 kνJ νjkν;q2 1 2Jνqj kν;q2 1 2=Jνqj kν;q2 1 2 j 3 2 kνJ νjkν;q2 1 2Pnj2 kν;q. From (6.2), Proposition 6.7 and (6.4) it follows that, for all n, lim k→∞ Pnj2 kν;q a(n,ν) n(j2 kν)n=1,and Pnj2 kν;q=Oqn(n+1−ν) j2n kνas k→∞. Combining Theorems A and C with Lemma 6.5, this allows to bound the second term in (6.25), Aqk2+νk+n(n+1−ν)−2kn =Aqn−k−ν−1 22−(ν−1)2 4qk≤Bqk, since (ν) k(q)>0 and (ν) k(q)=O(q2k)when k→∞(see (2.8) and (2.6)). Since the constants Aand Bare positive and independent of kand n, we finally conclude that a(ν) k(f)Jνqn+1jkν;q2=Oqkas k→∞. 123