SAMPLING THEOREMS ASSOCIATED WITH SINGULAR Q-STURM ...

SAMPLING THEOREMS ASSOCIATED WITH SINGULAR Q-STURM LIOUVILLE PROBLEMS M.H. ANNABY1 , H.A. HASSAN2 AND Z. S. MANSOUR3

Abstract. In this paper we introduce three sampling theorems for transformations defined in terms of Jackson q-integration when the kernel of the transformation is a solution or the Green’s function of singular q-Sturm–Liouville problems. We consider the problem when the q-Sturm– Liouville problem is singular either at infinity or zero with detailed investigations when the singular point is infinity. This approach allows the derivation of sampling representations for transforms whose kernels are linear combinations of q-Bessel functions, not just a single one as previously established.

1. Introduction and Notations Let 0 < q < 1, we say that A is q-geometric if for every x ∈ A, qx ∈ A. Let f be a real or complex valued function defined on a q-geometric set A. The q-difference operator is defined by Dq f (x) :=

f (x) − f (qx) , x(1 − q)

x 6= 0.

(1.1)

If 0 ∈ A, the q-derivative at zero is defined by f (xq n ) − f (0) , x ∈ A, (1.2) n→∞ xq n if the limit exists and does not depend on x. Since the formulation of self-adjoint eigenvalue problems requires the definition of Dq−1 , we define Dq−1 in a same manner to be  −1   f (x) − f (q x) , x ∈ A \ {0} ,  −1 x(1 − q ) Dq−1 f (x) :=    Dq f (0), x = 0, Dq f (0) := lim

provided that Dq f (0) exists. A right inverse, q-integration, of the q-difference operator Dq is defined by F. H. Jackson, cf. [14], Z x ∞ X f (t)dq t := x(1 − q) q n f (xq n ), x ∈ A, (1.3) 0

n=0

provided that the series converges. In general, Z b Z b Z f (t)dq t := f (t)dq t − a

0

a

f (t)dq t,

a, b ∈ A.

0

In [10], W. Hahn defined the q-integration for a function f over [0, ∞) by Z ∞ ∞ X f (t) dq t = (1 − q) q n f (q n ). 0

Let

L2q (0, ∞)

−∞

be the space of all functions, f , defined on [0, ∞) for which Z ∞ |f (t)|2 dq t < ∞, 0

2000 Mathematics Subject Classification. 34B24, 39A13, 94A20. Key words and phrases. Singular q-Sturm–Liouville equations, Sampling theory, q-Bessel function. The work of Z.S. Mansour is supported by King Saud University, Riyadh, through grant DSFP/ MATH 01. 1

2

ANNABY, HASSAN AND MANSOUR

and we associate with L2q (0, ∞) the inner product defined by Z ∞ hf, gi := f (x)g(x) dq x, f, g ∈ L2q (0, ∞).

(1.4)

0

Hence L2q (0, ∞) is a Hilbert space. Let Cq2 (0, ∞) be the space of all functions y defined on [0, ∞[ such that y(·) and Dq y(·) are continuous at zero. In [1], a q-Bessel analogue of the sampling theorem is derived by considering q-Hankel transform whose kernel contains the third Jackson q-Bessel function Jν(3) (x; q)

n(n+1) ∞ (q ν+1 ; q)∞ X q 2 n x2n , := x (−1) (q; q)∞ n=0 (q; q)n (q ν+1 ; q)n

ν

ν > −1,

see [11, 13]. The sampling theorem reads; Theorem 1.1. If g(·) ∈ L2q (0, 1) and Z f (x) =

1

1

(xt) 2 g(t)Jν(3) (xt; q 2 )dq t,

(1.5)

0

then f can be represented in the form 1

∞ X

(3)

2(xqjnν (q 2 )) 2 Jν (xq −1 ; q 2 ) i h f (x) = f (qjnν (q )) 2 (q 2 )) d J (3) (xq −1 ; q 2 ) (x2 − q 2 jnν ν n=0 dx 2

.

(1.6)

x=qjnν (q 2 )

(3)

where jnν (q 2 ) are the positive zeros of Jν (x; q 2 ). The convergence of (1.6) is uniform on any compact subset of (0, ∞). The sampling representation (1.6) is slightly different from that in [1] which has minor error; the value of the third Jackson q-Bessel function at xq −1 not at x. This paper is organized as follows. The next section contains the preliminary notations and notions from q−calculus in addition to the main results of [5] which will be needed in the remaining sections. In section 3, we derive sampling theorems associated with kernel arising from solutions or Green’s function of the singular q-Sturm– Liouville problems and in section 4, we give an example involving Jackson q−Bessel function but (3) the sampled points will be only the positive zeros of Jν (x; q). 2. q singular problems To the best we know, the q-analogue of the singular Sturm–Liouville problem are firstly introduced in [5]. The authors of [5] studied the singular q-Sturm Liouville problem 1 Lx y := − Dq−1 Dq y(x) + u(x)y(x) = λy(x), q y(0) cos α + Dq−1 y(0) sin α = 0,

0 6 x < ∞,

λ∈C

(2.1) (2.2)

where u(·) is a real-valued function defined on [0, ∞[ and continuous at zero and Dq is the q-difference operator defined in (1.1) below. They followed Titchmarsh technique established in [18, 19] to define the limit point and the limit circle classification in the q-setting and to give a classification criterion. They also constructed Green’s function and derive eigenfunction expansion theorem. We state here in brief the limit and the limit circle classification in the q-setting and the main results of [5] which we shall need throughout our investigation of the q-sampling theorem. Let α ∈ R be fixed and {φ(·, λ), θ(·, λ)} be a fundamental set of solutions of (2.1) that satisfy φ(0, λ)

=

cos α,

Dq−1 φ(0, λ) = sin α,

(2.3)

θ(0, λ)

=

sin α,

Dq−1 θ(0, λ) = − cos α.

(2.4)

For each n ∈ N, let lq−n (λ, z) be the transformation defined by lq−n (λ, z) = −

φ(q −n , λ)z + Dq φ(q −n , λ) . θ(q −n , λ)z + Dq θ(q −n , λ)

SAMPLING AND q-SINGULAR PROBLEMS

3

It is proved in [5] that lq−n (λ, z) is a one to one conformal mapping in z for every λ. Therefore if =(λ) 6= 0, then lq−n (λ, z) varies on a circle Cq−n (λ) with a finite radius. They also proved that n o Cq−n (λ) is a decreasing sequence, where Cq−n (λ) is the set composed of the circle Cq−n (λ) and its interior. Set \ Cq−n (λ). C∞ (λ) := n∈N

Then x = ∞ is a limit point case or a limit circle case according to C∞ is a point or a circle. The treatment when the singular point at zero is similar. In this case we shall consider the conformal mapping lqn (λ) and Cqn (λ) instead of lq−n (λ) and Cq−n (λ). In this case we define C0 to be \ C0 (λ) := Cqn (λ), n∈N

and x = 0 is a limit point or a limit circle case according to C0 is a point or a circle, respectively. For every non real λ, an L2q (0, ∞) solution of (2.1) is ψ(x, λ) = φ(x, λ) + m(λ) θ(x, λ),

(2.5)

where m(λ) is the limit point or any point on the limit circle case. This limit represents an analytic function in the upper and lower half of the λ−plane and it has poles on the real axis, they are simple. Also m(λ) = m(λ). The upper and lower forms of m(λ) will be assumed to be analytic continuation of each other through this work. Hence m(λ) is a meromorphic function which has all singularities on the real axis as simple poles. Let {λn }∞ n=0 be the poles of m(λ) with residues {rn }∞ . These residues are positive and the poles are the eigenvalues of (2.1)–(2.2) in the limit n=0 point case or limit circle case. Therefore the solution (2.5) can be defined for all λ 6= λn . The eigenvalues of (2.1)–(2.2) form an increasing sequence with ∞ as its unique limit point, see [5]. In the following lemma we summarize all results of [5] which are essential in our investigations. Lemma 2.1. (1) The functions n = 0, 1, · · ·

θn (x) := θ(x, λn )

are the eigenfunctions corresponding to the eigenvalues λn , and they consist orthogonal basis in L2q (0, ∞), and kθn (·)k2 =

1 , rn



1 . λ − λn

 ψ(·, λ), θn (·) =

(2.6)

(2) If u(x, λ) is a solution of the initial value problem Lx u = λu,

u(0) = γ,

Dq u(0) = δ,

0 6 x < ∞.

in the limit circle case, then for any λ0 ∈ C, =(λ0 ) 6= 0, (  ∞ X ku(x, λ)k 6 ku(·, λ0 )k 1 + |λ − λ0 |n n=1

εn n!q n−1

(2.7)

) ,

(2.8)

where ε is a positive constant. (3) The solution of Lx y − λy(x) = f (x),

0 6 x < ∞,

λ∈C

y(0) cos α + Dq−1 y(0) sin α = 0, in the limit point or limit circle cases is Z ∞ y(x, λ, f ) = G(x, qy, λ)f (qy) dq y, 0

 x ∈ 0, q −n , n ∈ N ,

(2.9) (2.10)

4

ANNABY, HASSAN AND MANSOUR

where G(x, y, λ) is the Green’s function of (2.9). So in our notation   ψ(x, λ)θ(y, λ), 0 6 y 6 x < ∞, G(x, y, λ) =  θ(x, λ)ψ(y, λ), 0 6 x 6 y < ∞. (4) For any two functions y and z in Cq2 (0, ∞), we have the Green’s identity Z x (yLt z − z Lt y) dq t = Wq (y, z)(0) − Wq (y, z)(xq −1 ), for all x ∈ (0, ∞),

(2.11)

(2.12)

0

where Wq (y, z)(x) := y(x)Dq z(x) − z(x)Dq y(x), is the q-Wronskian, cf. [4, 16]. (5) For fixed x0 = q −m0 , m0 ∈ N and fixed non real λ we have for some positive constant K, Z ∞ Z x0 Z ∞ 2 2 2 2 |G(x0 , t, λ)| dq t = |ψ(x0 , λ)| |θn (t)| dq t + |θn (x0 )| |ψ(y, λ)|2 dq y 6 K, (2.13) 0

0

x0

This means that the function G(x0 , ·, λ) is L2q (0, ∞)−function. Also Z ∞ θn (x) . G(x, y, λ) θn (y) dq y = λn − λ 0 Hence

∞ √ X rn θn (x) 2 kG(x, ·, λ)k = λn − λ 6 K.

(2.14)

=(m(λ)) , v

(2.15)

2

n=0

(6) For non-real λ Z 0

∞

|ψ(x, λ)|2 dq x = −

where v = =(λ).

Remark 2.2. Since ψ(x, λ) is defined for any λ 6= λn , and m(λ) is real when λ is real, the relation (2.15) is also valid when λ is real, λ 6= λn , in the sense Z ∞ ∂  |ψ(x, λ)|2 dq x = − =(m(λ)) . ∂v 0 v=0 This implies that kψ(·, λ)k is uniformly bounded on any compact subset of C which does not contain any of the λn ’s. Similarly kG(x0 , ·, λ)k and the series in (2.14) are uniformly bounded on each compact subset of C which is clear from (2.13). 3. Sampling Theorems In this section we give three sampling theorems. The first two theorems are valid in both of limit point and limit circle cases and the last one is only valid in the limit circle case. These theorems are the q-analogy of the classical case, see [20–22]. We state and prove here the theorems in detail when infinity is the singular point. Therefore all the transformations are defined on (0, ∞) and the Hilbert space under consideration is L2q (0, ∞). As we mentioned in section 1, the treatment when the q-Sturm–Liouville problem is singular at zero is similar. But the transformation will be defined on (0, a), a > 0, and the Hilbert space under consideration is L2q (0, a).

SAMPLING AND q-SINGULAR PROBLEMS

5

Let {λn }∞ n=1 be the nonzero eigenvalues of the the problem (2.1)-(2.2), and we set λ0 = 0 if zero is an eigenvalue. Consider the infinite product   ∞  Y λ   , if zero is not an eigenvalue, 1−    λn  n=1 P (λ) = (3.1)   ∞   Y  λ   , if zero is an eigenvalue. 1−  λ λn n=1 If the product in (3.1) does not converge, we will multiply the product by  
1
2
p 1 exp λ/λn + λ/λn + · · · + λ/λn , 2 p
p+1 P∞ where p is the smallest integer that satisfies n=1 1/λn converges. Consider the function   P (λ) ψ(x, λ),

Ψ(x, λ) =

 

x ∈ [0, ∞),

λ ∈ C − {λn }∞ n=0 (3.2)

0

P (λn ) rn θn (x),

x ∈ [0, ∞),

λ = λn .

Note that the definition of Ψ(x, λn ) is a limit case as λ → λn . Since ψ(·, λ) and θn (·) are L2q (0, ∞)−functions, then Ψ(·, λ) is L2q (0, ∞)−function for any λ ∈ C. Theorem 3.1. Let Z

∞

F1 (λ) =

g(x) Ψ(x, λ) dq x,

g ∈ L2q (0, ∞).

(3.3)

0

Then F1 (λ) admits the expansion F1 (λ) =

∞ X

F1 (λn )

n=0

P (λ) . (λ − λn )P 0 (λn )

(3.4)

The series (3.4) converges absolutely on C and uniformly on compact subsets of C. Proof. First we show that (3.3) is well defined. Since Ψ(·, λ) is L2q (0, ∞)−function, then using Cauchy-Schwarz inequality for (3.3) we have |F1 (λ)|2 ≤ kg(·)k2 kΨ(·, λ)k2 , which is defined for any λ ∈ C. Using Parseval’s equality for (3.3), we get F1 (λ) =

∞ X hΨ(·, λ), θn (·)i hθn (·), g(·)i 0

Now using (3.2) we obtain Z F1 (λn ) =

kθn (·)k2

.

(3.5)

∞

g(x) P 0 (λn ) rn θn (x) dq x = P 0 (λn ) rn hθn (·, λ), g(·)i .

(3.6)

0

Also from (2.6) we have rn P (λ) hΨ(·, λ), θn (·)i = . 2 kθn (·)k λ − λn

(3.7)

Combining (3.5),(3.6) and (3.7), we obtain (3.4). The uniform convergence can be achieved as follows. From Remark 2.2 and equation (3.2), for any compact subset D of C , there is a positive constant K such that kΨ(·, λ)k ≤ K,

for all λ ∈ D.

(3.8)

6

ANNABY, HASSAN AND MANSOUR

Therefore since g ∈ L2q (0, ∞), then using Cauchy-Schwarz’ inequality we obtain N ∞ X X hΨ(·, λ), θn (·)i hθn (·), g(·)i P (λ) F (λ) − ≤ F (λ ) 1 1 n (λ − λn )P 0 (λn ) kθn (·)k2 n=1 n>N !1/2 X !1/2 ∞ X hΨ(·, λ), θn (·)i 2 hθn (·), g(·)i 2 ≤ kθn (·)k kθn (·)k n>N n>N !1/2 2 X hθn (·), g(·)i ≤ K −→ 0, as N → ∞, kθn (·, λn )k n>N

independent of λ, which shows the uniform convergence. For the absolute convergence since Ψ, g are L2q (0, ∞)−functions, then we get ∞ ∞ X X hΨ(·, λ), θn (·)i hθn (·), g(·)i P (λ) = F1 (λn ) 0 (λ ) 2 (λ − λ )P kθ (·)k n n n n=0 n=0 !1/2 ∞ !1/2 ∞ X X hθn (·), g(·)i 2 hΨ(·, λ), θn (·)i 2 ≤ kθn (·)k kθn (·)k n=0 n=0 = kΨ(·, λ)k kg(·)k < ∞, for any λ ∈ C, and the proof is complete.  Now we give a sampling theorem associated with Green’s function. Let x0 = q −m0 , m0 ∈ N be fixed and consider the function Φ(y, λ) = P (λ) G(x0 , y, λ), which will be considered as limit at λ = λn . The function Φ(·, λ) is L2q (0, ∞)−function. Theorem 3.2. Let Z F2 (λ) =

∞

g ∈ L2q (0, ∞).

g(y) Φ(y, λ) dq y,

(3.9)

0

Then F2 (λ) admits the expansion F2 (λ) =

∞ X n=0

F2 (λn )

P (λ) . (λ − λn )P 0 (λn )

(3.10)

The series (3.4) converges absolutely on C and uniformly on compact subsets of C. Proof. Again we can show as in Theorem 3.1 that (3.9) is well defined. Using Parseval’s equality for (3.9), we get ∞ X hΦ(·, λ), θn (·)i hθn (·), g(·)i F2 (λ) = . (3.11) kθn (·)k2 n=0 From Lemma 2.1, the equation
Lx − λI θn (x) = (λn − λ)θn (x),

λ 6= λn ,

has the solution ∞

Z

G(x, qy, λ)(λn − λ)θn (qy) dq y =

θn (x) = 0

1 (λn − λ) q

Z

∞

G(x, y, λ)θn (y) dq y, 0

then D E D E q P (λ)θ (x ) n 0 Φ(·, λ), θn (·) = P (λ) G(x0 , ·, λ), θn (·) = . (λn − λ)

(3.12)

SAMPLING AND q-SINGULAR PROBLEMS

7

Since G(x0 , y, λ) is L2q (0, ∞)−function, then using (3.12) we obtain  ∞

∞ X X G(x0 , ·, λ), θn (·) q θn (x0 ) G(x0 , y, λ) = θ (y) = θ (y). n 2 2 (λ − λ) n kθ (·)k kθ (·)k n n n n=0 n=0 Therefore F2 (λk ) = lim P (λ) hG(x0 , ·, λ), g(·)i λ→λk

= lim

λ→λk

∞ E X q θn (x0 ) P (λ) D (λ − λk ) θ (·), g(·) n λ − λk kθn (·)k2 (λn − λ) n=0

(3.13)

 q θk (x0 )

θk (·), g(·) . 2 kθk (·)k

= −P 0 (λk )

Combining (3.11), (3.12) and (3.13), we obtain (3.10). For the uniform convergence, since P (λ) is an entire function and its zeros are the eigenvalues, then we get on compact subset D of C, ∞ √ X rn θn (x0 )P (λ) 2 ≤ K, for all λ ∈ D, (λ − λn ) n=0 for some non-negative constant K, cf. Remark 2.2. Thus N ∞ X X hθn (·), g(·)i √rn θn (x0 )P (λ) P (λ) F2 (λn ) F2 (λ) − ≤ kθn (·)k2 (λ − λn )P 0 (λn ) (λ − λn ) n=1 n>N !1/2 !1/2 ∞ √ X X hθn (·), g(·)i 2 rn θn (x0 )P (λ) 2 ≤ kθn (·)k (λ − λn ) n>N n>N !1/2 X hθn (·), g(·)i 2 −→ 0, as N → ∞. ≤ K kθn (·, λn )k n>N

The absolute convergence can be achieved in similar way as in Theorem 3.1.



It is proved in [5], that if x = ∞ is in the limit circle case, then all the solutions of (2.1)–(2.2) are in the limit circle case. Therefore the following theorem holds only on the limit circle case. In this theorem the kernel is the function θ(·, λ) and the integral transform will clearly defined for all λ ∈ C. Theorem 3.3. Let Z F3 (λ) :=

∞

g(x) θ(x, λ) dq x,

g ∈ L2q (0, ∞).

0

Then F3 (λ) admits the expansion F3 (λ) =

∞ X

F3 (λn )

n=0

Wn (λ) (λ − λn )Wn0 (λn )

(3.14)

where Wn (λ) := − lim Wq (θ, θn )(xq −m−1 ), m→∞

x > 0.

If the function θ(x, λ) is of order zero as a function of λ, then the function Wn (λ) can be replaced by P (λ) in (3.14) Moreover the series (3.14) converges absolutely on C and uniformly on compact subset of C. Proof. Using Parseval’s equality for (3.3), we get F3 (λ) =

∞ ∞ X hθ(·, λ), θn (·)i hθn (·), g(·)i X hθ(·, λ), θn (·)i = F3 (λn ) . 2 kθ (·)k kθn (·)k2 n n=0 n=0

(3.15)

8

ANNABY, HASSAN AND MANSOUR

In Green’s identity (2.12), set y = θ(x, λ) and z = θn (x). Then we have for any x > 0 and m ∈ N Z xq−m (λ − λn ) θ(t, λ) θn (t) dq t = Wq (θ, θn )(0) − Wq (θ, θn )(xq −m−1 ). (3.16) 0

Since θ(·), θn (·) satisfy the same initial conditions (2.4), then Wq (θ, θn )(0) = 0. Since θ(t, λ) and θn (t) are L2q (0, ∞)- functions, then the right hand side of (3.16) is convergent. Consequently calculating the limit in (3.16) as m → ∞ gives (λ − λn ) hθ(·, λ), θn (·)i = − lim Wq (θ, θn )(xq −m−1 ) = Wn (λ). m→∞

(3.17)

From which we get 2

kθn (·)k = Wn0 (λn ).

(3.18)

Using (3.17) and (3.18) in (3.15), we obtain (3.14). Now assume that for each fixed x, θ(x, λ) is a function of order zero. We shall show that in this case Wn (λ) can be replaced by P (λ). It is clear from (3.17) that every eigenvalue λm , m ∈ N, is a zero of Wn (λ). Also if Wn (λ∗ ) = 0, λ∗ 6= λm , then θ(·, λ∗ ) satisfies the conditions (2.4) and lim Wq (θ, θn )(xq −m−1 ) = 0.

m→∞

Thus θ(·, λ∗ ) satisfies the problem (2.1)-(2.2) in the limit circle case, see [5], and then θ(·, λ∗ ) is an eigenfunction of the problem, which is a contradiction. Therefore the zeros of Wn (λ) are only the eigenvalues. Since θ(x, λ) and θn (x) are of order zero, so is W (λ) and hence we obtain from Hadamard Factorization Theorem that Wn (λ) = αn P (λ), for some nonzero constant αn . The statement is now clear since Wn (λ)/Wn0 (λn ) = P (λ)/P 0 (λn ). Since kθ(·, λ)k is bounded on compact subset D of C from (2.8), then the absolute and uniform convergence proof is similar to the one of Theorem 3.1 and is omitted.  4. Applications As an application on the sampling expansion theorems of this paper, we shall study the second order q-difference equation ! ν+ 21 ν− 21 (1 − q )(1 − q ) 1 q 1−ν s2 − q −1 y(qx) = 0, 0 < x ≤ 1. (4.1) Dq2 y(x) + (1 − q)2 x2 Here the function u(·) has a singularity at x = 0 when ν is any real number different from ±1/2 and the classification of the singular point is as we mentioned in Section 1. The Hilbert space of consideration in this case is the space L2q (0, 1). It is worthy to mention that if we replace x by q −1 x in (4.1) and use that 2 −1 Dq,q x) = Dq−1 ,x Dq,x y(x), −1 x y(q then (4.1) can be written as 1 Dq−1 Dq y(x) + (1 − q)2

1

q

1

(1 − q ν+ 2 )(1 − q ν− 2 ) s −q x2

1−ν 2

! y(x) = 0.

The latter equation is in the form of Equation (2.1). A fundamental set of solutions of (4.1) is 1

y1 (x) = x 2 Jν(3) (sx; q),

1

y2 (x) = x 2 Yν (sx; q),

√ where s := λ is defined with respect to the principal branch and Yν (x; q) is the function defined for ν 6∈ Z by o Γq (ν)Γq (1 − ν) n ν/2 (3) Yν (x; q) = q cos πνJν(3) (x; q) − J−ν (xq −ν/2 ; q) , (4.2) π and for n ∈ Z, Yn (x; q) = lim Yν (x; q). ν→n

SAMPLING AND q-SINGULAR PROBLEMS

9

Also we have, q ν(ν−1) (1 + q) , πx Let θ(x, λ), φ(x, λ) be the solutions which satisfy Wq (Jν (·; q 2 ), Yν (·; q 2 ))(x) ≡

x ∈ R \ {0} .

φ(1, λ) = 1, Dq φ(1, λ) = 0, θ(1, λ) = 0, Dq θ(1, λ) = −1. Hence n o 1 1 θ(x, λ) = c1 q − 2 x 2 Jν(3) (sx; q 2 )Yν (s; q 2 ) − Yν (sx; q 2 )Jν(3) (s; q 2 ) , n o 1 θ(x, λ) φ(x, λ) = c1 x 2 s Jν(3) (sx; q 2 )Dq Yν (s; q 2 ) − Yν (sx; q 2 )Dq Jν(3) (s; q 2 ) + √ , (1 + q) where c1 :=

(4.3)

πq −ν(ν−1) . 1+q

In the following we shall give the sampling expansion when ν ∈ (0, 1) − {1/2}, and when ν ∈ (1, ∞) in two different cases. We start with the case of ν ∈ (1, ∞). Case I: ν > 1 : In the above notation, we have (3)

m(λ) ψ(x, λ)

1

= −sq 2

Dq Jν (s; q 2 ) (3) Jν (s; q 2 )

−

1 √ (1 + q)

= φ(x, λ) + m(λ)θ(x, λ) √ (3) c1 s xJν (sx; q 2 ) = Wq (Jν(3) (·; q 2 ), Yν (·; q 2 ))(x) (3) Jν (s; q 2 ) =

√ Jν(3) (sx; q 2 ) , x (3) Jν (s; q 2 ) 1

1

= q − 2 c1 x 2 Yν(3) (sn ; q 2 )Jν(3) (sn x; q 2 ). √ (3) (3) √ (3) The eigenvalues λn are the zeros of ( λ)−ν Jν ( λ; q 2 ) = s−ν Jν (s; q 2 ). Since s−ν Jν (s; q 2 ) is an entire function of order zero its zeros are all simple, we have by Hadamard Factorization Theorem  ∞  Y s2 P (λ) = 1 − 2 = β s−ν Jν(3) (s; q 2 ), sn n=1 √ −ν (3) Ψ(x, λ) = β x s Jν (s x; q 2 ), θ(x, λn )

(3)

2 2 where {sn }∞ n=1 are the positive zeros of Jν (s; q ), sn = λn , and β is a constant. Write the 2 transform (3.3) as F (s) instead of F1 (λ) = F1 (s ). Thus Z 1 √ F (λ) = g(x) β x s−ν Jν(3) (s x; q 2 ) dq x, g ∈ L2q (0, 1), (4.4) 0

we obtain F (s) =

∞ X

(3)

F (sn )

n=1

2 sν+1 s−ν Jν (s; q 2 ) n h i (3) d (s2 − s2n ) dx Jν (x; q 2 )

.

x=sn

For h(x) = β g(x), redefine the transform (4.4) as Z 1 √ 1 sν+ 2 F (s2 ) = h(x) s x Jν(3) (s x; q 2 ) dq x := f (s),

(4.5)

0

which is the transform (1.5) and hence has the sampling representation f (s) =

∞ X n=1

(3)

f (sn )

2(s sn )1/2 Jν (s; q 2 ) h i (3) d (s2 − s2n ) dx Jν (x; q 2 )

x=sn

,

(4.6)

10

ANNABY, HASSAN AND MANSOUR

which is another representation for the transform (1.5) but the sampled points are different. The sampling expansion in (4.6) leads to the following corollaries. Corollary 4.1. (3)

Jν (qs; q 2 ) (3) Jν (s; q 2 )

=

∞ X

(3)

J (qsn ; q 2 ) h ν i (3) d 2) J (x; q ν dx

n=1

2 sn , (s2 − s2n )

(4.7)

x=sn

Proof. In (4.5) choose h(x) =

Thus we get f (s) =

√

(3) s Jν (qs; q 2 ).

 1    q √q(1 − q)

x = q,

   0

otherwise.

Substituting in (4.6), we obtain (4.7).



Corollary 4.2. (3)

Jν+1 (s; q 2 ) (3) Jν (s; q 2 )

=

∞ X n=1

(3)

J (sn ; q 2 ) i h ν+1 (3) d 2 dx Jν (x; q )

(s2

2s , − s2n )

(4.8)

x=sn

Proof. It is shown in [1] that the function 1

f (s) = sν−u+ 2 Ju(3) (s; q 2 ) ∈ P Wqν ,

where get

P Wqν

u > ν,
is the space of such functions of the form (1.5) or (4.5) . Substituting in (4.6), we (3)

sν−u

Ju (s; q 2 ) (3)

Jν (s; q 2 )

=

∞ X

(3)

2 sν−u+1 J (sn ; q 2 ) n i h u (3) 2 2 2 d n=1 (s − sn ) dx Jν (x; q )

.

x=sn

Put u = ν + 1, we obtain (4.8).



Remark 4.3. Equation (4.8) is similar to Equation (4.12) in [1]. But Equation (4.12), and hence Equation (4.3), in [1] are not true, one can check that the right side of the Equation (4.12) has √ singularities at x = q jnν (q), but the right one has singularities at jnν (q). In fact these equations can be corrected simply by achieving the modification we have done in (1.6). Furthermore, (3) Equation (4.3) is true only if |x| < j1ν (q), the smallest positive zero of Jν (s; q). Case II: 0 < ν < 1, ν 6= 21 : Here m(λ) = −sq

1 2

qν

2

/2 −ν

s

(3)

(3)

Dq Jν (s; q 2 ) − csν Dq J−ν (sq −ν/2 )

(3) q ν 2 /2 s−ν Jν (s; q 2 )

(3) csν J−ν (sq −ν/2 )

− where c is any constant. The eigenvalues are the zeros of qν

2

/2 −ν

s

−

1 √ , (1 + q)

(3)

Jν(3) (s; q 2 ) − csν J−ν (sq −ν/2 ).

(4.9)

Denoting these by λn . Since (4.9) is an entire function of order zero and its zeros are all simple, we have  2  (3) P (λ) = β2 q ν /2 s−ν Jν(3) (s; q 2 ) − csν J−ν (sq −ν/2 ) , where β2 is a constant. We use the sampling transform as in Theorem 3.3. Hence for the transform Z 1 n o 1 1 F (λ) = g(x) c1 q − 2 x 2 Jν(3) (xs; q 2 )Yν (s) − Yν (xs)Jν(3) (s; q 2 ) dq x, g ∈ L2q (0, 1), 0

we obtain F (λ) =

∞ X n=1

F (λn )

 2  (3) (3) β2 q ν /2 s−ν Jν (s; q 2 ) − c sν J−ν (s q −ν/2 ; q 2 ) (s2 − s2n )P 0 (λn )

.

SAMPLING AND q-SINGULAR PROBLEMS

11

References [1] L.D. Abreu. A q-samlping theorem related to the q-Hankel transform. Proc. Amer. Math. Soc, 133:1197–1203, 2005. [2] L.D. Abreu. Functions q-orthogonal with respect to their own zeros. Proc. Amer. Math. Soc, 134:2695–2701, 2006. [3] L.D. Abreu. Sampling theory associated with q-difference equations of the Sturm-Liouville type. J. Phy. A, 38:10311–10319, 2005. [4] M.H. Abu-Risha, M.H. Annaby, M.E.H. Ismail, and Z.S. Mansour. Linear q-difference equations. Z. Anal. Anwend., 26:481–494, 2007. [5] M. H. Annaby, Z. S. Mansour and I. A. Soliman, q−Titchmarsh-Weyl theory: Series Expansion, accepted for publications in Math Nach. [6] M.H. Annaby, J. Bustoz and M.E.H. Ismail. On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. J. Comput. Appl. Math., 206:73–85, 2007. [7] M.H. Annaby, Z.S. Mansour and O.A.Ashour. Sampling theorems associated with biorthogonal q−Bessel functions. J. phy. A, 43(10):15 pages , 2010. [8] M.H. Annaby. q−type sampling theorems. Results Math., 206:214–225, 2003. [9] W.N. Everitt, G. Sch¨ ottler and P. L. Butzer, Sturm-Liouville boundary value problems and Lagrange interpolation series, Rend. Math. Appl. 14: 87–126, 1994. [10] W. Hahn. Beitr¨ age zur Theorie der Heineschen Reihen. Math. Nachr., 2:340–379, 1949. [11] M.E.H. Ismail. The zeros of basic Bessel functions, the functions Jν+ax (x) and associated orthogonal polynomials. J. Math. Anal. Appl., 86:11–19, 1982. [12] M.E.H. Ismail and A.I. Zayed. A q-analogue of the Whittaker-Shannon-Kotel’nikov sampling theorem. Proc. Amer. Math. Soc., 131:3711–3719, 2003. [13] F.H. Jackson. The applications of basic numbers to Bessel’s and Legendre’s equations. Proc. Lond. Math. Soc., 2(2):192–220, 1905. [14] F.H. Jackson. On q-definite integrals. Quart. J. Pure and Appl. Math., 41:193–203, 1910. [15] B.M. Levitan and I.S. Sargsjan. Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators. Amer. Math. Soc., Providence, 1975. [16] R.F. Swarttouw. The Hahn-Exton q-Bessel Function. Ph. D thesis, The Technical University of Delft, 1992. [17] R.F. Swarttouw and H.G. Meijer. A q-analog of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation. Proc. Amer. Math. Soc., 120(3):855–864, 1994. [18] E. Titchmarsh. Eigenfunction Expansions Asociated with Second Order Differential Equations. Clarendon Press, 1962. [19] E.C. Titchmarsh. On the uniqueness of the Greens function associated with a second order differential equations. Canadian J. Math., 1:191–198, 1949. [20] A.I. Zayed, On integral transforms whose kernels are solutions of singular Sturm-Liouville problems, Proc. Roy. Soc. Edinburgh Sect. A 108 (1988), no. 3-4, 201228. [21] A.I. Zayed, G. Hinsen and P.L. Butzer, On Lagrange interpolations and Kramer-type sampling theorems associated with Sturm-Liouville problems, SIAM J. Appl. Math. 50:893–909, 1990. [22] A.I. Zayed, Ahmed and G.G. Walter, On the inversion of integral transforms associated with Sturm-Liouville problems, J. Math. Anal. Appl. 164(1):285-306, 1993. 1 Department of Mathematics, statistics & Physics, Qatar University, P.O. Box 2713 Doha, Qatar. Qatar, 2 Department of Mathematics, Faculty of Basic Education, PAAET, Shamiya, Kuwait, 3 Department of Mathematics, Faculty of Science, King Saudi University, Riyadh, P. O. Box 2455, Riyadh 11451, Kingdom of Saudi Arabia E-mail address: [email protected], [email protected], [email protected]