On the zeros of the big q -Bessel functions and applications
aa r X i v : . [ m a t h . C V ] N ov ON THE ZEROS OF THE BIG q -BESSEL FUNCTIONS ANDAPPLICATIONS FETHI BOUZEFFOUR AND HANEN BEN MANSOUR
Abstract.
This paper deals with the study of the zeros of the big q -Bessel functions.In particular, we prove a new orthogonality relations for this functions similar to the onefor the classical Bessel functions. Also we give some applications related to the samplingtheory. introduction The classical Bessel functions J α ( x ) which are defined by [14] J α ( x ) := ( x/ α Γ( α + 1) F (cid:18) − α + 1 ; − x (cid:19) , satisfy the orthogonality relations(1.1) Z J α ( j αn x ) J α ( j αm x ) xdx = δ nm J α +1 ( j αn x ) , where { j αn } n ∈ N are the zeros of J α ( x ).Moreover, a function f ∈ L ((0 , xdx ) , can be represented as the Fourier-Bessel series(1.2) f ( x ) = ∞ X n =0 c n J α ( j αn x ) , where(1.3) c n = 2 J α +1 ( j αn x ) Z f ( x ) J α ( j αn x ) xdx. In the literatures there are many basic extension of the Bessel functions J α ( x ), the oldestone have been introduced by Jackson in 1903 − q -analogues can be obtained as formal limit of the three q -analoguesof Jacobi polynomials, i.e., of little q -Jacobi polynomials, big q -Jacobi polynomials andAskey-Wilson polynomials for this reason we propose to speak about little q -Bessel func-tions, big q -Bessel functions and AW type q -Bessel functions for the corresponding limitcases.Recently, Koelink and Swartouw establish an orthogonality relations for the little q -Bessel(see [12]). Another orthogonality relations for Askey-Wilson functions founded by Bustozand Suslov (see, [6]). In this paper we shall discuss a new orthogonality relations for thebig q -Bessel functions [7](1.4) J α ( x, λ ; q ) = φ (cid:18) − /x q α +2 (cid:12)(cid:12)(cid:12)(cid:12) q ; λ x q α +2 (cid:19) . Mathematics Subject Classification.
Key words and phrases.
Basic hypergeometric functions, Completeness of sets of functions,interpolation.
In section 2, we define the big q -Bessel function, we give some recurrence relations andwe prove that the big q -Bessel is an eigenfunction of a q -difference equation of secondorder. The section 3, is devoted to study the zeros of the big q -Bessel functions. Insection 4, we show that if { j αn } ∞ n =1 are the zeros of J α +1 ( a, λ ; q ), the set of functions { J α +1 ( a, j αn ; q ) } ∞ n =1 is a complete orthogonal system in L q ((0 , t ( − t q ; q ) ∞ ( − t q α +4 ; q ) ∞ d q t ). Finallyin the last section, we give a version of the sampling theorem in the points j αn .2. The big q -bessel functions Throughout this paper we will fix q ∈ ]0 , a ∈ C and n ∈ N , the q -shifted factorials are defined by [8](2.1) ( a ; q ) := 1 , ( a ; q ) n := n − Y i =0 (1 − aq i ) , and(2.2) ( a , a , ..., a k ; q ) n := k Y j =1 ( a j ; q ) n . We also denote ( a ; q ) ∞ = lim n →∞ ( a ; q ) n . The basic hypergeometric series r φ s is defined by [8](2.3) r φ s (cid:18) a , a , ..., a r b , b , ..., b s (cid:12)(cid:12)(cid:12)(cid:12) q, z (cid:19) := + ∞ X k =0 ( a , a , ..., a r ; q ) k ( q, b , b , ..., b s ; q ) k ( − k (1+ s − r ) q (1+ s − r ) ( k ) z k . whenever the series converges.The q -derivative D q f ( x ) of a function f is defined by(2.4) ( D q f )( x ) = f ( x ) − f ( qx )(1 − q ) x , if x = 0 , and ( D q f )(0) = f ′ (0) provided f ′ (0) exists.The q -integral of a function f from 0 into a is defined by(2.5) Z a f ( t ) d q t = (1 − q ) a ∞ X n =0 f ( aq n ) q n . The q -integration by parts formula is given by(2.6) Z ba D q − [ g ( x )] f ( x ) d q x = q [ f ( x ) g ( q − x )] ba − q Z ba D q ( f ( x )) g ( x ) d q x. The big q -Bessel functions are defined by [7] J α ( x, λ ; q ) = φ (cid:18) − /x q α +2 (cid:12)(cid:12)(cid:12)(cid:12) q ; λ x q α +2 (cid:19) , (2.7) = ∞ X k =0 ( − k q k )+2 k ( α +1) ( q , q α +2 ; q ) k ( − x ; q ) k ( λx ) k . N THE ZEROS OF THE BIG q -BESSEL FUNCTIONS AND APPLICATIONS 3 For α > −
1, the functions J α ( x, λ ; q ) are analytic in C in their variables x and λ andsatisfying(2.8) lim q → J α ( x − q , (1 − q ) λ ; q ) = F (cid:18) − α + 1 ; − (2 λx ) (cid:19) = j α (2 λx )where j α ( . ) is the normalized Bessel function of order α given by j α ( x ) = ( x/ − α Γ( α + 1) J α ( x ) , = F (cid:18) − α + 1 ; − x (cid:19) . . Proposition 2.1.
The big q -Bessel functions satisfy the following recurrence differencerelations D q (cid:2) J α ( x, λ ; q ) (cid:3) = − λ q α +2 x (1 − q )(1 − q α +2 ) J α +1 ( x, λ ; q ) , (2.9) D q − (cid:2) ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, λ ; q ) (cid:3) = − x (1 − q α +2 )( − x q ; q ) ∞ (1 − q − )( − x q α +2 ; q ) ∞ J α ( x, λ ; q ) . (2.10) Proof.
A simple calculation show that D q (cid:2) ( − x ; q ) k x k (cid:3) = (1 − q k )(1 − q ) ( − x ; q ) k − x k − . Hence, D q (cid:2) J α ( x, λ ; q ) (cid:3) = ∞ X k =1 ( − k q k )+2 k ( α +1) ( q , q α +2 ; q ) k λ k D q (cid:2) ( − x ; q ) k x k (cid:3) , = 1(1 − q )(1 − q α +1 ) ∞ X k =1 ( − k q k )+2 k ( α +1) ( q , q α +4 ; q ) k − λ k ( − x ; q ) k − x k − . (2.11)Then, we obtain after making the change k → k + 1 in the second member of (2.11). D q (cid:2) J α ( x, λ ; q ) (cid:3) = − λ q α +2 x (1 − q )(1 − q α +2 ) ∞ X k =0 ( − k q k )+2 k ( α +2) ( q , q α +4 ; q ) k ( − x ; q ) k ( λx ) k , = − λ q α +2 x (1 − q )(1 − q α +2 ) J α +1 ( x, λ ; q ) . On the other hand from the following relation D q − [ ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ ( − x ; q ) k x k ] = q − k ( q α +2+2 k − − q − ( − x q ; q ) ∞ ( − x q α +2 ; q ) ∞ ( − x ; q ) k x k +1 , we obtain D q − (cid:2) ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, λ ; q ) (cid:3) = − x (1 − q α +2 )( − x q ; q ) ∞ (1 − q − )( − x q α +2 ; q ) ∞ ∞ X k =0 ( − k q k )+2 k ( α +1) ( q , q α +2 ; q ) k ( − x ; q ) k ( λx ) k , = − x (1 − q α +2 )( − x q ; q ) ∞ (1 − q − )( − x q α +2 ; q ) ∞ J α ( x, λ ; q ) . (cid:3) FETHI BOUZEFFOUR AND HANEN BEN MANSOUR
In the classical case the trigonometric functions sin( x ) and cos( x ) are related to theBessel function J α ( x ) by cos( x ) = r πx J − ( x ) , sin( x ) = r πx J ( x ) . Similarly, there are two big q -trigonometric functions associated to the big q -Bessel func-tion given cos( x, λ ; q ) = J − / ( x, λ ; q ) , = ∞ X k =0 ( − k q k )+ k ( q ; q ) k ( − x ; q ) k ( λx ) k , and, sin( x, λ ; q ) = 11 − q J / ( x, λ ; q ) , = ∞ X k =0 ( − k q k )+3 k ( q ; q ) k +1 ( − x ; q ) k ( λx ) k . We have D q (cid:2) cos( x, λ ; q ) (cid:3) = − λ qx − q sin( x, λ ; q ) ,D q − (cid:2) ( − x q ; q ) ∞ ( − x q ; q ) ∞ sin( x, λ ; q ) (cid:3) = − xq (1 − q ) ( − x q ; q ) ∞ ( − x q ; q ) ∞ cos( x, λ ; q ) . Let L be the linear q -difference operator defined by Lf ( x ) = ( − x q α +2 ; q ) ∞ ( − x q ; q ) ∞ x D q − (cid:2) ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ x D q f ( x ) (cid:3) . Theorem 2.2.
The big q -Bessel function is solution of the q -difference equation: Lf = − λ q α +3 (1 − q ) f. Proof.
By (2.9) we have J α +1 ( x, λ ; q ) = − − qλ q α +2 x D q (cid:2) J α ( x, λ ; q ) (cid:3) , and by (2.10) we can be write( − x q α +2 ; q ) ∞ ( − x q ; q ) ∞ x D q − (cid:2) ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ x D q [ J α ( x, λ ; q )] (cid:3) = − λ q α +3 (1 − q ) J α ( x, λ ; q ) . (cid:3) Proposition 2.3.
The big q -Bessel functions satisfy the recurrence relations i ) J α +1 ( x, λ ; q ) = (1 − q α +2 ) λ q α ( q α +2 x + 1) (cid:2) (1 − q α − λ q α x ) J α ( x, λ ; q ) − (1 − q α ) J α − ( x, λ ; q ) (cid:3) ,ii ) J α +1 ( xq − , λ ; q ) = (1 − q α +2 ) λ q α (1 + x ) (cid:2) (1 − q α ) J α ( x, λ ; q ) − J α − ( x, λ ; q ) (cid:3) . N THE ZEROS OF THE BIG q -BESSEL FUNCTIONS AND APPLICATIONS 5 Proof. i ) By (2.10) and (2.9) we get(2.12) (1 + x q α +2 ) J α ( xq, λ ; q ) − (1 + x q ) J α ( x, λ ; q ) = − x q (1 − q α ) J α − ( xq, λ ; q ) , and J α ( xq, λ ; q ) = J α ( x, λ ; q ) + λ x q α +2 − q α +2 J α +1 ( x, λ ; q ) . Hence, J α +1 ( x, λ ; q ) = (1 − q α +2 ) λ q α ( q α +2 x + 1) (cid:2) (1 − q α − λ q α x ) J α ( x, λ ; q ) − (1 − q α ) J α − ( x, λ ; q ) (cid:3) .ii ) Similarly, the equality (2.9) gives us(2.13) J α ( x, λ ; q ) = J α ( xq, λ ; q ) − λ x q α +2 − q α +2 J α +1 ( x, λ ; q ) , then by (2.12) and (2.13), we obtain J α +1 ( xq − , λ ; q ) = (1 − q α +2 ) λ q α (1 + x ) (cid:2) (1 − q α ) J α ( x, λ ; q ) − J α − ( x, λ ; q ) (cid:3) . (cid:3) On the zeros of the big q -Bessel functions By using a similar method as [12], we prove in this section that the big q -Bessel functionhas infinity of simple zeros on the real line and by an explicit evaluation of a q -integral,we established a new orthogonality relations for this function. Proposition 3.1.
Let α > − and a > . For every λ, µ ∈ C \ { } , we have (3.1) ( λ − µ ) Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, λ ; q ) J α +1 ( x, µ ; q ) d q x = (1 − q )(1 − q α +2 ) q α +2 ( − a ; q ) ∞ ( − a q α +2 ; q ) ∞ (cid:2) J α +1 ( aq − , µ ; q ) J α ( a, λ ; q ) − J α +1 ( aq − , λ ; q ) J α ( a, µ ; q ) (cid:3) . Proof.
Using the q -integration by parts formula (2.6) and relations (2.9) and (2.10), weget Z a x ( − x q ; q ) ∞ ( − x q α +2 ; q ) ∞ J α ( x, λ ; q ) J α ( x, µ ; q ) d q x = 1 − q − q α +2 (cid:2) ( − x ; q ) ∞ ( − x q α +2 ; q ) ∞ J α +1 ( xq − , λ ; q ) J α ( x, µ ; q ) (cid:3) a + λ q α +2 (1 − q α +2 ) Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, λ ; q ) J α +1 ( x, µ ; q ) d q x. Then the interchanging of λ and µ in the last equation, yields a set of two equations,which can be solved easily. (cid:3) Corollary 3.2.
Let α > − and a > . The zeros of the function J α ( a, λ ; q ) are real.Proof. Suppose λ = 0 is a zero of λ → J α ( a, λ ; q ) . We have J α ( a, λ ; q ) = J α ( a, λ ; q ) = 0 . FETHI BOUZEFFOUR AND HANEN BEN MANSOUR
The formula (3.1) with µ = λ yields( λ − λ ) Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | J α +1 ( x, λ ; q ) | d q x = 0 . Now λ = λ if and only if λ ∈ R or λ ∈ i R , then in all other cases we have(3.2) Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | J α +1 ( x, λ ; q ) | d q x = 0 . Using the definition of the q -integral we get Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | J α +1 ( x, λ ; q | d q x = (1 − q ) a ∞ X k =0 q k ( − a q k +2 ; q ) ∞ ( − a q k +2 α +4 ; q ) ∞ | J α +1 ( aq k , λ ; q ) | , then J α +1 ( aq k , λ ; q ) = 0 , k ∈ N ,and J α +1 ( ., λ ; q ) defines an analytic functions on C . Hence, J α +1 ( ., λ ; q ) = 0 . Now if λ = iµ , with µ ∈ R we have J α ( a, iµ ; q ) = ∞ X k =0 q k )+2 k ( α +1) ( q , q α +2 ; q ) k ( − a ; q ) k ( aµ ) k for α > − (cid:3) To obtain an expression for the q -integral in formula (3.1) with λ = µ, we use l’Hopital’srule. The result is Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ (cid:0) J α +1 ( x, λ ; q ) (cid:1) d q x = ( q − − q α +2 )( − a ; q ) ∞ λq α +2 ( − a q α +2 ; q ) ∞ h J α ( a, λ ; q ) (cid:2) ∂J α +1 ∂µ ( aq − , µ ; q ) (cid:3) µ = λ − J α +1 ( aq − , λ ; q ) (cid:2) ∂J α ∂µ ( a, µ ; q ) (cid:3) µ = λ i . This formula simplifies to(3.3) Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ ( J α +1 ( x, λ ; q )) d q x = (1 − q )(1 − q α +2 )2 λq α +2 ( − a ; q ) ∞ ( − a q α +2 ; q ) ∞ J α +1 ( aq − , λ ; q ) (cid:2) ∂J α ∂µ ( a, µ ; q ) (cid:3) µ = λ , for λ = 0 a real zero of J α ( a, λ ; q ) . Lemma 3.3.
The non-zero real zeros of λ → J α ( a, λ ; q ) , with α > − are simple zeros.Proof. Let λ be a non-zero real zero of J α ( a, λ ; q ) , with α > −
1. The integral Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | J α +1 ( x, λ ; q ) | d q x = Z a x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ ( J α +1 ( x, λ ; q )) d q x is strictly positive. If it were zero, this would imply that the big q -Bessel function isidentically zero as in the proof of Corollary 3.2 . Hence, (3.3) implies that (cid:2) ∂J α ∂µ ( a, µ ; q ) (cid:3) µ = λ = 0 , which proves the lemma. (cid:3) N THE ZEROS OF THE BIG q -BESSEL FUNCTIONS AND APPLICATIONS 7 Recall that the order ρ ( f ) of an entire function f ( z ) , see [3, 10], is given by ρ ( f ) = lim sup r →∞ ln ln M ( r ; f )ln r , M ( r ; f ) = max | z |≤ r | f ( z ) | . Lemma 3.4.
For α > − and a ∈ R , the big q -Bessel function λ → J α ( a, λ ; q ) hasinfinitely many zeros.Proof. We have J α ( a, λ ; q ) = ∞ X n =0 a n q n ( λa ) n , with a n = ( − n q (2 α +1) n ( q , q α +2 ; q ) n ( − a ; q ) n . By (Theorem 1.2.5, [10]) it suffices to show that ρ ( J α ( a, λ ; q )) = 0.Since α > − /
2, we have lim n →∞ a n = 0 , then there exist C > | a n | < C, and for | λ | < r, we have M ( r, λ → J α ( a, λ ; q )) ≤ C ∞ X n =0 q n ( ar ) n ,< C + ∞ X n = −∞ q n ( ar ) n . The Jacobi ’s triple identity leads M ( r, λ → J α ( a, λ ; q )) < C ( q , − ( ar ) q, − q/ ( ar ) ; q ) ∞ . Set r = q − ( N + ε ) a , for − ≤ ε < and N = 0 , , , ... . Clearly( − ( ar ) q ; q ) ∞ = ( − q − N + ε )+1 ; q ) ∞ , = ( − q − ε q − N ; q ) ∞ . We have ( − ( ar ) q ; q ) ∞ = ( − q − ε ; q ) ∞ ( − q − ε − N ; q ) N and ( − ( ar ) q ; q ) ∞ = q − ( N +2 Nε ) ( − q ε +1 ; q ) N ( − q − ε ; q ) ∞ , ≤ q − ( N +2 Nε ) ( − q ε +1 ; q ) ∞ ( − q − ε ; q ) ∞ , also ( − q/ ( ar ) ; q ) ∞ = ( − q N + ε )+1 ; q ) ∞ , = ( − q ) ∞ . Hence ln ln M ( r, λ → J α ( a, λ ; q ))ln q − ( N + ε ) , ≤ ln ln Aq − N ( N +2 ε ) ln q − ( N + ε ) , = ln N ( N + 2 ε ) + ln (cid:0) ln AN ( N +2 ε ) − ln q (cid:1) − ( N + ε ) ln q , FETHI BOUZEFFOUR AND HANEN BEN MANSOUR with A = C ( − q − ε , − q ε , − q ) ∞ . Which implies ρ ( λ → J α ( a, λ ; q )) = 0. (cid:3) Let α > − we order the positive zeros of λ → J α ( a, λ ; q ) as0 < j α < j α < j α < ... . Orthogonality relation and completeness
The Proposition 3.1 and relation (3.3) with a = 1 are useful to state the orthogonalityrelations for the big q -Bessel functions. Proposition 4.1.
Let α > − and < j α < j α < j α < ... the positive zeros of the big q -Bessel function J α (1 , λ ; q ) then Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, j αn ; q ) J α +1 ( x, j αm ; q ) d q x (4.1) = (1 − q )(1 − q α +2 )2 λq α +2 ( − q ) ∞ ( − q α +2 ; q ) ∞ J α +1 ( q − , j αn ; q ) (cid:2) ∂J α ∂µ (1 , µ ; q ) (cid:3) µ = j αn δ n,m . We consider the inner product giving by < f, g > = Z t ( − t q ; q ) ∞ ( − t q α +4 ; q ) ∞ f ( t ) g ( t ) d q t. Let G ( λ ) = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ g ( x ) J α +1 ( x, λ ; q ) d q x, and h ( λ ) = G ( λ ) J α (1 , λ ; q ) . Lemma 4.2. If α > − and g ( x ) ∈ L q ((0 , x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x ) then h ( λ ) is entire oforder .Proof. We first show that G ( λ ) is entire of order 0. From the definition of the q -integralwe have(4.2) G ( λ ) = (1 − q ) ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k ) J α +1 ( q k , λ ; q ) q k . The series in (4.2) converges uniformly in any disk | λ | ≤ R . Hence G ( λ ) is entire and wehave M ( r ; G ) ≤ M ( r ; λ → J α +1 ( x, λ ; q )) Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | g ( x ) | d q x. Since ρ ( λ → J α +1 ( x, λ ; q )) = 0 we have that ρ ( G ) = 0 . Both the numerator and the denominator of h ( λ ) are entire functions of order 0. Ifwe write a factor of G ( λ ) and J α (1 , λ ; q ) as canonical products, each factor of J α (1 , λ ; q )divides out with a factor of G ( λ ) by the hypothesis of theorem 4.4 h ( λ ) is thus entire oforder 0. (cid:3) N THE ZEROS OF THE BIG q -BESSEL FUNCTIONS AND APPLICATIONS 9 Lemma 4.3.
The quotient J α +1 ( q m ,λ ; q ) J α (1 ,λ ; q ) is bounded on the imaginary axis for m ∈ N . Proof.
We will make use of the simple inequalities( − q − m ; q ) n q nm = n − Y j =0 ( q m + q j ) ≤ ∞ Y j =0 (1 + q j )= ( − q ) ∞ , m ∈ N , − q n +2 α +2 > − q α +2 , and ( − q ) n ≥ . We get for λ = iµ, µ real, J α +1 ( q m , iµ ; q ) = ∞ X n =0 q n )+2 n ( α +2) ( q , q α +4 ; q ) n ( − q − m ; q ) n q nm µ n , ≤ ∞ X n =0 q n )+2 n ( α +1) ( q , q α +2 ; q ) n − q α +2 − q α +2+2 n ( − q ) ∞ µ n ,< ∞ X n =0 q n )+2 n ( α +1) ( q , q α +2 ; q ) n ( − q ) ∞ µ n .J α (1 , iµ ; q ) = ∞ X n =0 q n )+2 n ( α +1) ( q , q α +2 ; q ) n ( − q ) n µ n , ≥ ∞ X n =0 q n )+2 n ( α +1) ( q , q α +2 ; q ) n µ n . Thus we have 0 ≤ J α +1 ( q m , iµ ; q ) J α (1 , iµ ; q ) ≤ ( − q ) ∞ . (cid:3) Theorem 4.4.
Let α > − and g ( x ) ∈ L q (0 , . If Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ g ( x ) J α +1 ( x, j αn ; q ) d q x = 0 , n = 0 , , , ... , then g ( q m ) = 0 for m = 0 , , , ... .Proof. Lemma 4.2 implies that h ( iµ ) is bounded. Since h ( λ ) is entire of order 0, we canapply one of the versions of the Phragm´en-Lindel¨of theorem, see [3] and Lemma 4.2 andconclude that h ( λ ) is bounded in the entire λ -plane. Next by Liouville’s theorem weconclude that h ( λ ) is constant. Say that h ( λ ) = C . We will prove that C = 0. We have G ( λ ) = CJ α (1 , λ ; q ) , G ( λ ) = Z a ( − a q ; q ) ∞ ( − a q α +4 ; q ) ∞ g ( a ) J α +1 ( a, λ ; q ) d q a, = (1 − q ) ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k ) J α +1 ( q k , λ ; q ) , = (1 − q ) ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k ) ∞ X n =0 ( − n q n )+2 n ( α +2) ( q , q α +4 ; q ) n ( − q k ; q ) n q kn λ n , = (1 − q ) ∞ X n =0 ( − n q n )+2 n ( α +2) ( q , q α +4 ; q ) n ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k )( − q k ; q ) n q kn λ n , and J α (1 , λ ; q ) = ∞ X n =0 ( − n q n )+2 n ( α +1) ( q , q α +2 ; q ) n ( − q ) n λ n . It follows that(1 − q ) ( − n q n )+2 n ( α +1) ( q , q α +4 ; q ) n q n ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k )( − q k ; q ) n q kn = ( − n q n )+2 n ( α +1) ( q , q α +2 ; q ) n ( − q ) n C, n = 0 , , , ... . Dividing out common factors then we have(1 − q ) 1 − q α +2 − q α +2 n +2 q n ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +4 ; q ) ∞ g ( q k )( − q k ; q ) n q kn = ( − q ) n C, n = 0 , , , ... and letting n → ∞ gives ( − q ) ∞ C = 0 , then C = 0 . We can now conclude that G ( λ ) = 0 , or that is Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ g ( x ) J α +1 ( x, j αn ; q ) d q x = 0 . We complete the proof with a simple argument that gives g ( q m ) = 0 , m = 0 , , ... .If G ( λ ) = 0 , then ∞ X k =0 q k ( − q k +2 ; q ) ∞ ( − q k +2 α +2 ; q ) ∞ g ( q k )( − q k ; q ) n q kn = 0 . Letting n → ∞ gives g (1) = 0. Then dividing by q n and again letting n → ∞ gives g ( q ) = 0 . Continuing this process we have g ( q m ) = 0 and the proof of the theorem iscomplete. (cid:3) N THE ZEROS OF THE BIG q -BESSEL FUNCTIONS AND APPLICATIONS 11 Fourier big q -Bessel series Using the orthogonality relation (4.1), we consider the big q -Fourier-Bessel series, S ( α ) q [ f ], associated with a function f ,(5.1) S ( α ) q [ f ]( x ) = ∞ X k =1 a k ( f ) J α +1 ( x, j αk ; q ) , with the coefficients a k given by(5.2) a k ( f ) = 1 µ k Z t ( − t q ; q ) ∞ ( − t q α +4 ; q ) ∞ f ( t ) J α +1 ( t, j αk ; q ) d q t, where µ k = k J α +1 ( x, j αk ; q ) k L q (0 , , = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ (cid:2) J α +1 ( x, j αk ; q ) (cid:3) d q x. Sampling theorem
The classical Kramer sampling is as follows [9, 13]. Let K ( x ; λ ) be a function, contin-uous in λ such that, as a function of x ; K ( x ; λ ) ∈ L ( I ) for every real number λ , where I is an interval of the real line. Assume that there exists a sequence of real numbers λ n ,with n belonging to an indexing set contained in Z , such that K ( x ; λ n ) is a completeorthogonal sequence of functions of L ( I ) . Then for any F of the form F ( λ ) = Z I f ( x ) K ( x, λ ) dx, where F ∈ L ( I ) , we have(6.1) F ( λ ) = lim N →∞ X | n |≤ N F ( λ n ) S n ( λ ) , with S n ( t ) = R I K ( x, λ ) K ( x, λ n ) dx R I | K ( x, λ n ) | dx The series (6.1) converges uniformly wherever || K ( ., λ ) || L ( I ) is bounded.Now we give a q -sampling theorem for the q -integral transform of the form F ( λ ) = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ f ( x ) J α +1 ( x, λ ; q ) d q x, f ∈ L q (0 , , α > − . Theorem 6.1.
Let f be a function in L q ((0 , x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x ) . Then the q -integraltransform (6.2) F ( λ ) = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ f ( x ) J α +1 ( x, λ ; q ) d q x, α > − . has the point-wise convergent sampling expansion (6.3) F ( λ ) = ∞ X k =1 F ( j αk ) j αk J α +1 (1 , λ ; q )( λ − ( j αk ) )[ ∂J α +1 ∂λ (1 , λ ; q )] λ = j αk . The series (6.3) converges uniformly over any compact subset of C . Proof.
Set K ( x, λ ) = J α +1 ( x, λ ; q ) , and j αk is the k-th positive zero of J α +1 ( x, λ ; q ) , and { J α +1 ( x, j αk ; q ) } ∞ k =1 is a complete orthogonal sequence of function in L q ((0 , x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x ) . Hence applying Theorem 6.1, we get(6.4) F ( λ ) = ∞ X k =1 F ( j αk ) R x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ J α +1 ( x, j αk ; q ) J α +1 ( x, λ ; q ) d q x R x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ | J α +1 ( x, j αk ; q ) | d q x . But J α +1 ( x, λ ; q ) is analytic on C so is bounded on any compact subset of C andhence k J α +1 ( x, λ ; q ) k is bounded. Substituting from (3.1) with µ = j αk and from (3.4)we obtained (6.3) and the theorem follows. (cid:3) Example.
Define a function f on [0 ,
1] to be f ( t ) = (cid:26) − q t = 1 , otherwise. Then F ( λ ) = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ f ( x ) J α +1 ( x, λ ; q ) d q x = ( − q ; q ) ∞ ( − q α +4 ; q ) ∞ J α +1 (1 , λ ; q ) . Thus, applying Theorem 6.1 gives(6.5) 12 = ∞ X k =1 j αk J α +1 (1 , j αk ; q )( λ − ( j αk ) )[ ∂J α +1 ∂λ (1 , λ ; q )] λ = j αk , λ ∈ C \{± j αk , k ∈ N ∗ } . We define the Paley-Wiener space related to the big q -Bessel by P W B = {F B ( f ); f ∈ L q ((0 , x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x ) } , where the finite big q -Hankel transform F B ( f ) is defined by(6.6) F B ( f )( λ ) = Z x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ f ( x ) J α +1 ( x, λ ; q ) d q x, α > − . By a quite similar argument as in the proof of [Theorem 1, [2]] and Theorem 4.4, we seethat the space
P W B equipped with the inner product < f, g > P W = Z ( F B f )( x )( F B g )( x ) x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x is a Hilbert space and the finite big q -Hankel transform (6.6) becomes a Hilbert spaceisometry between L q ((0 , x ( − x q ; q ) ∞ ( − x q α +4 ; q ) ∞ d q x ) and P W A . Therefore, from [Theorem A, [2]]we deduce that the big q -Bessel function has an associated reproducing kernel. Acknowledgements
This research is supported by NPST Program of King Saud University, project number10-MAT1293-02.
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