Hypergeometric reproducing kernels and analytic continuation from a half-line

0.2 MB·English

by  Dmitry B. Karp

Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 1
Hypergeometric reproducing kernels and analytic continuation from
a half-line
6
0
0
Dmitrii Karp
2

n
a
J

0 Laboratory of Function Theory,
2 Institute of Applied Mathematics,

Far Eastern Branch of the Russian Academy of Sciences,
]
V 7 Radio Street, Vladivostok 690043, Russian Federation
C E-mail: [email protected]
.
h
t
a
m
[
Abstract. Indefinite inner product spaces of entire functions and functions

2 analytic inside a disk are considered and their completeness studied. Spaces
v induced by the rotation invariant reproducing kernels in the form of the gen-
9 eralized hypergeometric function are completely characterized. A particular
2
space generated by the modified Bessel function kernel is utilized to derive an
1
analytic continuation formula for functions on R+ using the best approxima-
8
0 tiontheoremofS.Saitoh. Asaby-productseveralnewareaintegralsinvolving
9 Bessel functions are explicitly evaluated.
9
/
h
2000 Mathematics Subject Classification: Primary 46E22, 46E50, 30B40 Secondary 33C20, 33C60,
t
a
33C10.
m
Key words and phrases. Indefinite inner product, holomorphic spaces, reproducing kernel, hyper-
:
v geometric function, analytic continuation, Bessel functions.
i
X
r 1. Bergman spaces with indefinite inner product
a
TheBergmanspaceA2 withradialweightw ontheunitdisk∆isdefinedasthesetoffunctions
w
analytic inside ∆ equipped with inner product
[f,g] = f(z)g(z)w(z )dσ(z), (1)
w
| |
Z
∆
where dσ is the planar Lebesgue measure, w : (0,1) R+ 0 and xw L (0,1). It seems to be
1
→ ∪{ } ∈
the folklore result that this space is complete iff
1
w(x)dx > 0, ξ (0,1). (2)
∀ ∈
Z
ξ
It is interesting to remark here that for non-radial non-negative weight no necessary and sufficient
conditions for completeness seem to be known. Now we relax the assumption of non-negativeness
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 2
of the weight w and consider the space A2 on the disk ∆ with the (possibly infinite) radius R
w R
determined by w as follows
∞
def
R = sup x: w(x)dx > 0 . (3)
 | | 
 Zx 
The space A2(∆ ) with sign-alternating weight w is of course no longer a Hilbert space. By
w R  
standard argument (which carries over with minor modifications to the sign-alternating case) the
inner product (1) in this space can be written as
R R
f(z)g(z)w(z )dσ = 2π ∞ f g r2k+1w(r)dr = ∞ c f g , c d=ef 2π r2k+1w(r)dr. (4)
k k k k k k
| |
∆ZR Xk=0 Z0 Xk=0 Z0
Define
def def def
I = k :c > 0 , I = k :c < 0 , I = k :c = 0 .
+ k k 0 k
{ } − { } { }
The space A2(∆ ) allows the fundamental decomposition
w R
A2 = A2 A2 , f = f +f , (5)
w w+⊕ w− + −
with
f (z) = f zk, f (z) = f zk, (6)
+ k k
−
kX∈I+ kX∈I−
and
[f ,f ]= c f 2 > 0, [f ,f ]= c f 2 < 0, [f ,f ] = 0.
+ + k k k k +
| | − − | | −
kX∈I+ kX∈I−
ThusA2(∆ )is aKreinspace(i.e. complete indefiniteinnerproductspace allowing decomposition
w R
(5), cf. [1, 3, 7]) if the spaces A2 and A2 are both complete. Define
w+ w
−
R
a d=ef 2π x2k+1 w(x)dx. (7)
k
| |
Z
0
Theorem 1 The space A2 is a (complete) Krein space if and only if the sequence a /c
is bounded. w { k k}k∈I−∪I+
The proof is straightforward and is given in [13]. Onecan also find an example there of weight that
violates the boundedness condition from Theorem 1. The space A2 admits the reproducing kernel
w
given by
1
ϕ(zu)= (zu)k. (8)
c
k
k∈XI−∪I+
For the general theory of reproducing kernels in Hilbert and Krein spaces see [1, 2, 19]. The
condition of Theorem 1 is obviously satisfied when there exists γ < R such that w 0 or w 0
≥ ≤
when x (γ,R). If this requirement is met and all moments (4) are finite we will call w an
∈
admissible weight on (0,R). Moreover, in the case of admissible weight at least one of the sets I
+
or I is finite implying that at least one of the spaces A2 or A2 is finite-dimensional and hence
w+ w
A2(−∆ ) becomes a Pontryagin space [1, 3, 7]. We call A2(∆ ) a−positive Pontryagin space when
w R w R
dimA2 < and a negative Pontryagin space when dimA2 < . The numbers dimA2 and
dimA2w− are∞called positive and negative (Pontryagin) indicews+of A∞2(∆ ), respectively. w+
w w R
−
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 3
2. Hypergeometric kernels
We now turn to the case when ϕ is the generalized hypergeometric function (cf. [18]):
ϕ(zu)= F a1,...,ap zu d=ef ∞ (a1)k...(ap)k (zu)k, (9)
p q b ,...,b (b ) ...(b ) k!
(cid:18) 1 q (cid:12) (cid:19) k=0 1 k q k
(cid:12) X
(cid:12)
where (a) = 1, (a) = a(a + 1)...(a+ k (cid:12)1) = Γ(a + k)/Γ(a) is the Pochhammer symbol or
0 k
−
shifted factorial. As shown below the weight w will be in this case expressed in terms of Meijer’s
G-function defined by
Gm,n z a1...ap = Gm,n z (ap) d=ef
p,q b ...b p,q (b )
1 q q
(cid:18) (cid:19) (cid:18) (cid:19)
1 Γ(b +s)...Γ(b +s)Γ(1 a s)...Γ(1 a s)z s
1 m 1 n −
− − − − ds.
2πi Γ(a +s)...Γ(a +s)Γ(1 b s)...Γ(1 b s)
n+1 p m+1 q
Z − − − −
LG
Here (m,n,p,q) is the order of G and 0 m q, 0 n p. The contour L is an infinite loop
G
≤ ≤ ≤ ≤
separating the poles of gamma functions of the form Γ(b +s), j = 1,2,...,m from those of gamma
j
functions of the form Γ(1 a s), j = 1,2,...,n. Details can be found in [15, 18]. We will only
j
− −
need G-functions with m = q, n = 0. On denoting
Γ(b +s)...Γ(b +s)
1 q
h(s) = , (10)
Γ(a +s)...Γ(a +s)
1 p
the residue theorem leads to the following expansion
Gqp,,0q(cid:18)x ((baqp))(cid:19)=jX=1,q (nxjkbj−+k1)! nXji=k−01Cinjk−1lni x1 [(s+bj +k)njkh(s)](|sn=jk−−b1j−−ki), (11)
k=0,
∞
where Ci are binomial coefficients, n is the multiplicity of the pole of h at s = b k or 0
j jk j,k − j −
if there is no pole at the point s and the convention 1/( 1)! = 0 is used. The basic property of
j,k
−
G-function is that it is the inverse Mellin transform of h:
∞ (a ) Γ(b +s)...Γ(b +s)
xs 1Gq,0 x p dx = 1 q . (12)
− p,q (b ) Γ(a +s)...Γ(a +s)
Z (cid:18) q (cid:19) 1 p
0
We will consider two cases p < q and p = q separately.
Lemma 1 Let p < q, b > 1, i = 1,q, possibly except for b of the form b = a +k for some
i i i j
−
(a )
j 1,2,...,p and k N ; θ > 0 and a R. Then Gq,0 θx2 p is an admissible weight
∈ { } ∈ 0 i ∈ p,q (bq)
(cid:18) (cid:19)
on (0, ).
∞
The proof hinges on the following asymptotic relations:
(a )
Gq,0 θx2 p x2blnj 1x, x 0, (13)
p,q (bq) ∼ − →
(cid:18) (cid:19)
Gq,0 x (ap) = (2π)12(µ−1)x(1 α)/µe µx1/µ 1+O(x 1/µ) , x , (14)
p,q (bq) √µ − − − → ∞
(cid:18) (cid:19)
h i
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 4
where µ = q p, α = p a q b + 1(q p+1). Here the symbol means that the ratio
− i=1 i − i=1 i 2 − ∼
of the quantities on the left and on the right is bounded from above an from below by positive
P P
constants independent of x. The first relation is a consequence of (11); the second is a particular
case of the formula on page 289 that follows (7.8) in [8].
Let [x] denote the largest integer not exceeding x. Having made these preparations we arrive
at the following
Theorem 2 Let (a ), (b ) be real none of them being a non-positive integer, p q are positive
p q
≤
integers, θ > 0, m is the number of negative elements of the set (a ),(b ) , d is the negative
p q i
{ }
element with i-th smallest absolute value and
m
def
ν = [d ].
k
k=1
X
Then the space θ induced by the reproducing kernel (9) is the Pontryagin space of entire
F(ap),(bq)
functions with inner product
∞ (b1)k...(bq)kk!
[f,g] = f g , (15)
(a ) ...(a ) θk k k
1 k p k
k=0
X
where f , g are Taylor’s coefficients of f and g, respectively. It is a positive space with negative
k k
index
m−1 |[di+1]| i
[d ]+k(m i) mod 2, (16)
j
 − 
Xi=0 k=|X[di]|+1 Xj=1
 
iff ν is even, and a negative space with positive index
m−1 |[di+1]| i
1 [d ]+k(m i) mod 2 +1, (17)
j
 − −  
Xi=0 k=|X[di]|+1 Xj=1
   
iff ν is odd. Let
l = min 0,[ min b ] .
i
(cid:26) bi6=aj+k (cid:27)
Then the inner product (15) may be represented in the form
θ1 l (a ) (a )+l 1,l
[f,g]= − Γ p f(l)(z)g(l)(z)Gq+2,0 θ z 2 p − dσ+
π (bq) p+1,q+2 | | (bq)+l 1,0,0
(cid:20) (cid:21)ZC (cid:18) (cid:12) − (cid:19)
(cid:12)
(cid:12)
(cid:12)
l 1
+ − (b1)k...(bq)k f(k)(0)g(k)(0). (18)
(a ) ...(a ) θkk!
1 k p k
k=0
X
In particular if l = 0
θ (a ) (a ) 1
[f,g] = Γ p f(z)g(z)Gq+1,0 θ z 2 p − dσ. (19)
π (bq) p,q+1 | | 0,(bq) 1
(cid:20) (cid:21)ZC (cid:18) − (cid:19)
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 5
Proof. The formulae (16) and (17) are derived by thorough counting of positive and negative
coefficients in (9). Expressions (18) and (19) are obtained by comparing (4) and (8) with (9), (12)
and (15). Existence of integral in (18) and (19) is guaranteed by Lemma 1. (cid:3)
When p = q = 0, the space θ becomes the classical Bargmann-Fock space [4, 5]. When
F(ap),(bq)
p = q = a = 1, b > 0 this space reduces to the generalized Bargmann-Fock space considered in
[9, 10]. Many more particular cases are given in [13].
For formulating a similar result in the case p = q the following lemma plays the key role.
Lemma 2 If x > 1 and ( p b ) < ( p a ), then
ℜ i=0 i ℜ i=0 i
P P (a )
Gp,0 x p = 0. (20)
p,p (b )
p
(cid:18) (cid:12) (cid:19)
(cid:12)
(cid:12)
p,0
For a proof the reader is again referred to(cid:12)[13]. Since G (x) has a (finite or infinite) limit as
p,p
x 1, it is an admissible weight on (0,1). Thus we get
→
Theorem 3 Let (a ), (b ), θ > 0, m, d and ν have the same meaning as in Theorem 2. Then
p p 1 i
the space θ induc−ed by the reproducing kernel
B(ap),(bp 1)
−
a ,...,a 1
K(z,u) = F 1 p zu (21)
p p−1(cid:18) b1,...,bp−1 (cid:12)θ (cid:19)
(cid:12)
is a Pontryagin space comprising functions holomorphic insid(cid:12)e ∆ with inner product
(cid:12) √θ
∞ (b1)k...(bp 1)kk!θk
[f,g] = − fkgk, (22)
(a ) ...(a )
1 k p k
k=0
X
where f , g are Taylor’s coefficients off and g, respectively. This space ispositive and has negative
k k
index given by (16) iff ν is even and is negative with positive index given by (17) iff ν is odd. Define
l = min 0,[ min b ] ,
1 i
(cid:26) bi6=aj+k (cid:27)
and
p 1 p
−
l = min k N : b +1 < a +2k .
2 0 i i
∈
( )
i=0 i=0
X X
The inner product (22) can be then written as
θ1 l (a ) z 2 (a )+l 1,l
[f,g]= − Γ p f(l)(z)g(l)(z)Gp+1,0 | | p − dσ+
π (b ) p+1,p+1 θ (b )+l 1,0,0
p 1 p 1
(cid:20) − (cid:21)∆Z (cid:18) − − (cid:19)
√θ
+l−1 (b1)k...(bp−1)kθkf(k)(0)g(k)(0), (23)
(a ) ...(a ) k!
1 k p k
k=0
X
with l =max(l ,l ).
1 2
If l = 0, then the inner product (22) can also be written as
1
1 (a ) 1 z 2 (a ),(2)
[f,g]= Γ p (Dz)lf(z) ((Dz)lg(z)) Gp+2l,0 | | p dσ. (24)
π (b ) z 2 p+2l,p+2l θ (b ),(1)
p 1 p 1
(cid:20) − (cid:21)∆Z (cid:16) (cid:17) | | (cid:18) − (cid:19)
√θ
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 6
In particular if l =l = 0
1 2
1 (a ) z 2 (a ) 1
[f,g]= Γ p f(z)g(z)Gp,0 | | p − dσ. (25)
θπ (b ) p,p θ 0,(b ) 1
p 1 p 1
(cid:20) − (cid:21)∆Z (cid:18) − − (cid:19)
√θ
The spaces θ contain, as particular cases, the classical Bergman (q = 0, a = 2), the
B(ap),(bp 1)
Hardy H (q = 0, a = −1) and the Bergman-Selberg (q = 0, a > 1) spaces, as well as many others
2
(see [13] for other examples).
3. Topologies of hypergeometric kernel spaces
In this section we consider a generalization of the kernel (9) given by
K(z,u) = ∞ Γ(B1k+b1)···Γ(Bqk+bq)(zu)k. (26)
Γ(A k+a ) Γ(A k+a )
1 1 p p
k=0 ···
X
The series on the right is called the Wright psi-function [20, 21]. For simplicity we limit ourselves
to the Hilbert space case A ,B ,a ,b > 0. Formula (12) is generalized as follows (cf.[16, 18])
i i i i
∞ (a ,A ) Γ(b +B s)...Γ(b +B s)
xs 1Hq,0 x p p dx = 1 1 q q . (27)
− p,q (b ,B ) Γ(a +A s)...Γ(a +A s)
Z (cid:18) q q (cid:19) 1 1 p p
0
q,0
Here H is Fox’s H-function [8, 16, 18]. Thus one can construct the analogues of Theorems 2 and
p,q
3 for the spaces generated by the kernel (26). Thepurposeof this section, however, is different. We
show that while the geometries of the spaces generated by the kernel (26) dependon all parameters
A ,B ,a ,b their topologies are only dependent on three numbers
i i i i
q p
p q
α= b a + − ,
i i
− 2
i=1 i=1
X X
q p
µ = B A 0
i i
− ≥
i=1 i=1
X X
and
p q
ν = Ai−Ai BiBi > 0.
i=1 i=1
Y Y
Indeed, the norms in the Hilbert spaces H , ε 1,2 induced by the kernels K(1), K(2) of the
K(ε) ∈ { }
form (26) are equivalent when
(1) (1) (1) (1) (2) (2) (2) (2)
Γ(B k+b ) Γ(B k+b ) Γ(B k+b ) Γ(B k+b )
1 1 ··· q q 1 1 ··· q q , k .
Γ(A(1)k+a(1)) Γ(A(1)k+a(1)) ∼ Γ(A(2)k+a(2)) Γ(A(2)k+a(2)) → ∞
1 1 ··· p p 1 1 ··· p p
From Stirling’s formula one easily derives
Γ(B k+b ) Γ(B k+b )
1 1 ··· q q kαe−µkkµkνk, k , (28)
Γ(A k+a ) Γ(A k+a ) ∼ → ∞
1 1 p p
···
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 7
which proves our conjecture about the topologies of H . Thus we obtain two ”model” spaces for
K
two distinct cases µ > 0 and µ = 0. If µ > 0 the space induced by (26) is topologically equivalent
to the space of entire functions equipped with the norm
l 1
−
f(l)(z)2exp µν µ z 2/µ z (2α+4l+2µl+1)/µ 2dσ+ f(k)(0)2,
− −
| | − | | | | | |
ZC (cid:16) (cid:17) Xk=0
where l = max(0,[( α 1)/µ]+1). In particular, when α > 1/2 the norm reduces to
− − 2 −
2α+1
f(z)2exp µν µ z 2/µ z (2α+1)/µ 2dσ,
− −
2πµ | | − | | | |
ZC (cid:16) (cid:17)
and the reproducing kernel is E (ν 1µµzu), where
µ,α+1/2 −
∞ tk
E (t) =
µ,α
Γ(µk+α)
k=0
X
is the two-parameter Mittag-Leffler function [12].
If µ = 0 the space induced by (26) is topologically equivalent to the space of functions analytic
inside ∆ equipped with the norm
√ν
l 1
−
f(l)(z)2(1 z 2/ν)2l α 1dσ+ f(k)(0)2,
− −
| | −| | | |
Z k=0
∆ X
√ν
where l = max(0,[α] + 1). In particular, when α < 0 it is the Bergman-Selberg (or weighted
Bergman) space with the norm
α
− f(z)2(1 z 2/ν)−α−1dσ
πν | | −| |
Z
∆
√ν
and reproducing kernel
α 1
zu −
K(z,u) = 1 .
− ν
(cid:18) (cid:19)
4. Analytic continuation from the positive half-line
In this section we use a hypergeometric kernel space to derive an analytic extension formula for
functions defined on R+. As a by-product several new area integrals involving Bessel functions are
explicitly evaluated.
We shall use the theory of best approximation in reproducing kernel Hilbert spaces (RKHS)
developed by S. Saitoh as set forth in [19]. Let E be an arbitrary set, and let H be a RKHS
K
comprising functions E C and admitting the reproducing kernel K. For X E we consider a
→ ⊂
Hilbert space H(X) comprising functions X C. We additionally assume that restrictions f of
X
→ |
f H belong to the Hilbert space H(X) and the restriction operator T is continuous from H
K K
∈
into H(X). In this setting Saitoh has proved the following result.
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 8
Theorem 4 For a given d H(X) there exists f˜ H such that
K
∈ ∈
inf Tf d = Tf˜ d (29)
H(X) H(X)
f∈HKk − k k − k
if and only if T d H , where RKHS H is induced by the kernel
∗ k k
∈
k(p,q) = (T TK(,q),T TK(,p)) on E E. (30)
∗ · ∗ · HK ×
Among all solutions of (29) there exists a unique extremal f with the minimum norm in H which
∗ K
is given by
f (p) =(T d,T TK(,p)) . (31)
∗ ∗ ∗ · Hk
This theorem has been applied in [11] to derive an analytic extension formula for function
defined on R. In [14] the author found some analytic continuability criteria for functions on R, R+,
and [ 1,1] without the application of Theorem 4.
To−obtain a result for functions on R+ similar to that of [11] consider the space Hα,ν induced
K
by the kernel
1 α2zu ∞ α2k(zu)k 2 ν
K (z,u) = F ;ν +1; = = (zu) ν/2I (α√zu),
α,ν Γ(ν +1)0 1 − 4 4kΓ(k+ν +1)k! α − ν
(cid:18) (cid:19) k=0 (cid:18) (cid:19)
X
(32)
where I is the modified Bessel function (cf. [6, 17]), ν > 1 and α > 0. By Theorem 2 this space
ν
−
consists of entire functions with finite norms
αν+2
f 2 = f(z)2 z νK (αz )dσ, (33)
k kν,α 2ν+1π | | | | ν | |
ZC
α,ν
where K is the MacDonald function (cf. [6, 17]). We consider the restriction operator T :H
ν K →
L (R+;W ), W (t) = e αttν.
2 α,ν α,ν −
Lemma 3 The restriction operator T is bounded.
Proof. From the formula
∞ 1 z2+u2 zu
J (zt)J (ut)e γt2tdt = exp I , γ > 0, (34)
ν ν − ν
2γ − 4γ 2γ
Z (cid:20) (cid:21) (cid:18) (cid:19)
0
found for example in [6, 17], we get the following representation for the kernel
2 ν ∞
K (z,u) = eαz/2eαu/2 (zt) ν/2J (√zt)(ut) ν/2J (√ut)e t/(2α)tνdt. (35)
α,ν − ν − ν −
α
(cid:18) (cid:19) Z
0
In these formulae J is the Bessel function. According to the theory developed in [19], this repre-
ν
α,ν
sentation of the kernel implies that each f H is expressible in the form
∈ K
2 ν ∞
f(z) = eαz/2 (zt) ν/2J (√zt)F(t)e t/(2α)tνdt (36)
− ν −
α
(cid:18) (cid:19) Z
0
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 9
for some uniquely determined function F satisfying
∞
F(t)2e t/(2α)tνdt < ,
−
| | ∞
Z
0
and we have the isometric identity
2 ν ∞
f 2 = F(t)2e t/(2α)tνdt.
k kν,α α | | −
(cid:18) (cid:19) Z
0
On the other hand from the unitarity of the Hankel transform in L (R+) it follows by a simple
2
change of variable that the operator
1 ∞
A: g J √ t g(t)dt
ν
→ 2 ·
Z0 (cid:16) (cid:17)
is also unitary in L (R+). Hence from (36)
2
∞ 22ν+2 ∞ 2
f(t)2W (t)dt = F(t)e t/(2α)tν/2 dt
| | α,ν α2ν −
Z0 Z0 (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
2 2ν ∞ 2 ν
4 F(t)2e t/(2α)tνdt = 4 f 2 . (cid:3)
≤ α | | − α k kν,α
(cid:18) (cid:19) Z (cid:18) (cid:19)
0
Now we consider the existence of the best approximation f
∗
kTf∗−FkL2(R+,Wα,ν) = f iHnfα,νkTf −FkL2(R+,Wα,ν) = 0.
∈ K
The last equality is due to the density in L (R+;W ) of restrictions of functions from Hα,ν to R+.
2 α,ν K
T is expressible in the form (by definition of the adjoint operator):
∗
(T∗F(·))(u) = ((T∗F)(·),Kα,ν(·,u))HαK,ν = (F(·),TKα,ν(·,u))L2(R+;Wα,ν) =
∞ ∞
= F(t)K (t,u)W (t)dt = F(t)K (t,u)W (t)dt.
α,ν α,ν α,ν α,ν
Z Z
0 0
Then for s R+
∈
∞
(TT TK (,z))(s) = K (t,z)K (t,s)W (t)dt
∗ α,ν α,ν α,ν α,ν
·
Z
0
2 2ν ∞
= (sz) ν/2 I (α√tz)I (α√ts)e αtdt
− ν ν −
α
(cid:18) (cid:19) Z
0
2ν
1 2 α
= (sz) ν/2eα(z+s)/4I √sz = α ν 1eα(z+s)/4K (s,z).
α α − ν 2 − − α/2,ν
(cid:18) (cid:19)
(cid:16) (cid:17)
Journal of Integral Transforms and Special Functions, vol.14, no.6, 2003, pp. 485-498 10
Hence for the kernel k from Theorem 4 we calculate
k (z,u) = (T TK (,u),T TK (,z)) = (TK (,u),TT TK (,z))
α,ν ∗ α,ν · ∗ α,ν · HαK,ν α,ν · ∗ α,ν · L2(R+,Wα,ν)
∞
= α ν 1eαz/4 K (s,u)K (s,z)eαs/4W (s)ds
− − α,ν α/2,ν α,ν
Z
0
1 2 3ν ∞ α
= eαz/4(zu) ν/2 I (α√su)I √sz e 3αs/4ds
− ν ν −
α α 2
(cid:18) (cid:19) Z0 (cid:16) (cid:17)
4 2 3ν α√zu 4 ν+1
= eαz/3eαu/3(zu) ν/2I = α 2ν 2eαz/3eαu/3K (z,u).
3α2 α − ν 3 3 − − α/3,ν
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
α,ν
From the general theory of reproducing kernels [2, 19] and (33) we see that the space H induced
k
by the kernel k comprises entire functions with finite norms
α,ν
α3ν+4 2 αz
f(z)2exp α z z νK | | dσ.
3π23ν+3 | | −3 ℜ | | ν 3
ZC (cid:18) (cid:19) (cid:18) (cid:19)
This brings us to the main result of this section. In the following theorem analytic continuation
is understood in the sense that the extended and the original functions may differ on a set of zero
measure.
Theorem 5 A function F L (R+;W ) can be analytically continued to the entire complex
2 α,ν
∈ α,ν
plane to a function f from the space H if and only if
K
2
∞ αz
F(t)Iν(α√tz)e−αttν/2dt e−23αℜzKν | | dσ < . (37)
(cid:12) (cid:12) 3 ∞
ZC (cid:12)(cid:12)Z0 (cid:12)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
The analytic continuat(cid:12)ion is given by (cid:12)
(cid:12) (cid:12)
α3 α αu ∞
f(z) = 3π23eαz/4z−ν/2 Iν 2√zu e−α4u−α6ℜuKν 3| | F(t)Iν(α√tu)e−αttν/2dtdσ(u). (38)
ZC (cid:16) (cid:17) (cid:18) (cid:19)Z0
Formula (38) contains many interesting special cases. Taking F(t) = 1 we get
α2 α αu
e−αz/4zν/2 = 3π23+ν uν2Iν 2√zu e−16αℜuKν 3| | dσ(u).
ZC (cid:16) (cid:17) (cid:18) (cid:19)
In particular, when α = 2 and ν = 1 the last equality reduces to
2
1
z21e−z2 = u21 sinh √zu u−1e−32|u|−31ℜudσ(u),
4π√3 | |
ZC (cid:16) (cid:17)
which can be verified directly. If we put F(t) = tn, n N, then evaluation of the integral over R+
∈
in (38) leads to
n! α αu αu
zn+ν/2e−αz/4 = 3π2ν+3αn 2 uν2Iν 2√zu Lνn − 4 e−61αℜuKν 3| | dσ(u), (39)
− ZC (cid:16) (cid:17) (cid:16) (cid:17) (cid:18) (cid:19)