Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 2 (2012.), Pages 1-16
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
(COMMUNICATED BY PALLE JORGENSEN)
FETHI SOLTANI
Abstract. In this work, we introduce a class of Hilbert spaces Fq,α of entire
1
functions on the disk D(0, 1−q
), 0 < q < 1, with reproducing kernel given by
the q-Bessel-Struve kernel Sα (z; q 2 ). The definition and properties of the space
Fq,α extend naturally those of the well-known classical Fock space. Next, we
study the bounded of some operators on the Fock spaces Fq,α ; and we give an
application of the theory of reproducing kernels to the Tikhonov regularization,
which gives the approximate solutions for bounded linear operator equation
on the Fock spaces Fq,α .
1. Introduction
Fock space F (called
P∞ also Segal-Bargmann space [3]) is the Hilbert space of entire
functions f (z) = n=0 an z n on C such that
kf k2F :=
∞
X
|an |2 n! < ∞.
n=0
This space was introduced by Bargmann in [2] and it was the aim of many works
[3]. Especially, the differential operator D = d/dz and the multiplication operator
by z are densely defined, closed and adjoint-operators on F (see [2]).
In [7], Gasmi and Soltani introduced a Hilbert space Fα of entire functions on
C, where the inner product is weighted by the modified Macdonald function. On
Fα the Bessel-Struve operator
i
2α + 1 h
`α f (z) := D2 f (z) +
Df (z) − Df (0) , α > −1/2,
z
and the multiplication operator M by z 2 are densely defined, closed and adjointoperators.
2000 Mathematics Subject Classification. 32A36, 32A15, 46E22.
Key words and phrases. Generalized q-Fock spaces, q-Bessel-Struve kernel; q-Bessel-Struve
operator, q-translation operators, extremal function, Tikhonov regularization.
c
°2012
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted December 15, 2010. Accepted February 12, 2012.
Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.
1
2
FETHI SOLTANI
In this paper, we consider the q-Bessel-Struve kernel:
Sα (x; q 2 ) :=
∞
X
xn
,
c (α; q 2 )
n=0 n
where cn (α; q 2 ) are given later in section 2. We discuss some properties of a class of
Fock spaces associated to the q-Bessel-Struve kernel and we give some applications.
The contents of the paper are as follows. In section 2, building on the ideas of
Bargmann [2], Cholewinski [4] and as the same ofP
paper [18], we define the q-Fock
∞
1
space Fq,α as the space of entire functions f (z) = n=0 an z n on the disk D(0, 1−q
)
1
of center 0 and radius 1−q , and such that
kf k2Fq,α :=
∞
X
|an |2 cn (α; q 2 ) < ∞.
n=0
Let f and g be in Fq,α , such that f (z) =
inner product is given by
hf, giFq,α =
∞
X
P∞
n=0
an z n and g(z) =
P∞
n=0 bn z
n
, the
an bn cn (α; q 2 ).
n=0
The q-Fock space Fq,α has also a reproducing kernel Kq,α given by
Kq,α (w, z) = Sα (wz; q 2 );
w, z ∈ D(0,
1
).
1−q
w ∈ D(0,
1
).
1−q
Then, if f ∈ Fq,α , we have
hf, Kq,α (w, .)iFq,α = f (w),
Using this property, we prove that the space Fq,α is a Hilbert space and we give an
Hilbert basis.
In section 3, using the previous results, we consider the multiplication operator
M by z 2 and the q-Bessel-Struve operator `q,α on the Fock space Fq,α , and we
prove that these operators are continuous from Fq,α into itself, and satisfy:
k`q,α f kFq,α ≤
1
kf kFq,α ,
1−q
1
kf kFq,α .
1−q
Then, we prove that these operators are adjoint-operators on Fq,α :
kM f kFq,α ≤
hM f, giFq,α = hf, `q,α giFq,α ;
f, g ∈ Fq,α .
These properties are not true on the classical Fock spaces [2, 4, 7, 15, 16]. For
example on Fα , these properties are statued on D(`α ) = {f ∈ Fα : `α f ∈ Fα },
D(M ) = {f ∈ Fα : M f ∈ Fα } and D(`α ) ∩ D(M ), respectively.
Lastly, we define and study on the Fock space Fq,α , the q-translation operators:
Tz f (w) := Sα (z`q,α ; q 2 )f (w);
w, z ∈ D(0,
1
),
1−q
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
3
and the generalized multiplication operators:
1
).
1−q
Using the continuous properties of `q,α and M we deduce also that the operators
1
), are continuous from Fq,α into itself, and satisfy:
Tz and Mz , for z ∈ D(0, 1−q
Mz f (w) := Sα (zM ; q 2 )f (w);
kTz f kFq,α ≤ Sα (
w, z ∈ D(0,
|z|
; q 2 )kf kFq,α ,
1−q
|z|
; q 2 )kf kFq,α .
1−q
These properties are not true on the classical Fock spaces.
The last section of this paper is devoted to give an application of the theory of
reproducing kernels to the Tikhonov regularization, which gives the approximate
solutions for bounded linear operator equation on the Fock spaces Fq,α .
Let L : Fq,α → Fq,α be a bounded operator from Fq,α into itself. For λ > 0, we
define on the space Fq,α , the new inner product by setting
kMz f kFq,α ≤ Sα (
hf, giλ,Fq,α = λhf, giFq,α + hLf, LgiFq,α .
Building on the ideas of Saitoh, Matsuura and Yamada [11, 12, 14, 19], and using
the theory of reproducing kernels [1], we give best approximation of the operator
L. More precisely, for all λ > 0, h ∈ Fq,α , the infimum
n
o
inf λkf k2Fq,α + kh − Lf k2Fq,α ,
f ∈Fq,α
∗
is attained at one function fλ,h
, called the extremal function.
In particular for f ∈ Fq,α and h = Lf , the corresponding extremal functions
∗
{fλ,Lf
}λ>0 converges to f as λ → 0+ .
2. Preliminaries and the q-Fock spaces Fq,α
Let a and q be real numbers such that 0 < q < 1; the q-shifted factorial are
defined by
(a; q)0 := 1,
(a; q)n :=
n−1
Y
(1 − aq i ), n = 1, 2, ..., ∞.
i=0
Jackson [8] defined the q-analogue of the Gamma function as
Γq (x) :=
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
x 6= 0, −1, −2, ....
It satisfies the functional equation
Γq (x + 1) =
1 − qx
Γq (x),
1−q
Γq (1) = 1,
and tends to Γ(x) when q tends to 1− . In particular, for n = 1, 2, ..., we have
Γq (n + 1) =
(q; q)n
.
(1 − q)n
The q-derivative Dq f of a suitable function f (see [10]) is given by
Dq f (x) :=
f (x) − f (qx)
,
(1 − q)x
x 6= 0,
4
FETHI SOLTANI
and Dq f (0) = f 0 (0) provided f 0 (0) exists.
If f is differentiable then Dq f (x) tends to f 0 (x) as q → 1− .
Taking account of the paper [6, 7] and the same way, we define the q-BesselStruve kernel by
Sα (x; q 2 ) := jα (ix; q 2 ) − ihα (ix; q 2 ),
where jα (x; q 2 ) is the q-normalized Bessel function [5, 17] given by
jα (x; q 2 ) := Γq2 (α + 1)
∞
X
(−1)n x2n
,
(1 + q)2n Γq2 (n + 1)Γq2 (n + α + 1)
n=0
and hα (x; q 2 ) is the q-normalized Struve function given by
hα (x; q 2 ) := Γq2 (α + 1)
∞
X
(−1)n x2n+1
.
(1 + q)2n+1 Γq2 (n + 32 )Γq2 (n + α + 32 )
n=0
Furthermore, the q-Bessel-Struve kernel Sα (x; q 2 ) can be expanded in a power series
in the form
∞
X
xn
Sα (x; q 2 ) :=
,
c (α; q 2 )
n=0 n
where
cn (α; q 2 ) :=
If we put Un :=
1
cn (α;q 2 ) ,
(1 + q)n Γq2 ( n2 + 1)Γq2 ( n2 + α + 1)
.
Γq2 (α + 1)
(1)
then
Un
1
→
,
Un+1
(1 − q)2
q → 1− .
1
Thus, the q-Bessel-Struve kernel Sα (x; q 2 ) is defined on D(0, (1−q)
2 ) and tends to
−
the Bessel-Struve kernel Sα (x) as q → 1 .
We consider the q-Bessel-Struve operator `q,α defined by
i
[2α + 1]q h
`q,α f (x) := Dq2 f (x) +
Dq f (qx) − Dq f (0) ,
x
where
[2α + 1]q :=
1 − q 2α+1
.
1−q
The q-Bessel-Struve operator tends to the Bessel-Struve operator `α as q → 1− (see
[6, 7]).
1
Lemma 2.1. The function Sα (λ.; q 2 ), λ ∈ D(0, 1−q
), is the unique analytic solution of the q-problem:
`q,α y(x) = λ2 y(x),
Dq y(0) =
λΓq2 (α + 1)
,
(1 + q)Γq2 ( 32 )Γq2 (α + 32 )
y(0) = 1.
(2)
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
Proof. Searching a solution of (2) in the form y(x) =
Dq y(x) =
∞
X
P∞
n=0
5
an xn . Then
an [n]q xn−1 ,
n=1
and
∞
X
Dq2 y(x) = Dq (Dq y)(x) =
an [n]q [n − 1]q xn−2 .
n=2
Replacing in (2), we obtain
∞
X
an [n]q ([n − 1]q + q n−1 [2α + 1]q )xn = λ2
n=2
∞
X
an−2 xn .
n=2
Thus,
an [n]q ([n − 1]q + q n−1 [2α + 1]q ) = λ2 an−2 ,
Using the fact that [n − 1]q + q
an =
n−1
n = 2, 3, ...
[2α + 1]q = [n + 2α]q , we deduce that
λ2
an−2 ,
[n]q [n + 2α]q
n = 2, 3, ...
Since [2n]q = (1 + q)[n]q2 , we deduce
a2n =
λ2
(1 +
q)2 [n]q2 [n
+ α]q2
a2n−2 ,
λ2
a2n+1 =
(1 + q)2 [n +
1
2
2 ]q [n
+ α + 12 ]q2
a2n−1 .
This proves that
a2n =
a2n+1 =
λ2n Γq2 (α + 1)
,
(1 + q)2n Γq2 (n + 1)Γq2 (n + α + 1)
(1 +
λ2n+1 Γq2 (α + 1)
3
3 .
q 2 (n + 2 )Γq 2 (n + α + 2 )
q)2n+1 Γ
Therefore,
y(x) = Γq2 (α + 1)
+ Γq2 (α + 1)
∞
X
(λx)2n
(1 + q)2n Γq2 (n + 1)Γq2 (n + α + 1)
n=0
∞
X
n=0
(1 +
q)2n+1 Γ
(λx)2n+1
3
3
q 2 (n + 2 )Γq 2 (n + α + 2 )
= jα (iλx; q 2 ) − ihα (iλx; q 2 ),
which completes the proof of the lemma.
¤
Lemma 2.2. The constants bn (α; q 2 ), n ∈ N satisfy the following relation:
cn+2 (α; q 2 ) = [n + 2]q [n + 2α + 2]q cn (α; q 2 ).
Lemma 2.3. For n ∈ N, we have
`q,α z n =
cn (α; q 2 ) n−2
z
,
cn−2 (α; q 2 )
n ≥ 2.
6
FETHI SOLTANI
Proof. Since
Sα (λz; q 2 ) :=
∞
X
(λz)k
,
ck (α; q 2 )
k=0
then from equation (2) we obtain
∞
∞
X
`q,α z k k X
z k−2
λ
=
λk .
ck (α; q 2 )
ck−2 (α; q 2 )
k=2
k=2
This clearly yields the result.
¤
Definition 2.1. Let α ≥
The q-Fock space Fq,α is the prehilbertian space of
P−1/2.
∞
1
entire functions f (z) = n=0 an z n on D(0, 1−q
), such that
kf k2Fq,α :=
∞
X
|an |2 cn (α; q 2 ) < ∞,
(3)
n=0
where cn (α; q 2 ) is given by (1).
The inner product in Fq,α is given for f (z) =
by
hf, giFq,α =
∞
X
P∞
n=0
an z n and g(z) =
P∞
n=0 bn z
an bn cn (α; q 2 ).
n
(4)
n=0
Remark 1. If q → 1− , the space Fq,α agrees with the generalized Fock space
associated to the Bessel-Struve operator (see [7]).
The following theorem prove that Fq,α is a reproducing kernel space.
1
Theorem 2.1. The function Kq,α given for w, z ∈ D(0, 1−q
), by
Kq,α (w, z) = Sα (wz; q 2 ),
is a reproducing kernel for the q-Fock space Fq,α , that is:
1
(i) for all w ∈ D(0, 1−q
), the function z → Kq,α (w, z) belongs to Fq,α .
1
(ii) For all w ∈ D(0, 1−q ) and f ∈ Fq,α , we have
hf, Kq,α (w, .)iFq,α = f (w).
Proof. (i) Since
Kq,α (w, z) =
∞
X
wn
zn;
2)
c
(α;
q
n
n=0
z, w ∈ D(0,
1
),
1−q
(5)
then from (3), we deduce that
kKq,α (w, .)k2Fq,α
=
∞
X
|w|2n
= Sα (|w|2 ; q 2 ) < ∞,
c (α; q 2 )
n=0 n
which proves (i).
P∞
(ii) If f (z) = n=0 an z n ∈ Fq,α , from (4) and (5), we deduce
hf, Kq,α (w, .)iFq,α =
∞
X
an wn = f (w),
n=0
This completes the proof of the theorem.
w ∈ D(0,
1
).
1−q
¤
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
7
1
Remark 2. From Theorem 2.1 (ii), for f ∈ Fq,α and w ∈ D(0, 1−q
), we have
|f (w)| ≤ kKq,α (w, .)kFq,α kf kFq,α = [Sα (|w|2 ; q 2 )]1/2 kf kFq,α .
(6)
Proposition 2.1. The space
n Fq,α oequipped with the inner product h., .iFq,α is an
Hilbert space; and the set ξn (.; q 2 )
given by
n∈N
ξn (z; q 2 ) = p
zn
cn
(α; q 2 )
,
z ∈ D(0,
1
),
1−q
forms an Hilbert basis for the space Fq,α .
Proof. Let {fn }n∈N be a Cauchy sequence in Fq,α . We put
f = lim fn ,
n→∞
in Fq,α .
From (6), we have
|fn+p (w) − fn (w)| ≤ [Sα (|w|2 ; q 2 )]1/2 kfn+p − fn kFq,α .
This inequality shows that the sequence {fn }n∈N is pointwise convergent to f .
1
Since the function w → [Sα (|w|2 ; q 2 )]1/2 is continuous on D(0, 1−q
), then {fn }n∈N
1
converges to f uniformly on all compact set of D(0, 1−q ). Consequently, f is an
1
entire function on D(0, 1−q
), then f belongs to the space Fq,α .
On the other hand, from the relation (4), we get
hξn (.; q 2 ), ξm (.; q 2 )iFq,α = δn,m ,
where δn,m is the Kroneckernsymbol. o
is an orthonormal set in Fq,α .
This shows that the family ξn (.; q 2 )
n∈N
P∞
Let f (z) = n=0 an z n be an element of Fq,α such that
hf, ξn (.; q 2 )iFq,α = 0,
∀ n ∈ N.
From the relation (4), we deduce that
an = 0,
This completes the proof.
Remark 3.
n
(a) The set Sα (w.; q 2 ),
∀ n ∈ N.
o
1
w ∈ D(0, 1−q
) is dense in Fq,α .
¤
1
(b) For all z, w ∈ D(0, 1−q
), we have
Sα (wz; q 2 ) = hSα (z.; q 2 ), Sα (w.; q 2 )iFq,α .
3. Multiplication and translation operators on Fq,α
On Fq,α , we consider the multiplication operators M and Nq given by
M f (z) := z 2 f (z),
f (z) − f (qz)
.
1−q
the q-Bessel-Struve operator defined for entire functions on
Nq f (z) := zDq f (z) =
We denote also by `q,α
1
D(0, 1−q
).
We write
[`q,α , M ] = `q,α M − M `q,α .
Then by straightforward calculation we obtain.
8
FETHI SOLTANI
Lemma 3.1. [`q,α , M ] = (1 + q)[2α + 2]q Bq2 + Wq,α ,
where
Bq f (z) := f (qz),
Wq,α f (z) := (1 + q)(1 + q 2α )qzDq f (qz) + [2α + 1]q zDq f (0).
(7)
Remark 4. The constant (1 + q)[2α + 2]q equals to c2 (α; q 2 ), and the Lemma 3.1
is the analogous commutation rule of [7]. When q → 1− , then [`q,α , M ] tends to
4(α + 1)I + W , where I is the identity operator and W is the operator given by
W f (z) := 4zDf (z) + (2α + 1)zDf (0).
Lemma 3.2. If f ∈ Fq,α then Bq f , Nq f and Wq,α f belong to Fq,α , and
(i) kBq f kFq,α ≤ kf kFq,α ,
1
(ii) kNq f kFq,α ≤ 1−q
kf kFq,α ,
2α
)
kf kFq,α .
(iii) kWq,α f kFq,α ≤ (1+q)(1+q
1−q
P∞
Proof. Let f (z) = n=0 an z n ∈ Fq,α , then
Bq f (z) = f (qz) =
∞
X
an q n z n ,
(8)
n=0
Nq f (z) =
∞
f (z) − f (qz) X
=
an [n]q z n ,
1−q
n=0
(9)
and from (3), we obtain
kBq f k2Fq,α
=
∞
X
2 2n
2
|an | q cn (α; q ) ≤
n=0
∞
X
|an |2 cn (α; q 2 ) = kf k2Fq,α ,
n=0
and
kNq f k2Fq,α =
∞
X
|an |2 ([n]q )2 cn (α; q 2 ).
n=0
Using the fact that [n]q ≤
kNq f k2Fq,α ≤
1
1−q ,
we deduce
∞
X
1
1
|an |2 cn (α; q 2 ) =
kf k2Fq,α .
2
(1 − q) n=0
(1 − q)2
On the other hand from (7), we have
Wq,α f (z) = [2α + 1]q a1 z + (1 + q)(1 + q 2α )
∞
X
an [n]q q n z n ,
(10)
n=1
and
kWq,α f k2Fq,α = (q+q 2 +[2α+3]q )2 |a1 |2 c1 (α; q 2 )+[(1+q)(1+q 2α )]2
∞
X
n=2
Using the fact that [n]q ≤
1
1−q ,
kWq,α f k2Fq,α ≤
we deduce that
∞
[(1 + q)(1 + q 2α )]2 X
|an |2 cn (α; q 2 ).
(1 − q)2
n=1
|an |2 ([n]q )2 q 2n cn (α; q 2 ).
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
9
Therefore, we conclude that
kWq,α f kFq,α ≤
(1 + q)(1 + q 2α )
kf kFq,α .
1−q
which completes the proof of the Lemma.
¤
In the classical Fock spaces Fα , if f ∈ Fα , the functions `α f and M f are
not necessarily elements of Fα . For thus, the authors statued these operators on
D(`α ) = {f ∈ Fα : `α f ∈ Fα } and D(M ) = {f ∈ Fα : M f ∈ Fα }, respectively.
But on Fq,α , we can study the continuous property of the operators `q,α and M
from Fq,α into itself.
Theorem 3.1. If f ∈ Fq,α then `q,α f and M f belong to Fq,α , and we have
1
(i) k`q,α f kFq,α ≤ 1−q
kf kFq,α ,
1
(ii) kM f kFq,α ≤ 1−q kf kFq,α .
P∞
Proof. Let f (z) = n=0 an z n ∈ Fq,α .
(i) From Lemma 2.3,
`q,α f (z) =
∞
X
an
n=2
∞
cn (α; q 2 ) n−2 X
cn+2 (α; q 2 ) n
z
=
an+2
z .
2
cn−2 (α; q )
cn (α; q 2 )
n=0
(11)
Then from (11), we get
k`q,α f k2Fq,α =
∞
X
|an+2 |2
n=0
cn+2 (α; q 2 )
cn+2 (α; q 2 ).
cn (α; q 2 )
Using Lemma 2.2, we obtain
k`q,α f k2Fq,α
∞
X
=
|an+2 |2 [n + 2]q [n + 2α + 2]q cn+2 (α; q 2 ),
n=0
and consequently,
k`q,α f k2Fq,α =
∞
X
|an |2 [n]q [n + 2α]q cn (α; q 2 ).
(12)
n=2
Using the fact that [n]q [n + 2α]q ≤
k`q,α f kFq,α ≤
1
(1−q)2 ,
we obtain
∞
i1/2
1 hX
1
|an |2 cn (α; q 2 )
=
kf kFq,α .
1 − q n=0
1−q
(ii) On the other hand, since
M f (z) =
∞
X
an−2 z n ,
(13)
n=2
then
kM f k2Fq,α =
∞
X
|an−2 |2 cn (α; q 2 ) =
n=2
∞
X
|an |2 cn+2 (α; q 2 ).
n=0
By Lemma 2.2, we deduce
kM f k2Fq,α =
∞
X
n=0
|an |2 [n + 2]q [n + 2α + 2]q cn (α; q 2 ).
(14)
10
FETHI SOLTANI
Using the fact that [n + 2]q [n + 2α + 2]q ≤
kM f kFq,α ≤
1
(1−q)2 ,
we obtain
1
kf kFq,α ,
1−q
which completes the proof of the theorem.
We deduce also the following norm equalities.
¤
Theorem 3.2. If f ∈ Fq,α then
(i) hf, Wq,α f iFq,α = (1+q)(1+q 2α )hNq f, Bq f iFq,α +[2α+1]q |Dq f (0)|2 c1 (α; q 2 ),
(ii) k`q,α f k2Fq,α = kNq f k2Fq,α +[2α]q hNq f, Bq f iFq,α −[2α+1]q |Dq f (0)|2 c1 (α; q 2 ),
(iii) kM f k2Fq,α = kNq f k2Fq,α + (1 + q)[2α + 2]q kBq f k2Fq,α + (1 + q + [2α +
2]q )hNq f, Bq f iFq,α ,
(iv) kM f k2Fq,α = k`q,α f k2Fq,α + (1 + q)[2α + 2]q kBq f k2Fq,α + hf, Wq,α f iFq,α .
P∞
Proof. Let f (z) = n=0 an z n ∈ Fq,α .
(i) Follows from (8), (9) and (10).
(ii) From (12), we get
k`q,α f k2Fq,α =
∞
X
|an |2 [n]q [n + 2α]q cn (α; q 2 ) − [2α + 1]q |Dq f (0)|2 c1 (α; q 2 ).
n=0
Using the fact [n + 2α]q = [n]q + q n [2α]q , we deduce
k`q,α f k2Fq,α = kNq f k2Fq,α + [2α]q hNq f, Bq f iFq,α − [2α + 1]q |Dq f (0)|2 c1 (α; q 2 ).
(iii) By (14) and using the fact that
[n + 2]q [n + 2α + 2]q = ([n]q )2 + (1 + q + [2α + 2]q )q n [n]q + (1 + q)[2α + 2]q q 2n ,
we obtain
kM f k2Fq,α = kNq f k2Fq,α +(1+q)[2α+2]q kBq f k2Fq,α +(1+q+[2α+2]q )hNq f, Bq f iFq,α .
(iv) Follows directly from (i), (ii) and (iii).
¤
P∞
Remark 5.
(a) Let f (z) = n=0 an z n ∈ Fq,α . Since hf, Wq,α f iFq,α ≥ 0, then
kM f k2Fq,α ≥ (1 + q)[2α + 2]q kBq f k2Fq,α .
Therefore M f = 0 implies that f = 0. Then M : Fq,α → Fq,α is injective
continuous operator on Fq,α .
(b) In the classical Fock spaces Fα the norm inequalities of Theorem 3.2 are
realized on D(M ), and therefore M : D(M ) → Fq,α is injective continuous
operator on D(M ).
In the classical Fock spaces Fα , for f ∈ D(M ) and g ∈ D(`α ), we have
hM f, giFα = hf, `α giFα .
But in Fq,α , and since D(M ) = D(`q,α ) = Fq,α , we obtain the following.
Proposition 3.1. The operators M and `q,α are adjoint-operators on Fq,α ; and
for all f, g ∈ Fq,α , we have
hM f, giFq,α = hf, `q,α giFq,α .
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
Proof. Consider f (z) =
and (13),
P∞
n=0
an z n and g(z) =
`q,α g(z) =
∞
X
bn+2
n=0
and
M f (z) =
∞
X
P∞
n=0 bn z
n
11
in Fq,α . From (11)
cn+2 (α; q 2 ) n
z ,
cn (α; q 2 )
an−2 z n .
n=2
Thus from (4), we get
hM f, giFq,α =
∞
X
an−2 bn cn (α; q 2 ) =
n=2
∞
X
an bn+2 cn+2 (α; q 2 ) = hf, `q,α giFq,α ,
n=0
which gives the result.
¤
In the next part of this section we study a generalized translation and multiplication operators on Fq,α . We begin by the following definition.
1
Definition 3.1. For f ∈ Fq,α and w, z ∈ D(0, 1−q
), we define:
- The q-translation operators on Fq,α , by
∞
X
`nq,α f (w) n
z .
c (α; q 2 )
n=0 n
Tz f (w) :=
(15)
- The generalized multiplication operators on Fq,α , by
Mz f (w) :=
∞
X
M n f (w) n
z .
c (α; q 2 )
n=0 n
(16)
1
), the function Sα (.; q 2 ) satisfies the following product forFor w, z ∈ D(0, 1−q
mulas:
Tz Sα (.; q 2 )(w) = Sα (z; q 2 )Sα (w; q 2 ),
Mz Sα (.; q 2 )(w) = Sα (zw2 ; q 2 )Sα (w; q 2 ).
P∞
1
). Then
Proposition 3.2. Let f (z) = n=0 an z n ∈ Fq,α and z, w ∈ D(0, 1−q
h
i
n
P∞
P[n/2]
(
(α+1)Γ
Γ
+α+1)
2
2
q
q
2
z k
2
n
(i) Tz f (w) = n=0 an
k=0 γ(n, k; q ) Γ ( k +α+1)Γ ( n −k+α+1) ( w2 ) w ,
q2
where [n/2] is the integer part of n/2 and
γ(n, k; q 2 ) =
(ii) Mz f (w) =
P∞ h P[n/2]
n=0
k=0
q2
2
(1 + q)k Γq2 ( n2 + 1)
Γq2 ( k2 + 1)Γq2 ( n2 − k + 1)
i
an−2k
k
z
wn .
2
ck (α;q )
2
.
P∞
Proof. Let f (z) = n=0 an z n ∈ Fq,α .
(i) From (15), we have
∞
X
`nq,α f (w) n
z ;
Tz f (w) =
c (α; q 2 )
n=0 n
w, z ∈ D(0,
1
).
1−q
Since from Lemma 2.3,
`nq,α wk =
ck (α; q 2 )
wk−2n ,
ck−2n (α; q 2 )
k ≥ 2n,
12
FETHI SOLTANI
we can write
`nq,α f (w) =
∞
X
ak
k=2n
ck (α; q 2 )
wk−2n .
ck−2n (α; q 2 )
Thus we obtain
Tz f (w) =
∞
X
n=0
[n/2]
an
X
k=0
cn (α; q 2 )
wn−2k z k .
ck (α; q 2 )cn−2k (α; q 2 )
On the other hand from (1), we get
Γq2 (α + 1)Γq2 ( n2 + α + 1)
cn (α; q 2 )
2
=
γ(n,
k;
q
)
,
ck (α; q 2 )cn−2k (α; q 2 )
Γq2 ( k2 + α + 1)Γq2 ( n2 − k + α + 1)
which gives the (i).
(ii) From (16), we have
Mz f (w) =
∞
X
M n f (w) n
z ;
c (α; q 2 )
n=0 n
But from (13), we have
M n f (w) =
∞
X
w, z ∈ D(0,
1
).
1−q
ak−2n wk .
k=2n
Thus we obtain
Mz f (w) =
∞ h [n/2]
i
X
X an−2k
k
z
wn ,
2)
c
(α;
q
k
n=0
k=0
which completes the proof of the proposition.
¤
In the classical Fock spaces Fα , if f ∈ Fα , then τz f and Mz f , z ∈ C are not
necessarily elements of Fα . But on Fq,α , and according to Theorem 3.1 we can study
1
the continuous property of the operators Tz and Mz on Fq,α , for z ∈ D(0, 1−q
).
1
Theorem 3.3. If f ∈ Fq,α and z ∈ D(0, 1−q
), then Tz f and Mz f belong to Fq,α ,
and
|z|
; q 2 )kf kFq,α ,
(i) kTz f kFq,α ≤ Sα ( 1−q
|z|
(ii) kMz f kFq,α ≤ Sα ( 1−q
; q 2 )kf kFq,α .
Proof. From (15) and Theorem 3.1 (i), we deduce
∞
∞
X
X
|z|n
|z|n
kTz f kFq,α ≤
k`nq,α f kFq,α
≤
kf kFq,α .
cn (α; q 2 ) n=0 (1 − q)n cn (α; q 2 )
n=0
Therefore,
|z|
; q 2 )kf kFq,α ,
1−q
which gives the first inequality, and as in the same way we prove the second inequality of this theorem.
¤
From Proposition 3.1 we deduce the following results.
kTz f kFq,α ≤ Sα (
1
Proposition 3.3. For all f, g ∈ Fq,α and z ∈ D(0, 1−q
), we have
hMz f, giFq,α = hf, Tz giFq,α ,
hTz f, giFq,α = hf, Mz giFq,α .
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
13
1
We denote by Rz , z ∈ D(0, 1−q
) the following operator defined on Fq,α by
Rz := Tz Mz − Mz Tz = Sα (z`q,α ; q 2 )Sα (zM ; q 2 ) − Sα (zM ; q 2 )Sα (z`q,α ; q 2 ).
Then, we prove the following theorem.
1
Theorem 3.4. For all f ∈ Fq,α and z ∈ D(0, 1−q
), we have
(i) kMz f k2Fq,α = kTz f k2Fq,α + hf, Rz f iFq,α ,
|z|
(ii) kRz f kFq,α ≤ 2[Sα ( 1−q
; q 2 )]2 kf kFq,α .
Proof. (i) From Proposition 3.3, we get
kMz f k2Fq,α = hf, Tz Mz f iFq,α = hf, (Mz Tz + Rz )f iFq,α = kTz f k2Fq,α + hf, Rz f iFq,α .
(ii) From Theorem 3.3, we have
kRz f kFq,α ≤ kTz Mz f kFq,α + kMz Tz kFq,α ≤ 2[Sα (
|z|
; q 2 )]2 kf kFq,α ,
1−q
which completes the proof.
¤
4. Application: Extremal function on Fq,α
In this section we shall give an application of the theory of reproducing kernels
to the Tikhonov regularization, which gives the approximate solutions for bounded
linear operator equation on the Fock spaces Fq,α .
Definition 4.1. Let λ > 0 and let L : Fq,α → Fq,α be a bounded linear operator
from Fq,α into itself. We denote by h., .iλ,Fq,α the inner product defined on the space
Fq,α by
hf, giλ,Fq,α := λhf, giFq,α + hLf, LgiFq,α ,
(17)
p
and the norm kf kλ,Fq,α := hf, f iλ,Fq,α .
As examples of the operator L we can choose the precedent operators Bq , Nq ,
1
Wq,α , `α,q , M , Tz , Mz and Rz , when z ∈ D(0, 1−q
).
Remark 6. Let λ > 0 and let f ∈ Fq,α . The two norms k.kFq,α and k.kλ,Fq,α are
equivalent, and
p
√
λ kf kFq,α ≤ kf kλ,Fq,α ≤ λ + kLk2 kf kFq,α .
Lemma 4.1. Let λ > 0. The Fock space (Fq,α , h., .iλ,Fq,α ) possesses a reproducing
1
kernel KL (w, z); w, z ∈ D(0, 1−q
) which satisfying the equation
(λI + L∗ L)KL (w, .) = Kq,α (w, .),
(18)
where L∗ is the adjoint of L in (Fq,α , h., .iFq,α ).
Proof. Let f ∈ Fq,α . From relation (6) and Remark 6, we have
h S (|w|2 ; q 2 ) i1/2
α
|f (w)| ≤ kKq,α (w, .)kFq,α kf kFq,α =
kf kλ,Fq,α .
λ
1
Then, the map f → f (w), w ∈ D(0, 1−q
) is a continuous linear functional on
(Fq,α , h., .iλ,Fq,α ). Thus from [1], (Fq,α , h., .iλ,Fq,α ) has a reproducing kernel denoted
14
FETHI SOLTANI
by KL (w, z).
On the other hand,
f (w) = λhf, KL (w, .)iFq,α + hLf, L[KL (w, .)]iFq,α
= hf, (λI + L∗ L)[KL (w, .)]iFq,α .
Thus,
(λI + L∗ L)KL (w, .) = Kq,α (w, .).
This clearly yields the result.
Example 4.1. Let w, z ∈ D(0,
¤
1
1−q ).
(a) If L = M , then
KL (w, z) =
∞
X
(wz)n
.
(λ + [n + 2]q [n + 2α + 2]q )cn (α; q 2 )
n=0
(b) If L = `q,α , then
∞
KL (w, z) =
X
1
wz
(wz)n
+
+
.
λc0 (α; q 2 ) λc1 (α; q 2 ) n=2 (λ + [n]q [n + 2α]q )cn (α; q 2 )
(c) If L = Bq , then
KL (w, z) =
∞
X
(wz)n
.
(λ + q 2n )cn (α; q 2 )
n=0
(d) If L = Nq , then
KL (w, z) =
∞
X
(wz)n
.
(λ + ([n]q )2 )cn (α; q 2 )
n=0
(e) If L = Wq,α , then
∞
KL (w, z) =
where
X
1
wz
(wz)n
+
+
,
2
2
2
λc0 (α; q ) ([2α + 1]q ) c1 (α; q ) n=1 (λ + η([n]q )2 q 2n )cn (α; q 2 )
η = (1 + q)2 (1 + q 2α )2 .
The main result of this section can then be stated as follows.
Theorem 4.1. For any h ∈ Fq,α and for any λ > 0, there exists a unique function
∗
fλ,h
, where the infimum
n
o
inf λkf k2Fq,α + kh − Lf k2Fq,α
(19)
f ∈Fq,α
∗
is given by
is attained. Moreover, the extremal function fλ,h
∗
fλ,h
(w) = hh, L[KL (w, .)]iFq,α ,
where KL is the kernel given by (18).
∗
Proof. The existence and unicity of the extremal function fλ,h
satisfying (19), is
given by [9, 11, 13]. Moreover, by Lemma 4.1 we deduce that
∗
fλ,h
(w) = hh, L[KL (w, .)]iFq,α ,
where KL is the kernel given by (18).
This clearly yields the result.
(20)
¤
FOCK SPACES FOR THE q-BESSEL-STRUVE KERNEL
15
∗
Remark 7. The extremal function fλ,h
satisfies the following inequalities
∗
|fλ,h
(w)| ≤ kLkkhkFq,α kKL (w, .)kFq,α
kLk
≤ √ khkFq,α kKL (w, .)kλ,Fq,α
λ
kLk
≤ √ [KL (w, w)]1/2 khkFq,α .
λ
If we take in (20), h = Lf , where f ∈ Fq,α , we obtain the following Calderón’s
reproducing formula.
Theorem 4.2. (Calderón’s formula). Let λ > 0 and f ∈ Fq,α . The extremal
function fλ∗ given by
fλ∗ (w) = hLf, L[KL (w, .)]iFq,α ,
satisfies
f (w) = lim+ fλ∗ (w) = lim+ hLf, L[KL (w, .)]iFq,α .
λ→0
λ→0
Proof. Let f ∈ Fq,α , h = Lf and
fλ∗
∗
= fλ,Lf
. Then
fλ∗ (w) = hf, L∗ L[KL (w, .)]iFq,α .
(21)
But from (18), we have
lim L∗ L[KL (w, .)] = Kq,α (w, .).
λ→0+
Thus,
lim fλ∗ (w) = hf, Kq,α (w, .)iFq,α = f (w),
λ→0+
which ends the proof.
¤
1
Remark 8. Let w ∈ D(0, 1−q
). From (18) and (21), the extremal function fλ∗
satisfies
fλ∗ (w) = f (w) − λhf, KL (w, .)iFq,α .
Thus we obtain
lim λKL (w, .) = 0,
λ→0+
and
|fλ∗ (w) − f (w)| ≤ λkf kFq,α kKL (w, .)kFq,α
√
≤ λkf kFq,α kKL (w, .)kλ,Fq,α
√
≤ λ[KL (w, w)]1/2 kf kFq,α .
Acknowledgements. The author is very grateful to the referee for many critical
comments on this paper.
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Higher College of Technology and Informatics, Street of the Employers 45, Charguia 2 - 2035 Tunis, Tunisia
E-mail address: fethisoltani10@yahoo.com