R E S E A R C H
Open Access
Basic Fourier transform on the space of entire
functions of logarithm order 2
F Bouzeffour
**Correspondence:
fbouzaff[email protected] Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
Abstract
We study in this paper the basic Fourier transform,q-translation andq-convolution associated to theq-difference operator in the space of entire functions with logarithmic order 2 and finite logarithmic type and their dual.
1 Introduction
The concept of the basic Fourier transform is related to the quantum group which is a q-deformation of the Lie group. The deformation parameterqis always assumed to satisfy <q< . The basic Fourier transform was defined firstly in [] and studied after that from the point of view of harmonic analysis in [–], . . . .
In this work, we are interested in the basic Fourier transforms of entire functions with logarithmic order and finite logarithmic type which is introduced by []. This notion of logarithmic order is a refinement order of entire functions of order zero which is used to study the growth of order of some basic hypergeometric series. Our investigation is in-spired by the ideas developed by [, ] and []. Some of the arguments used here are similar to the one considered in [] and []. However, we need to introduce new proce-dures to prove the results in theq-theory setting.
The paper is organized as follows. In Section , we give a brief introduction and recall some known results aboutq-shift factorial,q-derivative andq-exponential function. In Section , firstly we describe the space of entire functions and its dual. Also, we give a new q-Taylor expansion of an entire function. Secondly, we introduce the logarithmic order and logarithmic type. In Section , we study a newq-translation operator and its related q-convolution and we give several characterizations from the space of entire functions into itself that commute with theq-translation. Finally, in Section , we define theq-Fourier transform on the dual space of entire functions and we establish aq-Paley-Wiener theorem type.
2 Preliminaries
We assume thatz∈Cand <q< , unless specified otherwise. We recall some notations []. For an arbitrary complex numbera,
(a,q)n:=
⎧ ⎨ ⎩
forn= ,
( –a)( –aq)· · ·( –aqn–) forn≥,
(a,q)∞:= lim
n→∞(a,q)n,
and
n k
q
:= (q,q)n (q,q)k(q,q)n–k
.
Theq-derivative of a functionf(z) is defined as in [] by
Dqf(z) :=
f(z) –f(qz)
( –q)z , z= ,
and we say thatf has theq-derivative at zero if the limit
lim
n→∞
f(zqn) –f()
zqn
exists and does not depend onz. We define the operator
qf(z) :=f
q–z. Obviously,
qDq=qDqq
and for an arbitrary positive integern,
( –q)k
(q;q)k
(qDz)kzn=
⎧ ⎨ ⎩
n k
qq( k+
)–nkzn–k for ≤k≤n,
fork>n.
()
Consider theq-exponentials (see []) defined by
eq(z) :=
∞
n=
zn (q,q)n
=
(z,q)∞, |z|<
and
Eq(z) :=
∞
n=
q(n)zn
(q,q)n
= (–z,q)∞. ()
Theq-exponentials functionseq(z) andEq(z) satisfy
Dqeq(z) =eq(z) and qDqEq(z) =Eq(z).
3 Space of entire functions of finite logarithmic order
We denote byAthe space of entire functions onC. This space is endowed with the topol-ogy of the uniform convergence on compact subsets ofC. Thus,Ais a Frechet space.A denotes the strong dual space ofA. IfT∈A, there exist a constantC> and a compact subsetKofCsuch that
T,f≤Csup
z∈K
According to the Hahn-Banach theorem, the mapping T can be extended to the space C(K) of continuous functions onK, as an element of the dual space ofC(K). Then the Riesz representation theorem implies there exists a regular complex Borel measureγsupported inKsuch that
T,f=
Cf(z)dγ(z), f ∈A.
Proposition The operatorqDqis continuous fromAinto its self.
Proof According to the Cauchy integral formula, for|z|<qr, we can write
qDqf(z) =
iπ
|u|=r
f(u)
(u–q–z)(u–z)du.
Then
|qDqf|qr≤
( –q)( –q)r|f|r,
where|f|r=sup|z|≤r|f(z)|. Thus, we conclude that the operatorqDqis continuous from
Ainto itself.
In the following theorem, we establish a newq-Taylor formula different from the one considered in [] and [].
Theorem If f ∈A,then f admits the representation
f(z) = ∞
n=
( –q)nq(n )zn
(q;q)n
(qDq)nf().
Proof Suppose that
f(z) = ∞
n=
anzn.
Then fork= , , . . . , we have
( –q)k
(q;q)k
(qDq)kf(z) =
∞
n=k
an
n k
q
q(k+)–nkzn–k. ()
By takingz= , we obtain fork= , , . . . that
ak=
( –q)k
(q;q)k
q(k)(
qDq)kf(), ()
This proves the result.
We recall (see []) that an entire functionf of order zero has logarithmic orderif
:= lim
r→+∞sup ln ln|f|r
ln lnr , ()
and when the logarithmic orderoff is finite, we define the logarithmic typeτ as
τ:= lim
r→+∞sup ln|f|r
(lnr). ()
It is easy to see that:
f is of logarithmic orderρif and only if, for everyε> ,
|f|r=O
e(lnr)ρ+ε
.
Iff is of logarithmic orderρ, then its logarithmic type is equal toτ if and only if
|f|r=O
e(τ+ε)(lnr)ρ).
We denote byEτ,q(C) the space of entire functions of logarithmic order and logarithmic
typeτ. In particular, whenτ=lnq–,Eτ,q(C) is denoted simply byEq(C). That is, an entire
functionf is inEτ,q(C) if and only if for everyε> ,
f(z)=Oe(τ+ε)(ln|z|), as|z| → ∞.
The spaceEτ,q(C) is endowed by the topology associated to the family{ρτ,ε}of seminorms,
where for everyε> ,
ρτ,ε(f) =sup z∈C
f(z)e–(τ+ε)ln|z|.
Thus,Eτ,q(C) is a Banach space. The dual ofEτ,q(C) is denoted, as usual, byEτ,q(C).
Lemma ([]) Let f(z) =n∞=anznbe an entire function of order.Its logarithmic order ρand logarithmic typeτ satisfy
ρ= +lim sup
n→∞
lnn
ln ln(√n|a n|)
()
and
τ=(ρ– )
ρ–
ρρ lim sup n→∞
n (ln(√n|a
n|))
ρ–. ()
Proposition The logarithmic order of the q-exponential function Eq(z)is equal toand
its logarithmic type is–ln(q).
Proposition Letχbe a functional on Eτ,q(C),thenχ∈Eτ,q(C)if and only if there exists
Proof Suppose thatχ admits the representation () for a certain complex regular Borel measureγ onCandε> . Then we have
it by the usual topology ofC. By using Hanh-Banach and Riesz representation theorems
in a standard way, we can conclude thatχadmits a representation like () for a certain complex regular Borel measureγ onCandε> .
4 q-translation andq-convolution
In this section, we define a newq-translation operator related toq-difference operator
qDq, on the space of entire functions of logarithm order .
Definition Theq-translationTξ is defined on monomialsznby
Tξzn:=
Proposition Forξ∈C,the q-translation operator Tξcan be extended on the entire
func-tion f(z) =∞n=anznof logarithmic orderand logarithmic type lesser than –lnq in the
Proof It suffices to prove the convergence of the following series:
For everyn∈Nandk= , . . . ,n, we can write
On the other hand, the function
ϕ(x) =
Byq-binomial theorem, we get
Then
|A| ≤C ∞
n=
q(–β)nqn+zn
–|ξ|
|z|;q
n
<∞.
This shows the result.
Proposition For z,ξ∈C,the q-exponential function satisfies the product formula
TξEq(z) =Eq(z)Eq(ξ).
Proof Letf be an entire function of logarithmic order and logarithmic type lesser than
–lnq. The series defined in () is absolutely convergent. Then, after interchanging the
double sum, we can write
Tξf(z) =
∞
k=
( –q)k
(q;q)k
q–(n–k)kq(k)ξk
∞
n=k
an
n k
q
q(k+)–nkzn–k.
By (), the operatorTξ can be represented by the form
Tξf(z) =
∞
n=
( –q)n
(q;q)n
q(n)ξn(qDq)n(f)(z). ()
The result follows from the fact that theq-exponential functionEq(z) is an eigenfunction
of the operatorqDqcorresponding to the eigenvalue .
Proposition For f ∈Eτ,q(C),withτ lesser than –lnqand z,ξ∈C,we have
Tzf(ξ) =Tξf(z), Tf =f,
Tz◦Tξf =Tξ◦Tzf, (qDq)Tzf=Tz(qDq)f.
Now, we define theq-convolution ofχ∈Eτ,q(C) andf ∈Eτ,q(C) as
χ f(ξ) = χ;Tξf. ()
In the following theorem, we obtain several characterizations of the continuous linear mappingsLfromAinto itself that commute with theq-translation operators.
Theorem Let L be a continuous linear mapping fromAinto itself.The following asser-tions are equivalent:
(i) Lcommutes with theq-translation operatorsTz,z∈C,that is,
TzL=LTz, z∈C.
(ii) Lcommutes with theq-difference operatorqDq.
(iv) There exists a complex Borel regular measure having compact support onC,for which,for allf ∈A,we have
L(f)(z) =
Cσzf(w)dγ(w).
(v) There exists an entire functionof logarithmic ordersuch that
Lf(z) =(qDq)f(z),
where
(z) = ∞
n=
anzn.
Proof (i)⇒(ii) Letg∈A, we have
Tzg(ξ) –g(ξ)
z =
∞
n=
( –q)n (q;q)n
q(n)zn–(qDq)n(g)(ξ). ()
Hence,
lim
z→
Tzg(ξ) –g(ξ)
z =qDqg(ξ).
The operatorLis continuous, then
qDqLg(ξ) =lim z→
TzLg(ξ) –Lg(ξ)
z =Lzlim→
Tzg(ξ) –g(ξ)
z =LqDqg(ξ).
Hence, (i)⇒(ii).
(ii)⇒(i) The operatorLcommutes withqDq, then for everyn∈N,
L(qDq)n= (qDq)nL.
Hence,
LTzg(ξ) =
∞
n=
( –q)n
(q;q)n
q(n)zn–L(qDq)n(g)(ξ)
= ∞
n=
( –q)n
(q;q)n
q(n)zn–(
qDq)nL(g)(ξ)
=TzLg(z).
(i)⇒(iii) Assume that (i) holds. We define the functionalϑonAas follows:
ϑ;f=Lf().
It is clear thatϑ∈Aand
(iii)⇒(iv) It follows immediately from Hahn-Banach and Riesz representation theo-rems.
(iv)⇒(v) Suppose that for allf, we have
Lf(ξ) =
CTξf(z)dγ(z),
whereγ is a complex Borel regular measure with compact support. According to [], we obtain
L(f)(z) = ∞
n=
(qDq)nf(z)
C
( –q)n
(q;q)n
q(n)ξndγ(ξ).
Hence,
Lf(z) =(qDq)f(z),
where
(z) = ∞
n=
anzn, withan=
C
( –q)n
(q;q)n
q(n)ξndγ(ξ).
Sinceγ has compact support onC, for certainaandC, we have
|an| ≤
( –q)n
(q;q)n
q(n)an.
Then we get
(z)≤Eq
a|z|,
andis of logarithmic order .
(iv)⇒(i) Suppose that for everyf ∈A, we obtain
L(f)(z) = ∞
n=
(qDq)nf(z)
C
( –q)n
(q;q)n
q(n)ξndγ(ξ).
Hence, iff ∈AsinceTzqDqf =qDqTzf,
TzL(f)(z) =
∞
n=
Tz(qDq)nf(z)
= ∞
n=
(qDq)nTzf(z)
5 Theq-Fourier transform on the spaceA We introduce theq-Fourier transform onAby
Fq(T)(ξ) =
T(z),Eq(–iξz)
, ξ,z∈C. ()
From () theq-Fourier transform takes the form
Fq(T)(ξ) =
Theorem (q-Paley-Wiener theorem) The q-Fourier transformFqis a topological
iso-morphism fromAonto Eq(C).
Proof Assume firstly thatT ∈A. Then there exists a complex regular Borel measureγ
onCand a > such that the support of it is contained in the open ballB(,a) of center and radiusaand
T,f=
The inequality () shows that
lim
n→∞
ln| T;zn|
Then
lim
n→∞
ln ln(√n|a n|) lnn = .
Furthermore, the logarithmic orderρofFq(T) is equal to
ρ= +lim sup
n→∞
lnn
ln ln(√n|a n|)
= .
Similarly, the logarithmic type ofτ ofFq(T) is given by
τ=lim sup
n→∞
n
ln(√n|a n|)
=
–lnq.
Hence, we have provedFq(T)∈Eq(C). Moreover,Fqis a continuous mapping fromAinto
Eq(C). Indeed, sinceAis a bornological space, it is sufficient to see thatFq(T)∈Eq(C).
Theq-Fourier transformFqis a one-to-one mapping fromAintoEq(C). Indeed, assume
thatT∈Ais such thatFq(T) = ,z∈C. Then from (), we infer that
T(w),wn= , ∀n∈N.
Hence, () implies also that
T(w),f= , ∀f ∈A.
Thus, the transformFqis one-to-one. Suppose now thatgis a function inEq(C). There
existsC> such that for everyr> ,
|g|r≤Ce
ln|r| lnq–.
From Theorem , we have
g(z) = ∞
n=
( –q)nq(n )zn
(q;q)n
(qDq)ng(), z∈C.
Then, for everyn∈N,
( –q)nq(n )
(q;q)n
(qDq)ng() =
CR
g(w) wn+dγ(w),
whereCRrepresents, for everyR> , the circular pathCR:w=Reiθ,θ∈[, ). Hence, for
everyn∈N, we have
(qDq)ng()≤C
(q;q)∞ ( –q)nRnq(n)
e ln|R|
In particular, if we takeR=q–n, we obtain
(qDq)ng()≤Can, a=
q/
–q.
On the other hand, from (), we have
iπ
Ca
Eq(–izξ)ξ––ndξ=
( –q)n
(q;q)n
q(n)(–iz)n.
For|z|>a, we put
f(z) = ∞
m=
(–i)–mz––m(qDq)m(g)().
Then the series converges absolutely inCa. Hence, for everyz∈C, we have
iπ
Ca
Eq(–izξ)f(ξ)dξ=
∞
m=
(–iz)n(qDq)n(g)()
iπ
Ca
Eq(–izξ)ξ––ndξ
= ∞
m=
( –q)nq(n )
(q;q)n
zn(qDq)n(g)()
=g(z).
We now define the functionalTonAby
χ;ϕ= iπ
Ca
f(z)ϕ(z)dz.
It is obvious thatχ∈A. Moreover,Fq(T) =g. Also, from (), we deduce that for every
bounded setofA, there existsC> such that
F–(g);ϕ≤Csup z∈
g(z). ()
We conclude thatF–is continuous fromE
q(C) ontoA.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
Received: 29 May 2012 Accepted: 1 October 2012 Published: 24 October 2012 References
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doi:10.1186/1687-1847-2012-184
Cite this article as:Bouzeffour:Basic Fourier transform on the space of entire functions of logarithm order 2.
