Basic Fourier transform on the space of entire functions of logarithm order 2
© Bouzeffour; licensee Springer 2012
Received: 29 May 2012
Accepted: 1 October 2012
Published: 24 October 2012
We study in this paper the basic Fourier transform, q-translation and q-convolution associated to the q-difference operator in the space of entire functions with logarithmic order 2 and finite logarithmic type and their dual.
The concept of the basic Fourier transform is related to the quantum group which is a q-deformation of the Lie group. The deformation parameter q is always assumed to satisfy . The basic Fourier transform was defined firstly in  and studied after that from the point of view of harmonic analysis in [2–7], ….
In this work, we are interested in the basic Fourier transforms of entire functions with logarithmic order 2 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 inspired by the ideas developed by [9, 10] and . Some of the arguments used here are similar to the one considered in  and . However, we need to introduce new procedures to prove the results in the q-theory setting.
The paper is organized as follows. In Section 2, we give a brief introduction and recall some known results about q-shift factorial, q-derivative and q-exponential function. In Section 3, 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 4, we study a new q-translation operator and its related q-convolution and we give several characterizations from the space of entire functions into itself that commute with the q-translation. Finally, in Section 5, we define the q-Fourier transform on the dual space of entire functions and we establish a q-Paley-Wiener theorem type.
3 Space of entire functions of finite logarithmic order
Proposition 1 The operatoris continuous frominto its self.
where . Thus, we conclude that the operator is continuous from into itself. □
This proves the result. □
It is easy to see that:
Thus, is a Banach space. The dual of is denoted, as usual, by .
Lemma 3 ()
Proposition 4 The logarithmic order of the q-exponential functionis equal to 2 and its logarithmic type is.
Proof The result follows from Lemma 3. □
We consider the mapping I from into defined by and the functional λ from into ℂ given by .
It is clear that λ is continuous when on we consider the topology induced in it by the usual topology of . By using Hanh-Banach and Riesz representation theorems in a standard way, we can conclude that χ admits a representation like (9) for a certain complex regular Borel measure γ on ℂ and . □
4 q-translation and q-convolution
In this section, we define a new q-translation operator related to q-difference operator , on the space of entire functions of logarithm order 2.
This shows the result. □
The result follows from the fact that the q-exponential function is an eigenfunction of the operator corresponding to the eigenvalue 1. □
In the following theorem, we obtain several characterizations of the continuous linear mappings L from into itself that commute with the q-translation operators.
- (i)L commutes with the q-translation operators , , that is,
L commutes with the q-difference operator .
There exists a unique such that , .
- (iv)There exists a complex Borel regular measure having compact support on ℂ, for which, for all , we have
- (v)There exists an entire function Ψ of logarithmic order 2 such that
Hence, (i) ⇒ (ii).
- (i)⇒ (iii) Assume that (i) holds. We define the functional ϑ on as follows:
⇒ (iv) It follows immediately from Hahn-Banach and Riesz representation theorems.
- (iv)⇒ (v) Suppose that for all f, we have
- (iv)⇒ (i) Suppose that for every , we obtain
5 The q-Fourier transform on the space
Theorem 11 (q-Paley-Wiener theorem)
The q-Fourier transformis a topological isomorphism fromonto.
We conclude that is continuous from onto . □
This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
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