- Open Access
Basic Fourier transform on the space of entire functions of logarithm order 2
Advances in Difference Equations volume 2012, Article number: 184 (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.
We assume that and , unless specified otherwise. We recall some notations . For an arbitrary complex number a,
The q-derivative of a function is defined as in  by
and we say that f has the q-derivative at zero if the limit
exists and does not depend on z. We define the operator
and for an arbitrary positive integer n,
Consider the q-exponentials (see ) defined by
The q-exponentials functions and satisfy
3 Space of entire functions of finite logarithmic order
We denote by the space of entire functions on ℂ. This space is endowed with the topology of the uniform convergence on compact subsets of ℂ. Thus, is a Frechet space. denotes the strong dual space of . If , there exist a constant and a compact subset K of ℂ such that
According to the Hahn-Banach theorem, the mapping T can be extended to the space of continuous functions on K, as an element of the dual space of . Then the Riesz representation theorem implies there exists a regular complex Borel measure γ supported in K such that
Proposition 1 The operatoris continuous frominto its self.
Proof According to the Cauchy integral formula, for , we can write
where . Thus, we conclude that the operator is continuous from into itself. □
Theorem 2 If, then f admits the representation
Proof Suppose that
Then for , we have
By taking , we obtain for that
This proves the result. □
We recall (see ) that an entire function f of order zero has logarithmic order ϱ if
and when the logarithmic order ϱ of f is finite, we define the logarithmic type τ as
It is easy to see that:
f is of logarithmic order ρ if and only if, for every ,
If f is of logarithmic order ρ, then its logarithmic type is equal to τ if and only if
We denote by the space of entire functions of logarithmic order 2 and logarithmic type τ. In particular, when , is denoted simply by . That is, an entire function f is in if and only if for every ,
The space is endowed by the topology associated to the family of seminorms, where for every ,
Thus, is a Banach space. The dual of is denoted, as usual, by .
Lemma 3 ()
Letbe an entire function of order 0. Its logarithmic order ρ and logarithmic type τ satisfy
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. □
Proposition 5 Let χ be a functional on, thenif and only if there exists a Borel measure γ on ℂ andsuch that for every,
Proof Suppose that χ admits the representation (9) for a certain complex regular Borel measure γ on ℂ and . Then we have
Then . Conversely, let , then there exists C such that
We denote by the space of continuous functions in ℂ vanishing at infinity. If , then . Indeed for , we have
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.
Definition 6 The q-translation is defined on monomials by
Proposition 7 For, the q-translation operatorcan be extended on the entire functionof logarithmic order 2 and logarithmic type lesser thanin the following manner:
Proof It suffices to prove the convergence of the following series:
For every and , we can write
On the other hand, the function
attains its minimum on . For such a value , the minimum value over is
By q-binomial theorem, we get
The logarithmic type τ of the function f is lesser than , then there exists such that . By applying Cauchy estimates, we find, for every and ,
In particular, for , we get
This shows the result. □
Proposition 8 For, the q-exponential function satisfies the product formula
Proof Let f be an entire function of logarithmic order 2 and logarithmic type lesser than . The series defined in (11) is absolutely convergent. Then, after interchanging the double sum, we can write
By (3), the operator can be represented by the form
The result follows from the fact that the q-exponential function is an eigenfunction of the operator corresponding to the eigenvalue 1. □
Proposition 9 For, with τ lesser thanand, we have
Now, we define the q-convolution of and as
In the following theorem, we obtain several characterizations of the continuous linear mappings L from into itself that commute with the q-translation operators.
Theorem 10 Let L be a continuous linear mapping frominto itself. The following assertions are equivalent:
L commutes with the q-translation operators , , that is,
L commutes with the q-difference operator .
There exists a unique such that , .
There exists a complex Borel regular measure having compact support on ℂ, for which, for all , we have
There exists an entire function Ψ of logarithmic order 2 such that
Proof (i) ⇒ (ii) Let , we have
The operator L is continuous, then
Hence, (i) ⇒ (ii).
(ii) ⇒ (i) The operator L commutes with , then for every ,
⇒ (iii) Assume that (i) holds. We define the functional ϑ on as follows:
It is clear that and
⇒ (iv) It follows immediately from Hahn-Banach and Riesz representation theorems.
⇒ (v) Suppose that for all f, we have
where γ is a complex Borel regular measure with compact support. According to , we obtain
Since γ has compact support on ℂ, for certain a and C, we have
Then we get
and Ψ is of logarithmic order 2.
⇒ (i) Suppose that for every , we obtain
Hence, if since ,
5 The q-Fourier transform on the space
We introduce the q-Fourier transform on by
From (2) the q-Fourier transform takes the form
Theorem 11 (q-Paley-Wiener theorem)
The q-Fourier transformis a topological isomorphism fromonto.
Proof Assume firstly that . Then there exists a complex regular Borel measure γ on ℂ and a > 0 such that the support of it is contained in the open ball of center 0 and radius a and
so that for every , there exists a constant such that
By the expansion (21) of the q-Fourier of T, we can write
The inequality (22) shows that
Furthermore, the logarithmic order ρ of is equal to
Similarly, the logarithmic type of τ of is given by
Hence, we have proved . Moreover, is a continuous mapping from into . Indeed, since is a bornological space, it is sufficient to see that . The q-Fourier transform is a one-to-one mapping from into . Indeed, assume that is such that , . Then from (20), we infer that
Hence, (21) implies also that
Thus, the transform is one-to-one. Suppose now that g is a function in . There exists such that for every ,
From Theorem 2, we have
Then, for every ,
where represents, for every , the circular path , . Hence, for every , we have
In particular, if we take , we obtain
On the other hand, from (2), we have
For , we put
Then the series converges absolutely in . Hence, for every , we have
We now define the functional T on by
It is obvious that . Moreover, . Also, from (22), we deduce that for every bounded set Ω of , there exists such that
We conclude that is continuous from onto . □
Kempf A, Majid S: Algebraic q -integration and Fourier theory on quantum and braided spaces. J. Math. Phys. 1994, 35: 6802–6837. 10.1063/1.530644
Koornwinder T: Special functions and q -commuting variables. Fields Inst. Commun. 14. In Special Functions, q-Series and Related Topics (Toronto, ON, 1995). Am. Math. Soc., Providence; 1997:131–166. q-alg/9608008
Olshanetsky, MA, Rogov, V-BK: The q-Fourier transform of q-distributions. Preprint IHES, 91440-Bures-sur-Yvette (1998)
Carnovale G, Koornwinder TH: A q -analogue of convolution on the line. Methods Appl. Anal. 2000, 7: 705–726.
Bonet J: Hypercyclic and chaotic convolution operators. J. Lond. Math. Soc. 2000, 62: 252–262.
Annaby MH, Hassan HA, Mansour ZS: Sampling theorems associated with singular q -Sturm Liouville problems. Results Math. 2012, 62: 121–136. 10.1007/s00025-011-0134-9
Abreu LD: A q -sampling theorem related to the q -Hankel transform. Proc. Am. Math. Soc. 2005, 133: 1197–1203. 10.1090/S0002-9939-04-07589-6
Berg C, Pedersen HL, Hayman W: Logarithmic order and type of indeterminate moment problems. In Proceedings of the International Conference ‘Difference Equations, Special Functions and Orthogonal Polynomials’. Edited by: Elaydi S, Cushing J, Lasser R, Papageorgiou V, Ruffing A, Assche W. World Scientific, Singapore; 2007:51–79.
Ehrenpreis L: Solution of some problems of division, III. Am. J. Math. 1956, 78: 685–715. 10.2307/2372464
Belhadj M, Betancor JJ: Hankel convolution operators on entire functions and distributions. J. Math. Anal. Appl. 2002, 276: 40–63. 10.1016/S0022-247X(02)00350-5
Betancor JJ, Betancor JD, M’endez JM: Hypercyclic and chaotic convolution operators in Chebli-Trimeche hypergroups. Rocky Mt. J. Math. 2004, 34: 1207–1238. 10.1216/rmjm/1181069796
Gasper G, Rahman M Encyclopedia of Mathematics and Its Applications 96. In Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.
Jackson FH: On a q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193–203.
Ismail MEH, Stanton D: Applications of q -Taylor theorems. J. Comput. Appl. Math. 2003, 153: 259–272. 10.1016/S0377-0427(02)00644-1
Ismail MEH, Stanton D: q -Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theory 2003, 123: 125–146. 10.1016/S0021-9045(03)00076-5
Bouzeffour F: Interpolation of entire functions, product formula for basic sine function. J. Nonlinear Math. Phys. 2006, 13(2):293–301. 10.2991/jnmp.2006.13.2.11
Bouzeffour F: A q -sampling theorem and product formula for continuous q -Jacobi functions. Proc. Am. Math. Soc. 2007, 135: 2131–2139. MR 2299491 (2008k:33055) 10.1090/S0002-9939-07-08717-5
Bouzeffour F: A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Askey-Wilson functions. J. Nonlinear Math. Phys. 2007, 14(3):367–380. MR 2350096 (2008h:94031)
Abreu LD, Bouzeffour F: A Paley-Wiener theorem for the Askey-Wilson function transform. Proc. Am. Math. Soc. 2010, 138: 2853–2862. 10.1090/S0002-9939-10-10327-X
Ramis JP: About the growth of entire functions solutions of linear algebraic q -difference equations. Ann. Fac. Sci. Toulouse, Math. 1992, 1(6):53–94.
This research is supported by NPST Program of King Saud University, project number 10-MAT1293-02.
The author declares that they have no competing interests.
About this article
Cite this article
Bouzeffour, F. Basic Fourier transform on the space of entire functions of logarithm order 2. Adv Differ Equ 2012, 184 (2012). https://doi.org/10.1186/1687-1847-2012-184
- Entire Function
- Continuous Linear Mapping
- Riesz Representation Theorem
- Cauchy Integral Formula
- Basic Hypergeometric Series