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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 530473, 12 pages
http://dx.doi.org/10.5402/2012/530473
Research Article

Resolution of the Identity of the Classical Hardy Space by Means of Barut-Girardello Coherent States

Department of Mathematics, Faculty of Sciences and Technics (M'Ghila), Sultan Moulay Slimane University, BP 523, Beni Mellal, Morocco

Received 11 April 2012; Accepted 31 May 2012

Academic Editors: V. Moretti and W.-H. Steeb

Copyright © 2012 Zouhaïr Mouayn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We construct a one-parameter family of coherent states of Barut-Girdrardello type performing a resolution of the identity of the classical Hardy space of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis.

1. Introduction

The study of Hardy spaces, which originated during the 1910s in the setting of Fourier series and complex analysis in one variable, has over time transformed into a rich and multifaceted theory, providing basic insights into such topics as maximal functions, Hankel operators, Hilbert transforms, and wavelets analysis [1]. In physics, Hardy spaces are central in the rigged Hilbert space or Gel’fand triplet theory and play a crucial role in time-asymmetric quantum mechanics [2]. These spaces are usually involved in causality problems. Indeed, a Hardy function is important in signal processing because it may be used as signal filter [3].

In this paper, our aim is to construct an integral transform that connects the classical Hardy space 𝐻2+() of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semi-axis, with a one-parameter family of weighted Bergman spaces 𝔉𝜎() consisting of analytic functions on the complex plane, which are square integrable with respect to the measure (𝑧𝑧)𝜎1/2𝐾(1/2)𝜎(2𝑧𝑧)𝑑𝜇(𝑧) where 2𝜎=1,2,3,, 𝐾𝜈() denotes the MacDonald function and 𝑑𝜇 is the Lebesgue measure on . These spaces have been considered by Barut and Girardello while introducing a class of coherent states associated with noncompact groups [4]. The constructed integral transform enables us to obtain a resolution of the identity of the Hardy space by means of a set of coherent states of Barut-Girardello type.

The paper is organized as follows. In Section 2, we review briefly the formalism of coherent states we will be using. Section 3 deals with some basic facts on the classical Hardy space 𝐻2+(). In Section 4, we recall the definition of the weighted Bergman spaces 𝔉𝜎() as well as some of their needed properties. In Section 5, we construct a coherent state transform mapping the Hilbert space 𝐿2(+) of square integrable functions on the positive real half-line into the space 𝔉𝜎() and we compose it with a Fourier transform to get a new transform connecting the space 𝔉𝜎() with the Hardy space 𝐻2+(). In Section 6, we deduce a set of coherent states of Barut-Girardello type by means of which a resolution of the identity of the Hardy space is achieved. Section 7 is devoted to a summary.

2. Coherent States

The first model of coherent states was the “nonspreading wavepacket” of the harmonic oscillator, which has been constructed by Schrödinger [5]. In suitable units, wave functions of these states are of the form Φ𝔷(𝜏)=𝜏𝔷=𝜋1/21exp2𝜏2+12𝜏𝔷2𝔷212||𝔷||2,(2.1) for every 𝜏, where 𝔷 determines the mean values of coordinate ̂𝑥 and momentum ̂𝑝 according to ̂𝑥=Φ𝔷,𝑥Φ𝔷=2Re𝔷 and ̂𝑝=Φ𝔷,𝑝Φ𝔷=2Im𝔷. The variances 𝜎𝑥=̂𝑥2̂𝑥2=1/2 and 𝜎𝑝=̂𝑝2̂𝑝2=1/2 have equal values, so their product assumes the minimal value permitted by the Heisenberg uncertainty relation. The coherent state in (2.1) have been also obtained by Glauber [6] from the vacuum state |0 by means of the unitary displacement operator as Φ𝔷=exp𝔷𝐴||𝔷𝐴0,(2.2) where 𝐴, 𝐴 are annihilation and creation operators defined by 1𝐴=2(̂𝑥+𝑖̂𝑝),𝐴=12(̂𝑥𝑖̂𝑝).(2.3) Following [7], it was Iwata [8] who used the well-known expansion over the Fock basis |𝑛 to give an expression of Φ𝔷 as ||Φ𝔷=𝑒|𝔷|21/2+𝑛=0𝔷𝑛𝑛!|𝑛.(2.4) Actually, various generalizations of coherent states have been proposed. Here, we will focus on a generalization of (2.4), according to a construction starting from a measure space 𝑋as a set of data” as presented in [9] or [10, pages 72–76]. Precisely, let (𝑋,𝜇) be a measure space, and let 𝒜2𝐿2(𝑋,𝜇) be a closed subspace of infinite dimension. Let {Ψ𝑛}𝑛=0 be an orthogonal basis of 𝒜2 satisfying, for arbitrary 𝜁𝑋, 𝒩(𝜁)=𝑛=0𝜌𝑛1Ψ𝑛(𝜁)Ψ𝑛(𝜁)<+,(2.5) where 𝜌𝑛=Ψ𝑛2𝐿2(𝑋,𝜇). Therefore, the function 𝒦(𝜁,𝜉)=𝑛=0𝜌𝑛1Ψ𝑛(𝜁)Ψ𝑛(𝜉)(2.6) defined on 𝑋×𝑋 is a reproducing kernel of the Hilbert space 𝒜2 so that we have 𝒩(𝜁)=𝒦(𝜁,𝜁), 𝜁𝑋. Let be another (functional) Hilbert space with dim=dim𝒜2 and {𝜙𝑛}𝑛=0 an orthonormal basis of , which will play the role of a Fock basis. The coherent states labelled by points 𝜁𝑋 are defined as the ket-vectors ||𝜁=(𝒩(𝜁))1/2𝑛=0Ψ𝑛(𝜁)𝜌𝑛||𝜙𝑛.(2.7) Now, by definition (2.7), it is straightforward to show that 𝜁𝜁=1 and that the coherent state transform 𝑊𝒜2𝐿2(𝑋,𝜇)(2.8) defined by 𝑊[𝜙](𝜁)=(𝒩(𝜁))1/2𝜁𝜙(2.9) is an isometry. Thus, for 𝜙,𝜓, we have 𝜙𝜓[𝜙][𝜓]=𝑊𝑊𝐿2(𝑋,𝜇)=𝑋𝑑𝜎(𝜁)𝒩(𝜁)𝜙𝜁𝜁𝜓.(2.10) Thereby, we have a resolution of the identity of that can be expressed in Dirac’s bra-ket notation as 𝟏=𝑋||||,𝑑𝜇(𝜁)𝒩(𝜁)𝜁𝜁(2.11) where 𝒩(𝜁) appears as a weight function.

For introductory papers on coherent states, we refer to [6] by Glauber for the radiation field and [11] by Arecchi et al. for atomic states. For an overview of this theory, we refer to early papers by Klauder (as [12]) and the survey of Dodonov [7] that contains a list of 451 references.

3. The Hardy Space 𝐻2+()

Recall that the classical Hardy space (Π+) on the upper half of the complex plane Π+={𝑧=𝑥+𝑖𝑦,𝑥,𝑦>0} consists of all functions 𝐹(𝑧) analytic on Π+ such that sup𝑦>0||||𝐹(𝑥+𝑖𝑦)2𝑑𝑥<+.(3.1) Any function 𝐹(𝑥+𝑖𝑦) has a unique boundary value 𝑓(𝑥) on the real line . That is, lim𝑦0𝐹(𝑥+𝑖𝑦)=𝑓(𝑥),(3.2) which is square integrable on . Thus, a function 𝐹(Π+) uniquely determines a function 𝑓𝐿2(). Conversely, any function 𝐹 can be recovered from its boundary values on the real line by means of the Cauchy integral [13] as follows: 1𝐹(𝑧)=2𝜋𝑖𝑓(𝑥)𝑥𝑧𝑑𝑥,(3.3)𝑓(𝑥) being the function representing the boundary values of 𝐹(𝑧). The linear space of all functions 𝑓(𝑥) is denoted by 2+(). Since there is one-to-one correspondence between functions in (Π+) and their boundary values in 2+(), we identify these two spaces. Moreover, using a Paley-Wiener theorem [14, page 175], one can characterize Hardy functions 𝑓2+() by the fact that their Fourier transform [𝑓](1𝑡)=2𝜋𝑒𝑖𝑡𝑥𝑓(𝑥)𝑑𝑥(3.4) is supported in +=[0,+). That is, 2+()=𝑓𝐿2[𝑓].(),(𝑡)=0,𝑡<0(3.5) This last definition of the Hardy space has been used in the context of the affine group and wavelets analysis [15]. Finally, it is well known that the Fourier transform is a linear isometry from 𝐿2() onto 𝐿2() under which the Hardy space 2+() is mapped onto the Hilbert space 𝐿2(+). The latter admits a complete orthonormal system given by the functions 𝜓𝛼𝑛(𝑡)=𝑛!Γ(𝑛+𝛼+1)1/2𝑡(1/2)𝛼𝑒(1/2)𝑡𝐿𝑛(𝛼)(𝑡),(3.6) in terms of the associated Laguerre polynomial defined by 𝐿𝑛(𝛼)(𝑡)=𝑛𝑘=0(1)𝑘𝑡𝑛+𝛼𝑛𝑘𝑘𝑘!,𝛼>1,(3.7) which leads to the expansion of functions in 𝐿2(+,𝑑𝑡) with respect to the 𝜓𝛼𝑛() (see [16]).

4. The Bergman Space 𝔉𝜎()

In [4, page 51] Barut and Girardello have considered a countable set of Hilbert spaces 𝔉𝜎(),𝜎>0 with 2𝜎=1,2,3,, whose elements are analytic functions 𝜑 on . For each fixed 𝜎, the inner product is defined by 𝜑,𝜓𝜎=𝜑(𝑧)𝜓(𝑧)𝑑𝜇𝜎(𝑧),(4.1) with respect to the one-parameter measure 𝑑𝜇𝜎2(𝑧)=𝜌𝜋Γ(2𝜎)2𝜎1𝐾(1/2)𝜎(2𝜌)𝜌𝑑𝜃𝑑𝜌,(4.2) where 𝑧=𝜌𝑒𝑖𝜃 and the MacDonald function [17, page 183] 𝐾𝜈1(𝑤)=2𝑤2𝜈0+𝑡𝜈1𝑤exp𝑡24𝑡𝑑𝑡(4.3) is defined for Re(𝑤2)>0. Precisely, 𝔉𝜎() consists of entire functions 𝜑 with finite norm 𝜑𝜎=𝜑,𝜑𝜎<+. Note also that if 𝜑(𝑧) is an entire function with power series 𝑛𝑐𝑛𝑧𝑛, then the norm square in terms of the expansion coefficients is given by 𝜑2𝜎=(Γ(2𝜎))1+𝑛=0𝑐𝑛𝑐𝑛𝑛!Γ(2𝜎+𝑛).(4.4) An orthonormal basis function of 𝔉𝜎() is given by 𝜑𝜎𝑛(𝑧)=Γ(2𝜎)𝑧𝑛!Γ(2𝜎+𝑛)𝑛(4.5) for every 𝑧 and varying 𝑛=0,1,2,.

Lemma 4.1. Let 2𝜎=1,2,3,. Then, the diagonal function of the reproducing kernel of the Hilbert space 𝔉𝜎() can be expressed in terms of the modified Bessel function as 𝒦𝜎(𝑧𝑧,𝑧)=Γ(2𝜎)𝑧(1/2)𝜎𝐼2𝜎12𝑧𝑧,(4.6) for every 𝑧.

Proof. By the general theory [18], the reproducing kernel of 𝔉𝜎() can be obtained form the orthonormal basis in (4.5) as 𝒦𝜎(𝑧,𝑤)=+𝑛=0𝜑𝜎𝑛(𝑧)𝜑𝜎𝑛(𝑤).(4.7) Replacing the 𝜑𝜎𝑛(𝑧) by their expressions, then the sum in (4.7) reads 𝐾𝜎(𝑧,𝑤)=+𝑛=01(2𝜎)𝑛𝑧𝑤𝑛=𝑛!0ϝ12𝜎;𝑧𝑤.(4.8) Here, we recall the confluent hypergeometric limit function [17, page 100]: 0ϝ1(𝑎;𝑥)=+𝑛=01(𝑎)𝑛𝑥𝑛,𝑛!(4.9) in which (𝑎)𝑛 denotes the Pochhammer symbol defined by (𝑎)0=1 and (𝑎)𝑛=𝑎(𝑎+1)(𝑎+𝑗1)=Γ(𝑎+𝑛).Γ(𝑎)(4.10) Making use of the relation [17, page 77] 0ϝ11𝜈+1;4𝑢21=Γ(𝜈+1)2𝑢𝜈𝐽𝜈(𝑢),(4.11)𝐽𝜈() being the Bessel function of order 𝜈, and recalling the definition of the modified Bessel function of the first kind 𝐼𝜈1(𝑢)=exp2𝐽𝜈𝜋𝑖𝜈𝑒(1/2)𝜋𝑖𝑢,(4.12) we obtain, for 𝜈=2𝜎1, 𝑢=2𝑖|𝑧|, and 𝑧=𝑤, the expression of the diagonal function of the reproducing kernel as in (4.6).

5. A Coherent State Transform

We are now in a position to apply the formalism in Section 2 in order to define a set of coherent states for the data of ((𝑋,𝑑𝜇),𝒜2,{Ψ𝑛}) and (,{𝜙𝑛}) where(i)(𝑋,𝑑𝜇(𝜁))(,𝑑𝜇𝜎(𝑧)),  𝜁𝑧 and 𝑑𝜇𝜎 is the measure defined in (4.2),(ii)𝒜2𝔉𝜎() is the weighed Bergman space defined in Section 4,(iii){Ψ𝑛}{𝜑𝜎𝑛(𝑧)} is the orthonormal basis in (4.5),(iv)𝐿2(+,𝑑𝑡), (v){𝜙𝑛}{𝜓𝛼𝑛} is the orthonormal basis in (3.6) with 𝛼=2𝜎1.

Definition 5.1. For each fixed 2𝜎=1,2,, a set of coherent states labelled by points 𝑧 and belonging to the Hilbert space 𝐿2(+) can be defined according to (2.7) through the ket-vectors ||𝒩𝑧,𝜎=𝜎(𝑧)1/2𝑛=0𝜑𝜎𝑛(||𝜓𝑧)𝑛2𝜎1,(5.1) where the normalizing factor has the expression 𝒩𝜎(𝑧𝑧)=Γ(2𝜎)𝑧(1/2)𝜎𝐼2𝜎12𝑧𝑧,(5.2) which is the quantity 𝒦𝜎(𝑧,𝑧) given in Lemma 4.1.

Proposition 5.2. The coherent state transform (CST) associated with the coherent states in (5.1) is the isometry 𝑊𝜎𝐿2(+)𝔉𝜎() defined by 𝑊𝜎[𝜓](𝑧)=Γ(2𝜎)𝑧𝜎+1/2𝑒𝑧0+𝐽2𝜎12𝑧𝑡𝜓(𝑡)𝑒(1/2)𝑡𝑑𝑡(5.3) for every 𝜓𝐿2(+,𝑑𝑡) and 𝑧.

Proof. According to (2.9), the coherent state transform 𝑊𝜎[𝜓] of an arbitrary function 𝜓𝐿2(+,𝑑𝑡) is defined by 𝑊𝜎[𝜓]𝒩(𝑧)=𝜎(𝑧)1/2𝑧,𝜎𝜓𝐿2(+).(5.4) We make use of (5.1) to rewrite (5.4) as 𝑊𝜎[𝜓](𝑧)=𝑛=0𝜑𝜎𝑛||𝜓(𝑧)𝑛2𝜎1𝜓𝐿2(+).(5.5) Note that (5.5) can also be presented as𝑊𝜎[𝜓](𝑧)=0+𝑊𝜎(𝑧,𝑡)𝜓(𝑡)𝑑𝑡(5.6) in terms of the kernel function 𝑊𝜎(𝑧,𝑡)=𝑛=0𝜑𝜎𝑛(𝑧)𝜓𝑛2𝜎1(𝑡).(5.7) Replacing 𝜑𝜎𝑛(𝑧) and 𝜓𝑛2𝜎1(𝑡) by their expressions in (4.5) and (3.6), respectively, then (5.7) takes the form 𝑊𝜎(𝑧,𝑡)=Γ(2𝜎)𝑡𝜎1/2𝑒(1/2)𝑡𝑛=0𝑧𝑛𝐿Γ(2𝜎+𝑛)𝑛(2𝜎1)(𝑡).(5.8) Next, we make use of the formula [19, page 1002] 𝑛=0𝑧𝑛𝐿Γ(𝛽+1+𝑛)𝑛(𝛽)(𝑡)=(𝑧𝑡)((1/2)𝛽)𝑒𝑧𝐽𝛽2𝑧𝑡,𝛽>1(5.9) for 𝛽=2𝜎1. Therefore, (5.8) becomes 𝑊𝜎(𝑧,𝑡)=Γ(2𝜎)𝑧𝜎+(1/2)𝑒𝑧𝑒(1/2)𝑡𝐽2𝜎12𝑧𝑡.(5.10) Finally, inserting the last expression of 𝑊𝜎(𝑧,𝑡) into (5.6), we arrive at the announced result.

Now, with the help of the CST 𝑊𝜎 we construct the following integral transform.

Theorem 5.3. Let 2𝜎=1,2,3, be a fixed parameter. Then, the integral transform 𝑇𝜎2+()𝔉𝜎() defined by composing the CST 𝑊𝜎, with the Fourier transform as 𝑇𝜎=𝑊𝜎 is an isometric map having the explicit form 𝑇𝜎[𝑓](𝑧)=Γ(𝜎+1/2)𝑒𝑧2𝜋Γ(2𝜎)12𝑖𝑥1𝜎(1/2)ϝ11𝜎+2,2𝜎,𝑧(1/2)𝑖𝑥𝑓(𝑥)𝑑𝑥,(5.11) for every 𝑓2+() and 𝑧, where 1ϝ1 is the confluent hypergeometric function.

Proof. Let 𝑓2+(). Then, we have successively 𝑇𝜎[𝑓]𝑊(𝑧)=𝜎[𝑓](𝑧)=𝑊𝜎[[𝑓=]](𝑧)(5.12)Γ(2𝜎)𝑧𝜎+(1/2)𝑒𝑧0+𝑒(1/2)𝑡𝐽2𝜎12𝑧𝑡[𝑓](=𝑡)𝑑𝑡(5.13)Γ(2𝜎)𝑧𝜎+1/2𝑒𝑧0+𝑒(1/2)𝑡𝐽2𝜎121𝑧𝑡2𝜋𝑒𝑖𝑡𝑥𝑓(𝑥)𝑑𝑥𝑑𝑡.(5.14) Changing the order of integration, we rewrite (5.14) as 𝑇𝜎[𝑓](𝑧)=Γ(2𝜎)𝑧2𝜋𝜎+1/2𝑒𝑧𝑇𝜎(𝑧,𝑥)𝑓(𝑥)𝑑𝑥,(5.15) where we have introduced the integral 𝑇𝜎(𝑧,𝑥)=0+𝑒(1/2𝑖𝑥)𝑡𝐽2𝜎12𝑧𝑡𝑑𝑡.(5.16)=20+𝑢𝑒(1/2𝑖𝑥)𝑢2𝐽2𝜎12𝑧𝑢𝑑𝑢.(5.17) Next, we make appeal to the identity [19, page 706] 0+𝑢𝜚𝑒𝛼𝑢2𝐽𝜈(𝛽𝑢)𝑑𝑢=2𝛽𝜈Γ((1/2)𝜈+(1/2)𝜚+1/2)2𝜈+1𝛼(1/2)(𝜚+𝜈+1)Γ(𝜈+1)1ϝ1𝜚+𝜈+12,𝜈+1,𝛽24𝛼,(5.18)Re𝛼>0, Re(𝜚+𝜈)>1 for the parameters 𝜚=1, 𝛼=(1/2)𝑖𝑥, 𝜈=2𝜎1, and 𝛽=2𝑧. Here, 1ϝ1(𝑎,𝑏;𝑢)=+𝑛=0(𝑎)𝑛(𝑏)𝑛𝑢𝑛𝑛!(5.19) is the confluent hypergeometric function [19, page 1023]. Therefore, the integral in (5.16) reads 𝑇𝜎(𝑧,𝑥)=Γ(𝜎+(1/2))𝑧𝜎(1/2)Γ(2𝜎)((1/2)𝑖𝑥)𝜎+(1/2)1ϝ11𝜎+2,2𝜎,𝑧(.1/2)𝑖𝑥(5.20) Returning back to (5.16) and replacing 𝑇𝜎(𝑧,𝑥) by its expression (5.20), we arrive at the result (5.11).

6. A Resolution of the Identity of 2+()

Now, observe that if one starts with coherent states whose wave functions are expressed in a closed form, then one will be able to construct an isometric map between two functional Hilbert spaces 𝒜2 and as discussed in Section 2. Here, we will take the opposite direction in the sense that we proceed to extract a set of coherent states of Barut-Girardello type belonging to the Hardy space 𝐻2+() from the expression of the isometric map 𝑇𝜎 in (5.11). We precisely establish the following result.

Theorem 6.1. Let 2𝜎=1,2,3, be a fixed parameter. Then, the states |𝑤,𝜎 labelled by points 𝑤 and defined by ||𝒩𝑤,𝜎=𝜎(𝑤)1/2Γ(2𝜎)𝑧2𝜋𝜎+(1/2)𝑒𝑧𝑇𝜎(𝑤,),(6.1) with the wave functions𝑥𝑤,𝜎=1ϝ1𝜎+1/2,2𝜎,(𝑖𝑥1/2)1𝑤𝑒𝑤Γ(𝜎+1/2)Γ(2𝜎)𝑤2𝜋𝑤1/2𝜎𝐼2𝜎12𝑤𝑤((1/2)𝑖𝑥)𝜎+1/2(6.2) are coherent states of Barut-Girardello type in the classical Hardy space and satisfy the following resolution of the identity: 𝟏2+()=Γ(2𝜎)𝑤𝑑𝜇(𝑤)𝑤1/2𝜎𝐼2𝜎12𝑤𝑤||||.𝑤,𝜎𝑤,𝜎(6.3)

Proof. We first make use of the relation [16, page 349] 1ϝ1𝑎,𝑏,1𝑐𝑐𝑢𝑒𝑢=+𝑗=0𝑀𝑗𝑢(𝑎,𝑏,𝑐)𝑗𝑗!(6.4) for the parameters 𝑢=𝑤, 𝑎=(𝜎+(1/2)),   𝑏=2𝜎, and 𝑐=(𝑖𝑥(1/2))((1/2)+𝑖𝑥)1, where 𝑀𝑗(𝑎,𝑏,𝑐) denotes the Meixner polynomial. Therefore, the confluent hypergeometric function occurring in (6.2) can be expanded into the following series: 1ϝ11𝜎+2,2𝜎,𝑤𝑒(1/2)𝑖𝑥𝑤=+𝑗=0𝑀𝑗1𝜎2,2𝜎,𝑖𝑥(1/2)𝑤𝑖𝑥+(1/2)𝑗.𝑗!(6.5)
Next, we write the Meixner polynomial in terms of a terminating Gauss hypergeometric 2ϝ1sum [16, page 346]: 𝑀𝑗(𝑎,𝑏,𝑐)=2ϝ11𝑗,𝑎,𝑏;1𝑐.(6.6) Therefore, (6.5) takes the form 1ϝ11𝜎+2,2𝜎,𝑤𝑒(1/2)𝑖𝑥𝑤=+𝑗=0𝑤𝑗𝑗!2ϝ11𝑗,𝜎+21,2𝜎,.(1/2)𝑖𝑥(6.7) Taking into account (6.7), the wave function in (6.2) can be rewritten as 𝒩𝑥𝑤,𝜎=𝜎(𝑤)1/2+𝑗=0Γ(2𝜎)𝑤𝑗!Γ(2𝜎+𝑗)𝑗𝜙𝜎𝑗(𝑥)(6.8) in terms of the functions 𝜙𝜎𝑗(𝑥)=Γ(𝜎+1/2)2𝜋Γ(2𝜎)Γ(2𝜎+𝑗)1𝑗!Γ(2𝜎)2𝑖𝑥(𝜎+1/2)2ϝ11𝑗,𝜎+21,2𝜎,.(1/2)𝑖𝑥(6.9) To check that 𝜙𝜎𝑗2+(), one can use the fact [3, Prop.1] that a function of moderate decrease is of class Hardy if and only if all its poles lie in the lower half-plane. The orthogonality of the functions (𝜙𝜎𝑗) can be deduced from the orthogonality relations of the Laguerre polynomials by applying the integral representation 2ϝ11𝑗,𝛽+1,𝛼+1,𝑠=𝑠𝛽+1𝑗!Γ(𝛼+1)Γ(𝛽+1)Γ(𝛼+𝑗+1)0+𝑒𝑠𝑡𝑡𝛽𝐿𝑗(𝛼)(𝑡)𝑑𝑡,(6.10)Re𝛽>1,  Re𝑠>0, for the parameters 𝛼=2𝜎1, 𝛽=𝜎1/2, and 𝑠=(1/2)𝑖𝑥. In any way, if we use the Legendre duplication formula [19, page 896] 𝜋Γ(2𝜖)=22𝜖11Γ(𝜖)Γ𝜖+2,(6.11) satisfied by the gamma function, then the constant in (6.9) reads Γ(𝜎+1/2)=12𝜋Γ(2𝜎)2𝜎𝜋1/4Γ(𝜎+1/2)Γ(𝜎)1/2,(6.12) and, therefore, one can verify by [20, page 62] that the functions (𝜙𝜎𝑗) in (6.9) written in terms of the constant in the right-hand side of (6.12) constitute a complete orthonormal system of rational functions in the Hardy space 2+(). Finally, the resolution of the identity in (6.3) follows by a direct application of (2.11). This ends the proof.

7. Summary

We have constructed an integral transform that connects the Hardy space 𝐻2+() of complex-valued square integrable functions on the real line, whose Fourier transform is supported by the positive real semiaxis, with a one-parameter family of weighted Bergman spaces 𝔉𝜎() consisting of analytic functions on the complex plane, which are square integrable with respect to a measure involving the MacDonald function. These spaces are attached to the so-called Barut-Girardello coherent states. The constructed integral transform has enabled us to construct a set of coherent states that satisfy the resolution of the identity of the Hardy space.

Acknowledgment

The author is very grateful for Professor Josef Wichmann for sending him some remarks on the preprint version of this work.

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