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

Authors: Zouhaïr Mouayn
Journal: ISRN Mathematical Physics
 (2012)

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## 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.


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## Body

## 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 spaceH+2(ℝ) 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(ℂ) consisting of analytic functions on the complex plane, which are square integrable with respect to the measure (zz¯)σ-1/2K(1/2)-σ(2zz¯)dμ(z) where 2σ=1,2,3,…, Kν(·) denotes the MacDonald function and dμ is the Lebesgue measure on ℂ. 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 Section2, we review briefly the formalism of coherent states we will be using. Section 3 deals with some basic facts on the classical Hardy space H+2(ℝ). In Section 4, we recall the definition of the weighted Bergman spaces 𝔉σ(ℂ) as well as some of their needed properties. In Section 5, we construct a coherent state transform mapping the Hilbert space L2(ℝ+) of square integrable functions on the positive real half-line into the space 𝔉σ(ℂ) and we compose it with a Fourier transform to get a new transform connecting the space 𝔉σ(ℂ) with the Hardy space H+2(ℝ). 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
(2.1)Φz(τ):=〈τ∣z〉=π-1/2exp⁡(-12τ2+2τz-12z2-12|z|2),
for every τ∈ℝ, where 𝔷∈ℂ determines the mean values of coordinate x^ and momentum p^ according to 〈x^〉:=〈Φ𝔷,xΦ𝔷〉=2Re𝔷 and 〈p^〉:=〈Φ𝔷,pΦ𝔷〉=2Im𝔷. The variances σx=〈x^2〉-〈x^〉2=1/2 and σp=〈p^2〉-〈p^〉2=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
(2.2)Φz=exp⁡(zA*-z¯A)|0〉,
where A, A* are annihilation and creation operators defined by
(2.3)A=12(x^+ip^),A*=12(x^-ip^).
Following [7], it was Iwata [8] who used the well-known expansion over the Fock basis |n〉 to give an expression of Φ𝔷 as
(2.4)|Φz〉=(e|z|2)-1/2∑n=0+∞‍znn!|n〉.
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 X “as a set of data” as presented in [9] or [10, pages 72–76]. Precisely, let (X,μ) be a measure space, and let 𝒜2⊂L2(X,μ) be a closed subspace of infinite dimension. Let {Ψn}n=0∞ be an orthogonal basis of 𝒜2 satisfying, for arbitrary ζ∈X,
(2.5)N(ζ):=∑n=0∞‍ρn-1Ψn(ζ)Ψn(ζ)¯<+∞,
where ρn=∥Ψn∥L2(X,μ)2. Therefore, the function
(2.6)K(ζ,ξ):=∑n=0∞‍ρn-1Ψn(ζ)Ψn(ξ)¯
defined on X×X is a reproducing kernel of the Hilbert space 𝒜2 so that we have 𝒩(ζ):=𝒦(ζ,ζ), ζ∈X. Let ℋ be another (functional) Hilbert space with dimℋ=dim𝒜2 and {ϕn}n=0∞ an orthonormal basis of ℋ, which will play the role of a Fock basis. The coherent states labelled by points ζ∈X are defined as the ket-vectors
(2.7)|ζ〉:=(N(ζ))-1/2∑n=0∞‍Ψn(ζ)ρn|ϕn〉.
Now, by definition (2.7), it is straightforward to show that 〈ζ∣ζ〉ℋ=1 and that the coherent state transform
(2.8)W:H⟶A2⊂L2(X,μ)
defined by
(2.9)W[ϕ](ζ):=(N(ζ))1/2〈ζ∣ϕ〉H
is an isometry. Thus, for ϕ,ψ∈ℋ, we have
(2.10)〈ϕ∣ψ〉H=〈W[ϕ]∣W[ψ]〉L2(X,μ)=∫Xdσ(ζ)N(ζ)〈ϕ∣ζ〉〈ζ∣ψ〉.
Thereby, we have a resolution of the identity of ℋ that can be expressed in Dirac’s bra-ket notation as
(2.11)1H=∫Xdμ(ζ)N(ζ)|ζ〉〈ζ|,
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 SpaceH+2(ℝ)

Recall that the classical Hardy spaceℋ(Π+) on the upper half of the complex plane Π+:={z=x+iy,x∈ℝ,y>0} consists of all functions F(z) analytic on Π+ such that
(3.1)sup⁡y>0∫R|F(x+iy)|2dx<+∞.
Any function F(x+iy) has a unique boundary value f(x) on the real line ℝ. That is,
(3.2)lim⁡y→0F(x+iy)=f(x),
which is square integrable on ℝ. Thus, a function F∈ℋ(Π+) uniquely determines a function f∈L2(ℝ). Conversely, any function F can be recovered from its boundary values on the real line by means of the Cauchy integral [13] as follows:
(3.3)F(z)=12πi∫Rf(x)x-zdx,f(x) being the function representing the boundary values of F(z). The linear space of all functions f(x) is denoted by ℋ+2(ℝ). Since there is one-to-one correspondence between functions in ℋ(Π+) and their boundary values in ℋ+2(ℝ), we identify these two spaces. Moreover, using a Paley-Wiener theorem [14, page 175], one can characterize Hardy functions f∈ℋ+2(ℝ) by the fact that their Fourier transform
(3.4)F[f](t)=12π∫Re-itxf(x)dx
is supported in ℝ+=[0,+∞). That is,
(3.5)H+2(R)={f∈L2(R),F[f](t)=0,∀t<0}.
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 ℱ is a linear isometry from L2(ℝ) onto L2(ℝ) under which the Hardy space ℋ+2(ℝ) is mapped onto the Hilbert space L2(ℝ+). The latter admits a complete orthonormal system given by the functions
(3.6)ψnα(t):=(n!Γ(n+α+1))1/2t(1/2)αe-(1/2)tLn(α)(t),
in terms of the associated Laguerre polynomial defined by
(3.7)Ln(α)(t)=∑k=0n‍(-1)k(n+αn-k)tkk!,α>-1,
which leads to the expansion of functions in L2(ℝ+,dt) with respect to the ψnα(·) (see [16]).


## 4. The Bergman Space𝔉σ(ℂ)

In [4, page 51] Barut and Girardello have considered a countable set of Hilbert spaces 𝔉σ(ℂ),σ>0 with 2σ=1,2,3,…, whose elements are analytic functions φ on ℂ. For each fixed σ, the inner product is defined by
(4.1)〈φ,ψ〉σ:=∫Cφ(z)ψ(z)¯dμσ(z),
with respect to the one-parameter measure
(4.2)dμσ(z):=2πΓ(2σ)ρ2σ-1K(1/2)-σ(2ρ)ρdθdρ,
where z=ρeiθ∈ℂ and the MacDonald function [17, page 183]
(4.3)Kν(w)=12(w2)ν∫0+∞t-ν-1exp⁡(-t-w24t)dt
is defined for Re(w2)>0. Precisely, 𝔉σ(ℂ) consists of entire functions φ with finite norm ∥φ∥σ=〈φ,φ〉σ<+∞. 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
(4.4)‖φ‖σ2=(Γ(2σ))-1∑n=0+∞‍cncn¯n!Γ(2σ+n).
An orthonormal basis function of 𝔉σ(ℂ) is given by
(4.5)φnσ(z):=Γ(2σ)n!Γ(2σ+n)zn
for every z∈ℂ and varying n=0,1,2,….Lemma 4.1.
Let2σ=1,2,3,…. Then, the diagonal function of the reproducing kernel of the Hilbert space 𝔉σ(ℂ) can be expressed in terms of the modified Bessel function as
(4.6)Kσ(z,z)=Γ(2σ)(zz¯)(1/2)-σI2σ-1(2zz¯),
for every z∈ℂ.Proof.
By the general theory [18], the reproducing kernel of 𝔉σ(ℂ) can be obtained form the orthonormal basis in (4.5) as
(4.7)Kσ(z,w)=∑n=0+∞‍φnσ(z)φnσ(w)¯.
Replacing the φnσ(z) by their expressions, then the sum in (4.7) reads
(4.8)Kσ(z,w)=∑n=0+∞‍1(2σ)n(zw¯)nn!=Ϝ01(2σ;zw¯).
Here, we recall the confluent hypergeometric limit function [17, page 100]:
(4.9)0Ϝ1(a;x)=∑n=0+∞‍1(a)nxnn!,
in which (a)n denotes the Pochhammer symbol defined by (a)0:=1 and
(4.10)(a)n:=a(a+1)⋯(a+j-1)=Γ(a+n)Γ(a).
Making use of the relation [17, page 77]
(4.11)0Ϝ1(ν+1;-14u2)=Γ(ν+1)(12u)-νJν(u),Jν(·) being the Bessel function of order ν∈ℝ, and recalling the definition of the modified Bessel function of the first kind
(4.12)Iν(u)=exp⁡(-12νπi)Jν(e(1/2)πiu),
we obtain, for ν=2σ-1, 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 Section2 in order to define a set of coherent states for the data of ((X,dμ),𝒜2,{Ψn}) and (ℋ,{ϕn}) where(i)
(X,dμ(ζ))≡(ℂ,dμσ(z)),  ζ≡z and dμσ is the measure defined in (4.2),(ii)
𝒜2≡𝔉σ(ℂ) is the weighed Bergman space defined in Section 4,(iii)
{Ψn}≡{φnσ(z)} is the orthonormal basis in (4.5),(iv)
ℋ≡L2(ℝ+,dt),(v)
{ϕn}≡{ψnα} is the orthonormal basis in (3.6) with α=2σ-1.Definition 5.1.
For each fixed2σ=1,2,…, a set of coherent states labelled by points z∈ℂ and belonging to the Hilbert space L2(ℝ+) can be defined according to (2.7) through the ket-vectors
(5.1)|z,σ〉:=(Nσ(z))-1/2∑n=0∞‍φnσ(z)|ψn2σ-1〉,
where the normalizing factor has the expression
(5.2)Nσ(z)=Γ(2σ)(zz¯)(1/2)-σI2σ-1(2zz¯),
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 Wσ:L2(ℝ+)→𝔉σ(ℂ) defined by
(5.3)Wσ[ψ](z)=Γ(2σ)z-σ+1/2ez∫0+∞J2σ-1(2zt)ψ(t)¯e-(1/2)tdt
for every ψ∈L2(ℝ+,dt) and z∈ℂ.Proof.
According to (2.9), the coherent state transform Wσ[ψ] of an arbitrary function ψ∈L2(ℝ+,dt) is defined by
(5.4)Wσ[ψ](z)=(Nσ(z))1/2〈z,σ∣ψ〉L2(R+).
We make use of (5.1) to rewrite (5.4) as
(5.5)Wσ[ψ](z)=〈∑n=0∞‍φnσ(z)|ψn2σ-1〉∣ψ〉L2(R+).
Note that (5.5) can also be presented as(5.6)Wσ[ψ](z)=∫0+∞Wσ(z,t)ψ(t)¯dt
in terms of the kernel function
(5.7)Wσ(z,t):=∑n=0∞‍φnσ(z)ψn2σ-1(t).
Replacing φnσ(z) and ψn2σ-1(t) by their expressions in (4.5) and (3.6), respectively, then (5.7) takes the form
(5.8)Wσ(z,t)=Γ(2σ)tσ-1/2e-(1/2)t∑n=0∞‍znΓ(2σ+n)Ln(2σ-1)(t).
Next, we make use of the formula [19, page 1002]
(5.9)∑n=0∞‍znΓ(β+1+n)Ln(β)(t)=(zt)-((1/2)β)ezJβ(2zt),β>-1
for β=2σ-1. Therefore, (5.8) becomes
(5.10)Wσ(z,t)=Γ(2σ)z-σ+(1/2)eze-(1/2)tJ2σ-1(2zt).
Finally, inserting the last expression of Wσ(z,t) into (5.6), we arrive at the announced result.Now, with the help of the CSTWσ we construct the following integral transform.Theorem 5.3.
Let2σ=1,2,3,… be a fixed parameter. Then, the integral transform Tσ:ℋ+2(ℝ)→𝔉σ(ℂ) defined by composing the CST Wσ, with the Fourier transform ℱ as Tσ:=Wσ∘ℱ is an isometric map having the explicit form
(5.11)Tσ[f](z)=Γ(σ+1/2)ez2πΓ(2σ)∫R(12-ix)-σ-(1/2)Ϝ11(σ+12,2σ,-z(1/2)-ix)f(x)¯dx,
for every f∈ℋ+2(ℝ) and z∈ℂ, where 1Ϝ1 is the confluent hypergeometric function.Proof.
Letf∈ℋ+2(ℝ). Then, we have successively
(5.12)Tσ[f](z)=(Wσ∘F)[f](z)=Wσ[F[f]](z)(5.13)=Γ(2σ)z-σ+(1/2)ez∫0+∞e-(1/2)tJ2σ-1(2zt)ℱ[f](t)¯‍dt(5.14)=Γ(2σ)z-σ+1/2ez∫0+∞e-(1/2)tJ2σ-1(2zt)(12π∫ℝeitxf(x)¯dx)dt.
Changing the order of integration, we rewrite (5.14) as
(5.15)Tσ[f](z)=Γ(2σ)2πz-σ+1/2ez∫RTσ(z,x)f(x)¯dx,
where we have introduced the integral
(5.16)Tσ(z,x):=∫0+∞e-(1/2-ix)tJ2σ-1(2zt)dt.(5.17)=2∫0+∞ue-(1/2-ix)u2J2σ-1(2zu)du.
Next, we make appeal to the identity [19, page 706]
(5.18)∫0+∞uϱe-αu2Jν(βu)du=2βνΓ((1/2)ν+(1/2)ϱ+1/2)2ν+1α(1/2)(ϱ+ν+1)Γ(ν+1)⋅Ϝ11(ϱ+ν+12,ν+1,-β24α),Reα>0, Re(ϱ+ν)>-1 for the parameters ϱ=1, α=(1/2)-ix, ν=2σ-1, and β=2z. Here,
(5.19)1Ϝ1(a,b;u)=∑n=0+∞‍(a)n(b)nunn!
is the confluent hypergeometric function [19, page 1023]. Therefore, the integral in (5.16) reads
(5.20)Tσ(z,x)=Γ(σ+(1/2))zσ-(1/2)Γ(2σ)((1/2)-ix)σ+(1/2)⋅Ϝ11(σ+12,2σ,-z(1/2)-ix).
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 ofℋ+2(ℝ)

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 spaces𝒜2 and ℋ 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 H+2(ℝ) from the expression of the isometric map Tσ in (5.11). We precisely establish the following result.Theorem 6.1.
Let2σ=1,2,3,… be a fixed parameter. Then, the states |w,σ〉 labelled by points w∈ℂ and defined by
(6.1)|w,σ〉:=(Nσ(w))-1/2Γ(2σ)2πz-σ+(1/2)ezTσ(w,⋅),
with the wave functions(6.2)〈x∣w,σ〉=1Ϝ1(σ+1/2,2σ,(ix-1/2)-1w)ewΓ(σ+1/2)Γ(2σ)2π(ww¯)1/2-σI2σ-1(2ww¯)((1/2)-ix)σ+1/2
are coherent states of Barut-Girardello type in the classical Hardy space and satisfy the following resolution of the identity:
(6.3)1H+2(R)=Γ(2σ)∫Cdμ(w)(ww¯)1/2-σI2σ-1(2ww¯)|w,σ〉〈w,σ|.Proof.
We first make use of the relation [16, page 349]
(6.4)1Ϝ1(-a,b,(1-cc)u)eu=∑j=0+∞‍Mj(a,b,c)ujj!
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:
(6.5)Ϝ11(σ+12,2σ,-w(1/2)-ix)ew=∑j=0+∞‍Mj(-σ-12,2σ,ix-(1/2)ix+(1/2))wjj!.
Next, we write the Meixner polynomial in terms of a terminating Gauss hypergeometricϜ21-sum [16, page 346]:
(6.6)Mj(a,b,c)=Ϝ21(-j,-a,b;1-1c).
Therefore, (6.5) takes the form
(6.7)Ϝ11(σ+12,2σ,-w(1/2)-ix)ew=∑j=0+∞‍wjj!⋅Ϝ21(-j,σ+12,2σ,1(1/2)-ix).
Taking into account (6.7), the wave function in (6.2) can be rewritten as
(6.8)〈x∣w,σ〉=(Nσ(w))-1/2∑j=0+∞‍Γ(2σ)j!Γ(2σ+j)wjϕjσ(x)
in terms of the functions
(6.9)ϕjσ(x):=Γ(σ+1/2)2πΓ(2σ)Γ(2σ+j)j!Γ(2σ)(12-ix)-(σ+1/2)⋅Ϝ21(-j,σ+12,2σ,1(1/2)-ix).
To check that ϕjσ∈ℋ+2(ℝ), 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 (ϕjσ) can be deduced from the orthogonality relations of the Laguerre polynomials by applying the integral representation
(6.10)2Ϝ1(-j,β+1,α+1,1s)=sβ+1j!Γ(α+1)Γ(β+1)Γ(α+j+1)∫0+∞e-sttβLj(α)(t)dt,Reβ>-1,  Res>0, for the parameters α=2σ-1, β=σ-1/2, and s=(1/2)-ix. In any way, if we use the Legendre duplication formula [19, page 896]
(6.11)πΓ(2ϵ)=22ϵ-1Γ(ϵ)Γ(ϵ+12),
satisfied by the gamma function, then the constant in (6.9) reads
(6.12)Γ(σ+1/2)2πΓ(2σ)=12σπ1/4(Γ(σ+1/2)Γ(σ))1/2,
and, therefore, one can verify by [20, page 62] that the functions (ϕjσ) 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 ℋ+2(ℝ). 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 spaceH+2(ℝ) 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(ℂ) 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.



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