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Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory

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Kayupe Kikodio Patrick

Submitted: June 23rd, 2020 Reviewed: November 1st, 2020 Published: February 24th, 2021

DOI: 10.5772/intechopen.94865

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Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also called cylindrical functions, or cylindrical harmonics. This chapter is devoted to the construction of the generalized coherent state (GCS) and the theory of Bessel wavelets. The GCS is built by replacing the coefficient zn/n!,z∈C of the canonical CS by the cylindrical Bessel functions. Then, the Paley-Wiener space PW1 is discussed in the framework of a set of GCS related to the cylindrical Bessel functions and to the Legendre oscillator. We prove that the kernel of the finite Fourier transform (FFT) of L2-functions supported on −11 form a set of GCS. Otherwise, the wavelet transform is the special case of CS associated respectively with the Weyl-Heisenberg group (which gives the canonical CS) and with the affine group on the line. We recall the wavelet theory on R. As an application, we discuss the continuous Bessel wavelet. Thus, coherent state transformation (CST) and continuous Bessel wavelet transformation (CBWT) are defined. This chapter is mainly devoted to the application of the Bessel function.


  • coherent state
  • Hankel transformation
  • Bessel wavelet transformation

1. Introduction

Coherent state (CS) was originally introduced by Schrödinger in 1926 as a Gaussian wavepacket to describe the evolution of a harmonic oscillator [1].

The notion of coherence associated with these states of physics was first noticed by Glauber [2, 3] and then introduced by Klauder [4, 5]. Because of their important properties these states were then generalized to other systems either from a physical or mathematical point of view. As the electromagnetic field in free space can be regarded as a superposition of many classical modes, each one governed by the equation of simple harmonic oscillator, the CS became significant as the tool for connecting quantum and classical optics. For a review of all of these generalizations see [6, 7, 8, 9].

Four main methods are well used in the literature to build CS, the so-called Schrödinger, Klauder-Perelomov, Barut-Girardello and Gazeau-Klauder approaches. The second and third approaches are based directly on the Lie algebra symmetries with their corresponding generators, the first is only established by means of an appropriate infinite superposition of wave functions associated with the harmonic oscillator whatever the Lie algebra symmetries. In [10, 11, 12] the authors introduced a new family of CS as a suitable superposition of the associated Bessel functions and in [13, 14, 15] the authors also use the generating function approach to construct a new type CS associated with Hermite polynomials and the associated Legendre functions, respectively. The important fact is that we do not use algebraic and group approaches (Barut-Girardello and Klauder-Perelomov) to construct generalized coherent states (GCS).

We first discuss GCS associated with a one-dimensional Schrödinger operator [16, 17] by following the work in [18, 19]. We build a family of GCS through superpositions of the corresponding eigenstates, say ψn,nN, which are expressed in terms of the Legendre polynomial Pnx [16]. The role of coefficients zn/n! of the canonical CS is played by


where ξR and Jn+12. denotes the cylindrical Bessel function [20]. When n=0, Eq. (1) becomes


where J0. denotes the spherical Bessel function of order 0. The choosen coefficients (1) and eigenfunctions (27) (see below) have been used in ([21], p. 1625). We proceed by determining the wavefunctions of these GCS in a closed form. The latter gives the kernel of the associated CS transform which makes correspondence between the quantum states Hilbert space L21121dx of the Legendre oscillator and a subspace of a Hilbert space of square integrable functions with respect to a suitable measure on the real line. We show that the kernel eixξ,ξR, of the L2-functions that are supported in 11 form a set of GCS.

There are in literature several approach to introducce Bessel Wavelets. We refer for instence to [22, 23]. Note that, for 11xcosy/n,nN, the Legendre polynomial Pnx and the Bessel function of order 0 are related by the Hansen’s limit


and the integral


Note that in [22, 23] the authors have introduced the Bessel wavelet based on the Hankel transform. The notion of wavelets was first introduced by J. Morlet a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by A. Grossmann and J. Morlet [24]. Harmonic analyst Y. Meyer and other mathematicians understood the importance of this theory and they recognized many classical results within (see [25, 26, 27]). Classical wavelets have several applications ranging from geophysical and acoustic signal analysis to quantum theory and pure mathematics. A wavelet base is a family of functions obtained from a function known as mother wavelet, by translation and dilation. This tool permits the representation of L2-functions in a basis well localized in time and in freqency. Wavelets are special functions with special properties which may not be satisfied by other functions. In the current context, our objective is to make a link between the construction of GCS and the theory of wavelets. Therefore, we will talk about coherent state transformation (CST) and the continuous Bessel wavelet transformation (CBWT).

The rest of this chapter is organized as follows: Section 2 is devoted to the generalized CS formalism that we are going to use. In Section 3, we briefly introduce the Paley-Wiener space PWΩ and some notions on Legendre’s Hamiltonian. We give in Section 4 a summary concept on the continuous wavelet transform on R. In Section 5, we have constructed a class of GCS related to the Bessel cylindrical function for the legendre Hamiltonian. In Section 6, we discuss the theory of CBWT where we show as an example that the function fLσ2R+


such that R+ftt=0 is the mother wavelet where t is an appropriate Legesgue’s measure on R. Finally in Section 7. we gives some concluding remarks on the chapter.


2. Generalized coherent states formalism

We follow the generalization of canonical coherent states (CCS) introduced in [18, 19]. The definition of CS as a set of vectors associated with a reproducing kernel is general, it encompasses all the situations encountered in the physical literature. For applications we will work with normalized vectors. Let Xμ be a measure space and let N2L2Xμ be a sub-closed space of infinite dimension. Let Cnn=0 be a satisfactory orthogonal basis of N2, for arbitrary xX


where ρnCnL2Xμ2. Define the kernel


Then, the expression Kxy is a reproducing kernel, N2 is the corresponding kernel Hilbert space and NxKxx,xX. Define






is an isometry. For ϕ,ψH, whe have




where Nx is a positive weight function.

Definition 1.LetHbe a Hilbert space withdimH=andφnn=0be an orthonormal basis ofH.The generalized coherent state (GCS) labeled by pointxXare defined as the ket-vectorϑxH, such that


By definition, it is straightforward to show thatϑxϑxH=1.

Definition 2.For each functionfH, the coherent state transform (CST) associated to the setϑxxXis the isometric map


Thereby, we have a resolution of the identity ofHwhich can be expressed in Dirac’s bra-ket notation as


where the rank one operatorTxϑxϑx:HHis define by


Nxappears as a weight function.

Next, the reproducing kernel has the additional property of being square integrable, i.e.,


Note that the formula (10) can be considered as generalization of the series expansion of the CCS [28].


with ϕnn=0 being an orthonormal basis of eigenstates of the quantum harmonic oscillator. Then, the space N2 is the Fock spaceFC and Nz=π1ezz¯,zC.


3. The Paley-wiener space PWΩ and the Legendre Hamiltonian: a brief overview

3.1 The Paley-wiener space PWΩ

The Paley-Wiener space is made up of all integer functions of exponential type whose restrictions on the real line is square integrable. We give in this Section a general overview on this notion ([29], pp. 45–47).

Definition 3.ConsiderFas an entire function. Then,Fis an entire function of exponential type if there exists constantsA,B>0such that, for allzC


Note that, if F satisfy Definition 3, we call Ω the type of F where


and where Mr=supz=rFz. The following conditions on an entire function F are verified:

  1. For all ε>0 there exists Cε such that


  2. There exists C>0 such that


  3. as z+


Then cleary, 321F is of exponential type at most Ω.

Definition 4.LetΩ>0and1p. The Paley-Wiener spacePWΩpis defined as


and we set


The Paley-Wiener PWΩp is the image via the Fourier transform of the Lp-function that are supported in ΩΩ. We will be interested in the case p=2, in which PWΩ to denote the Paley-Wiener space PWω2. From the Plancherel formula we have


Hence, by polarization, for f,φPWΩ,


Theorem 1.1 Let F be an entire function and Ω>0. Then the following are equivalent

  • FRL2R and


  • there exists fL2R with suppf̂ΩΩ such that


The function fPWΩ if and only if fL2R and f=FR (that is, f is the restriction to the real line of a function F), where F is an entire function of exponential type such that Fz=oeΩz for z+.

Theorem 1.2 The Paley-Wiener space PWΩ is a Hilbert space with reproducing kernel w.r.t the inner product (20). Its reproducing kernel is the function


wheresinct=sint/t. Hence, for every fPWΩ


where xR.

3.2 The Legendre Hamiltonian

The Legendre polynomials Pnx and the Legendre function ψnx are similar to the Hermite polynomials and the Hermite function in standard quantum mechanics. Based on the work of Borzov and Demaskinsky [16, 17] the Legendre Hamiltonian has the form


where X and P denotes respectively the position and momentum operators, a+ and a are the creation and annihilation operators. The eigenvalues of operators H are equal to


and the corresponding eigenfunctions reads


in terms of the Legendre polynomial Pn., which form an orthonormal basis ψnnn=0 in the Hilbert space HL21121dx. These functions satisfy the recurrence relations


with coefficients


The generalized position operator on the Hilbert space H connected with the Legendre polynomials Pnx is an operator of multiplication by argument Xψn=xψn. Taking into account of the relation (28), then


whee bn are coefficients defined by Eq. (29). Because n=01/bn=+, X is a self-adjoint operator on the Hilbert space H (see [30, 31, 32]). The momentum operator P by the way described in ([17], p. 126) acts on the basis elements in H, by the formula Pψn=ibnψn+1bn1ψn1. The usual commutator of operator X and P on the basis elements reads as


The creation and annihilation operators (25) are define by relations


these operators act as a+ψn=2bnψn+1andaψn=2bn1ψn1. They satisfy aa+=iXP, the commutation relations.


4. Wavelet theory on R and the reproduction of kernels

We briefly describe below some basis definitions and properties of the one-dimensional wavelet transform on R+, we refer to [22, 23, 33]. In the Hilbert space N=L2Rdx, the function ψ satisfying the so-called admissibility condition


where ψ̂ being the Hankel transform of ψ. Not every vector in N satisfies the above condition. A vector ψ satisfying (33) is called a mother wavelet. Combining dilatation and translation, one gets affine transformation


Thus baGaff=R×0, the affine group of the line. Specifically, for each pair ab of the real numbers, with a>0, from translations and dilatations of the function ψ, we obtain a family of wavelets ψa,bN as


Here a is the parameter of dilation (or scale) and b is the parameter of translation (or position). It is then easily cheked that


Moreover, in terms of the Dirac’s bracket notation it is an easy to show that the resolution of the identity


holds for these vectors (in the weak sense). Here IN is the identity operator on N. The continuous wavelet transform of an arbitrary vector (signal) fN at the scale a and the position b is given by


The wavelet transform Sfab has several properties [34]:

  • It is linear in the sense that:


  • It is translation invariant:


where τb refers to the translation of the function f by b given


  • It is dilatation-invariant, in the sense that, if f satisfies the invariance dilatation property fx=λfrx for some λ,r>0 fixed then


As in Fourier or Hilbert analysis, wavelet analysis provides a Plancherel type relation which permits itself the reconstruction of the analyzed function from its wavelet transform. More precisely we have


which in turns to reconstruct the analyzed function f in the L2- sense from its wavelet transform as


The function Sf is the continuous wavelet transform of the signal f. The parameter 1/a represents the signal frequency of f and b its time. The conservation of the energy of the signal is due to the resolution of the identity (37), so


Then, the transform Sf is a fonction in the Hilbert space L2R×R+dbdaa2. The reproducing kernel associated to the signal is


which satisfies the square integrability condition (13) with respect to the measure dbda/a2. The corresponding reproducing kernel Hilbert space Nψ, one see that this is the space of all signal transforms, corresponding to the mother wavelet ψ. If ψ and ψ are two mother wavelets such that ψψ0, then


The formula (41) generalizes to


The vector ψ is called the analyzing wavelet and ψ the reconstructing wavelet. The repoducing kernel Hilbert space NL2R×R+, consisting of all signal transforms with respect to the mother wavelet ψ. Then, we have


is the integral kernel of a unitary map between Nψ and Nψ. The properties of the wavelet transform can be understood in terms of the unitary irreductible representation of the one-dilensional affine group.It is important to note that the Wavelets built on the basis of the group representation theory have all the properties of CS. There is a wole body of work devoted to the study of CS arising from group representation theory [7, 33, 35].


5. Application 1: GCS for the Legendre Hamiltonian and CS transform

5.1 GCS for the Legendre Hamiltonian

By replacing the coefficients zn/n! of the canonical CS by the function Onξ in (1) as mentioned in the introduction. We construct in this section a class of GCS indexed by point ξR.

Definition 5.The GCS labeled by pointsξRis defined by the following superposition


hereNξis a normalization factor, the functionOnξΦnξρn1/2, with


whereJn+1/ the cylindrical Bessel function ([20], p. 626):


andρnare positive numbers given by


andψnis an orthonormal basis of the Hilbert spaceH=L21121dxdefined in(27).

Proposition 1.The normalization factor defined by the GCS(47)reads as


for everyξR.

Proof. From (47) and by using the orthonormality relation of basis elements ψnn=0+ in (27), then


In order to identify the above series, we make appeal to the formula ([36], p. 591):


we then obtain the result (51) by using the GCS condition ϑξϑξ=1.

Proposition 2.The GCS defined in(47)satisfy the following resolution of the identity


(in the weak sense) in terms of an acceptable measure


wherethe Lebesgue’s measure onR. The rank one operatorTξ=ϑξϑξ:HHis define as


Proof. We need to determine the function σξ. Let


where σξ is an auxiliary function. Let us writte Tm,nψmψn, defined as in (56). According to (56) and by writing


Hence, we need σξ such that


We make appeal to the integral ([36], p. 211):


with condition c>0. Then, for parameters c=1, we have


By comparing (62) with (66) we obtain finally the desired weight function σξ=1/π. Therefore, the measure (57) has the form (55) [37]. Indeed (59) reduces further to n=0Tn,n=1H, in other words


According to this construction, the state ϑξ form an overcomplete basis in the Hilbert space H (Figure 1).

Figure 1.

Plots of the probability distribution Pnξ versus ξ for various values of n.

When the GCS (47) describes a quantum system, the probability of finding the state ψn in some normalized state ϑξ of the state Hilbert space H is given by Pnξψnϑξ2. For the GCS (47) the probability distribution function is given by


5.2 Coherent state transform

To discuss coherent state transforms (CST), we will start by establishing the kernel of this transformation by giving the closed form of the GCS (47).

Proposition 3.For allx11, the wave functions of GCS in(47)can be written as


for allξR.

Proof. We start by the following expression


where the series


with the function Onξ=Φnξρn1/2, mentioned in Definition 5. To do this, we start by replacing the function Φnξ and the positive sequences ρn by their expressions in (48) and (50) thus Eq. (67) reads


Making use the explicit expression (27) of the eigenstates ψnx, then the sum (68) becomes


We now appeal to the Gegenbauer’s expansion of the plane wave in Gegenbauer polynomials and Bessel functions ([38], p. 116):


Then, for γ=1/2, y=x and by using the identity Γ1/2=π, we arrive at (65).

Corollary 1.When the variableξ1, the GCS in(47)becomes


Proof. The result follows immediately by using the formula ([20], p. 647):




is the spherical Bessel function [20]. This ends the proof.

The careful reader has certainly recognized in (70) the expression of nonlinear coherent states [38].

Let us note that, in view of the formula ([36], p. 667):


the reproducing kernel arising from GCS (47) can be written as


denotes the Dyson’s sine kernel, which is the reproducing kernel of the Paley-Wiener Hilbert space PW1. Then, the family πn+1/2/ξ1/2Jn+12ξ;nN0, forms an orthonormal basis of PW1 [39].

Once we have a closed form of GCS, we can look for the associated CST, this transform should map the space H=L21121dx spanned by eigenstates ψn in (27) onto PW1L2R as.

Proposition 4.ForφL21121dx, the CST is the unitary map


defined by means of(65)as


for allξR.

Corollary 2.The following integral



Proof. From (75), the image of the basis vector ψn under the transform W should exactly be


Now, by writing (75) as


and replacing ψn by their values given in (27), we obtain


the integral 78 can be evaluated by the help of the formula ([40], p. 456):


this ends the proof.

Note that, in view of ([28], p. 29), by considering hnξρn1/2Φnξ¯ and GCS Kξxxϑξ, the basis element ψnL21121dx has the integral representation


where the function Φnξ and the positive sequences ρn are given in (48) and (50) respectively, the measure ξ is given in (55), then the Legendre polynomial has the following integral representation


where the functionJn. is given in (72), which is recognized as the Fourier transform of the spherical Bessel function (72) (see [40], p. 267):


where Pn. the Legendre’s polynomial [40].

Remark 1.Also note that:

  • The usefulness expansion of GCS was made very clear in a paper authored by Ismail and Zhang, where it was used to solve the eigenvalue problem for the left inverse of the differential operator, onL2-spaces with ultraspherical weights [41,42].

  • Forx,ξR, the functionφξx=eixξ, is known as the Gabor’s coherent states introduced in signal theory where the propertyψξ=T̂ξψ, withψL2R, andT̂ξthe unitary transformation, is obtained by using the standard representation of the Heisenberg group in three dimensions, inL2R, for more information (see [43]).

Exercise 1.Show that the vectors


forms a set of GCS and gives the associated GCS transform.


6. Application 2: continuous Bessel wavelet transform

The continuous wavelet transform (CWT) is used to decompose a signal into wavelets. In mathematics, the CWT is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. There are several ways to introduce the Bessel wavelet [22, 23]. For 1p and μ>0, denote


and ψ,σ=ess0<x<supψx< and x is the measure defined as


Now, let us consider the function


where Jμ12x is the Bessel function of order lμ1/2 given by


For μ=1, the function jx=O0x coincides with equation 2 discussed in the introduction. For each function ϕL1,σ0, the Hankel transform of order μ is defined by


We know that from ([44], p. 316) that ϕ̂x is bounded and continuous on 0 and ϕ̂,σϕ1,σ. If ϕ,ϕ̂L1,σ0, then by inversion, we have


From ([45], p. 127) if ϕx and Φx are in L1,σ0, then the following Parseval formula also holds


Denoting therefore by


For a 1-variable function ψLσ2R+, we define the Hankel translation operator


Trime’che ([46], p. 177) has shown that the integral is convergent for almost all y and for each fixed x, and


The map yτyψ is continuous from 0 into 0. For a 2-variables the function ψ, we define a dilatation operator


From the inversion formula in (89), we have


for t=0 and μ1/2=0, we arrive at


The Bessel Wavelet copy ψa,b are defined from the Bessel wavelet mother ψLσ2R+ by


the integral being convergent by virtue of (92). As in the classical wavelet theory on R, let us define the continuous Bessel Wavelet transform (CBWT) of a function fLσ2R+, at the scale a and the position b by


The continuity of the Bessel wavelet follows from the boundedness property of the Hankel translation ([46], (104), p. 177). The following result is due to [22]:

Theorem 1.3 Let ψLσ2R+ and f,gLσ2R+. Then




For all μ>0.

Proof. For the function fLσ2R+, let us write the Bessel wavelet by using Eq. (38) as


Now observe that


Hence whe have that


In terms of the Parseval formula (90), we obtain


Now multiplying by a2μ1a and integrating, we get


The admissiblecondition (102) requires that ψ̂0=0. If ψ̂ is continuous then from (88) it follows that


6.1 Example

Let us consider the function


In the case μ=1/2, the measure (85) takes the form


and the function (86) reduces to


where J0x the Bessel’s function of the first kind. Also note that


The Bessel wavelet transform of ft is given by


Using the representation


then (122) becomes


Where the integral


In terms of the Legendre polynomial P2t, the function


Then (125) reads


The above equation can be evaluated by means of the formula ([47], p. 13):


For parameters n=2 and p=w0, we find that


In terms of the above result, the CBWT read as




To evaluated (131) we make appeal to the Lipschitz-Hankel integrals ([48], p. 389):


with conditions p±iq±ir>0 and μ+2ν>0, while ζ is written in place of q2+r22qrcosϕ1/2, where 2F1 denotes the hypergeometric function. For parameters p=w0/a,q=b/a,r=z,μ=3 and n=0, we arrive at


where ζ=a1b2+z22a1bzcosϕ1/2.

Next, by using the representation of the hypergeometric 2F1-sum ([49], p. 404, Eq. 209) (Figure 2):

Figure 2.

Plots of the mother wavelet ft defined in 6.34 versus t, for various values of the parameters w0.


Then (131) takes the form


This leads to the following CBWT


We have given an example of a signal ftLσ20 such that the CBWT is written as


According to Theorem 1.3, let ψLσ2R+ and f,gLσ2R+, then


Note that, for all w0>0, the given function


is the mother wavelet. The Hankel transform of ft is given by


and satisfy the admissible condition


The Hankel transformation f̂0=0, so by the help of (140) we obtain


Exercise 2

For which numbersnN, the following function


Is the mother wavelet wherePn.the Legendre’s polynomial.


7. Conclusions

In this chapter we are interested in the construction of the generalized coherent state (GCS) and the theory of wavelets. As it is well know wavelets constructed on the basis of group representation theory have the same properties as coherent states. In other words, the wavelets can actually be thought of as the coherent state associated with these groups. Coherent state is very important because of three properties they have: coherence, overcompleteness, intrinsic geometrization. We have seen that it is possible to construct coherent states without taking into account the theory of group representation. Throughout this chapter we have used the Bessel function to construct the coherent state transform and Bessel continuous wavelets transform. We have prove that the kernel of the finite Fourier transform (FFT) of L2-functions supported on 11 form a set of GCS. We therefore discussed another way of building a set of coherent states based on Wavelet’s theory makes it easier.

Building coherent states in this chapter is always not easy because it is necessary to find coefficients which will make it possible to find vectors which will certainly satisfy certain conditions but the procedure based on Wavelet’s theory makes it easier.

It should be noted that the theory of classical wavelets finds several applications ranging from the analysis of geophysical and acoustic signals to quantum theory. This theory solves difficult problems in mathematics, physics and engineering, with several modern applications such as data compression, wave propagation, signal processing, computer graphics, pattern recognition, pattern processing. Wavelet analysis is a robust technique used for investigative methods in quantifying the timing of measurements in Hamiltonian systems.


Conflict of interest

The authors declare no conflict of interest.


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Written By

Kayupe Kikodio Patrick

Submitted: June 23rd, 2020 Reviewed: November 1st, 2020 Published: February 24th, 2021