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

Written By

Kayupe Kikodio Patrick

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

DOI: 10.5772/intechopen.94865

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Abstract

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.

Keywords

  • 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

Onξinπ2n+12ξ12Jn+12ξ,n=0,1,2,,E1

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

O0=J0ξ=sinξξE2

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

limnPncosyn=0πeiycosϕ=J0y,

and the integral

0J0yJ0ydy=π2.E3

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+

ft:=2w0t22w02+t25/2,w0>0,E4

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.

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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

n=0ρn1Cnx2<+E5

where ρnCnL2Xμ2. Define the kernel

Kxyn=0ρn1CnxCny¯,x,yX.E6

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

ϑxNx1/2n=0ρn1/2Cnx¯φn.

Therefore,

ϑxϑx=Nx1n=0ρn1CnxCnx¯=1,

and

W:HN2withWϕ=N1/2ϑxϕ

is an isometry. For ϕ,ψH, whe have

ϕψH=WϕWψN2=XWϕx¯WψxxE7
=XϕϑxϑxψNxx,E8

and

XϑxϑxNxx=IH,E9

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

ϑxNx1/2n=0ρn1/2Cnx¯φn.E10

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

WfxNx1/2fϑxH.E11

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

1H=XTxNxxE12

where the rank one operatorTxϑxϑx:HHis define by

fTxf=ϑxfϑx.

Nxappears as a weight function.

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

XKxzKzyNzz=Kxy.E13

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

ϑz=πezz¯2k=0znn!ϕn,zCE14

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.

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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

FzAeBz.E15

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

Ω=limr+suplogMrrE16

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

    FzCεeΩ+εz;

  2. There exists C>0 such that

    FzCeΩz;

  3. as z+

    Fz=oeΩz.

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

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

PWΩp=fL2R:fx=ΩΩgyeixydywheregLp(ΩΩ)E17

and we set

fPWΩp=2πgLp.E18

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

fPWΩ2=ĝPWΩ2=2πgL2=f̂L2=fL2.E19

Hence, by polarization, for f,φPWΩ,

fφPWΩ=fφL2.E20

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

  • FRL2R and

Fz=oeΩzasz+,E21

  • there exists fL2R with suppf̂ΩΩ such that

Fz=12πRf̂ξeizξ.E22

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

Kxy=ΩπsincΩxy,E23

wheresinct=sint/t. Hence, for every fPWΩ

fx=ΩπRfysincΩxydy,E24

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

H=X2+P2=a+a+aa+,E25

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

λ0=23,λn=nn+112n+32n12,n=1,2,3,,E26

and the corresponding eigenfunctions reads

ψnx=2n+1Pnx,n=0,1,2,3,..,E27

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

xψnx=bn1ψn1x+bnψn+1x,ψ1x=0,ψ0x=1,E28

with coefficients

bn=n+122n+12n+3,n0.E29

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

Xψnx=bnψn+1x+bn1ψn1x,E30

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

XPψn=2ibn2bn12ψn=2i2n12n+12n+3ψn.E31

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

a+=12XiP;a=12X+iP,E32

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

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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

Cψψ̂ξ2ξ<,E33

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

y=baxax+b,a>0,bR,xR+.E34

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

ψa,bx=1aψxba,ψ1,0=ψ.E35

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

ψa,bxN2=ψxN2,foralla>0andbR.E36

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

1CψR×R+ψa,bψa,bdbdaa=INE37

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

Sfab=0ftψa,btdt.E38

The wavelet transform Sfab has several properties [34]:

  • It is linear in the sense that:

Sαf1+βf2ab=αSf1ab+βSf2ab,α,βRandf1,f2L2R+.

  • It is translation invariant:

Sτbfab=Sfabb

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

τbfx=fxb.

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

Sfab=λSfrarb.E39

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

fg=1Cψa>0bRSfabSgab¯dadba2,f,gL2RE40

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

fx=1Cψa>0bRSfabψa,bdadba2,whereSfab=ψa,bf.E41

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

Cψf2=R×R+Sfba2dbdaa2.E42

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

Kψbaba=1Cψψa,bψa,b.E43

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

1ψψR×R+ψa,bψa,bdbdaa2=IN,E44

The formula (41) generalizes to

f=1ψψR×R+Sfbaψa,bdbdaa2,whereSfab=ψa,bf.E45

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

Kψ,ψbaba=1CψCψ12ψa,bψa,bE46

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].

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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

ϑξ=Nξ1/2n=0Onξψn,ξRE47

hereNξis a normalization factor, the functionOnξΦnξρn1/2, with

Φnξ=inπ2ξJn+12ξ,E48

whereJn+1/2.is the cylindrical Bessel function ([20], p. 626):

Jn+12z=s=01ss!s+n+1/2!z22s+n+12,zCE49

andρnare positive numbers given by

ρn=12n+1,n=0,1,2,,E50

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

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

Nξ=1,E51

for everyξR.

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

ϑξϑξ=πξNξ1n=0n+12Jn+12ξJn+12ξ.E52

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

n=0n+12Jn+12ξJn+12ξ=π1ξ,E53

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

RTξξ=1H,E54

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

ξ=1π,E55

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

φTξφ=ϑξφϑξ.E56

Proof. We need to determine the function σξ. Let

ξ=σξ,E57

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

RTξξ
=n,m=0π21nin+mJm+12ξJn+12ξρmρnσξξTm,nE58
=n,m=0π21nin+m2m+12n+1Jm+12ξJn+12ξσξξTm,n.E59

Hence, we need σξ such that

Jn+12ξJm+12ξσξξ=2π2n+1δm,n.E60

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

1yJm+12cyJn+12cydy=22n+1δm,n,E61

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

1ξJm+12ξJn+12ξ=22n+1δm,n.E62

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

RTξξ=1H.E63

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

Pnξ=π2n+12ξJn+12ξ2,ξR+.E64

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

ϑξx=eixξ,E65

for allξR.

Proof. We start by the following expression

ϑξx=Nξ1/2Sxξ,E66

where the series

Sxξn=0Onξψnx,E67

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

Sxξ=π2ξn=01nin2n+1Jn+12ξψnx.E68

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

Sxξ=2πξn=01ninn+12Jn+12ξPnx.E69

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

eiξx=Γγξ2γn=0inn+γJn+γξCnγx

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

ϑξNξ1/2n=02πn22n+12n+1Γn+12ψn.E70

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

Jnξξn2n+1!!,ξ1E71

where

Jnξ=π2ξJn+12ξ,n=0,1,2,,E72

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):

n=0n+12Jn+12ηJn+12ξ=ηξπηξsinηξ,E73

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

KηξϑηϑξE74
=πn=0n+12Jn+12ηηJn+12ξξ=sinηξηξ,E75

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

WL2(1121dx=PW1,E76

defined by means of(65)as

Wφξ=Nξ1/2φξH=11eixξφx¯dx2,E77

for allξR.

Corollary 2.The following integral

inξJn+12ξ=12π11Pnxeiξxdx,ξR.E78

holds.

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

Wψnξ=inπ2n+12ξJn+12ξ.E79

Now, by writing (75) as

Wψnξ=11eixξψnxdx2,

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

Wψnξ=2n+1211eixξPnxdx,

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

11Pnxeiξxdx=in2πξJn+12ξ,E80

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

ψnx=hnξKξx¯ξE81

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

Pnx=inπJnteixξ,E82

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

eixtJntdt=πinPnx,1<x<112π±in,x=±1,0,±x>1E83

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

ϑξ=Nξ1/2n=02πn22n+12n+1Γn+12ψn.E84

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

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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

LσpR+ψsuchasψp,σp=0ψxpx<

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

x=x2μ2μ+12Γμ+32dx.E85

Now, let us consider the function

jx=2μ12Γμ+12x12μJμ12x,E86

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

Jlx=x2lk=01kk!Γk+l+1x22k.E87

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

ϕ̂x0jxtϕtt,0x<.E88

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

ϕx=0jxtϕ̂tt.E89

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

0ϕ̂tΦ̂tt=0ϕxΦxx.E90

Denoting therefore by

Dxyz=0jxtjytjztt.E91

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

τyψxψxy=0Dxyzψzz,x>0,y<.E92

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

ψx.2,σψ2,σ.E93

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

Daψxy=a2μ1ψxaya.E94

From the inversion formula in (89), we have

0jztDxyzz=jxtjyt,0<x,y<,0t<,

for t=0 and μ1/2=0, we arrive at

0Dxyzz=1.E95

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

ψa,bx:=Daτbψx=DaψbxE96
=a2μ10Dbaxazψzz,a>0,bR,E97

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

BbaBψfba=ftψb,atE98
=0ftψa,bt¯tE99
=a2μ100ftψz¯Dbatazzt.E100

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

00BψfbaBψgba¯ab=CψfgE101

whenever

Cψ=0t2μ1ψ̂t2t<.E102

For all μ>0.

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

Bψfba=0ftψa,bttE103
=1a2μ+100ftψ¯zDbatazzt.E104

Now observe that

Dbataz=0jbuajtuajzuu.E105

Hence whe have that

Bψfba=1a2μ+1R+3ftψzjbuajtuajzuuztE106
=1a2μ+1R+2f̂uaψzjbuajzuuzE107
=1a2μ+1R+f̂uaψ̂ujbuauE108
=R+f̂vψ̂avjbvvE109
=f̂vψ̂av̂b.E110

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

R+BψfbaBψf¯bab
=0f̂vψ̂av̂bĝvψ̂av¯¯̂buE111
=0f̂uψ̂au¯ĝuψ̂au¯¯uE112

Now multiplying by a2μ1a and integrating, we get

R+R+BψfbaBψf¯baa2μ1abE113
=0f̂uψ̂au¯ĝuψ̂au¯¯aa2μ+1uE114
=Rf̂uĝu¯Rψ̂au2aa2μ+1u=CψRf̂uĝu¯uE115
=Cψfg.E116

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

0ψxx=0.E117

6.1 Example

Let us consider the function

ft=2w02t22w02+t25/2,w0>0,tR+.E118

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

t=t2dtE119

and the function (86) reduces to

jt=J0t,E120

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

02w02t222w02+t25t<.E121

The Bessel wavelet transform of ft is given by

Bψ2w02t22w02+t25/2ba=a202w02t22w02+t25/2ψbatatE122
=a20ψz02w02t22w02+t25/2DbataztzE123

Using the representation

Dbataz=0J0bauJ0tauJ0zuuE124

then (122) becomes

a20ψz0J0bauJ0zuOa,w0uuz

Where the integral

Oa,w0u=02w02t22w02+t25/2J0taut.E125

In terms of the Legendre polynomial P2t, the function

2w02t22w02+t25/2=w02+t23/2P2w0w02+t21/2.E126

Then (125) reads

Oa,w0u=0w02+t23/2P2w0w02+t21/2J0taut.E127

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

1n!yn1/2epy=0x1/2p2+x212n12Pnpp2+x21/2xy1/2J0xydx.E128

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

Oa,w0u=14uexpw0ua.E129

In terms of the above result, the CBWT read as

Bψ2w02t22w02+t25/2ba=a20ψzMa,w0zzE130

where

Ma,w0z=081u2ew0auJ0bauJ0zudu.E131

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

0eptJνqtJνrttμ1dtE132
=qrνπpμ+2νΓμ+2ν2ν+12F1πμ+2ν2μ+2ν+12ν+1ζ2p2sin2νϕdϕ

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

Ma,w0z=a34πw032F1π3221aw01ζ2E133

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.

2F13221z=122+z1z5/2.E134

Then (131) takes the form

Ma,w0z=a38πw030π2w0121+w0125/2,E135

This leads to the following CBWT

Bψ2w02t22w02+t25/2ba=a4π0ψz0π2w0222w02+25/2dϕdσz.E136

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

Bψftba=a4π0π0ψzfz.E137

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

00BψfbaBψgba¯ab=1128w02fg.E138

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

ft=2w02t22w02+t25/2,tR+,E139

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

f̂y=02w02t22w02+t25/2J0xyt=14yew0y,0y<.E140

and satisfy the admissible condition

Cf=120f̂ξ2ξE141
=1128w02,w0>0.E142

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

0t2w02t2w02+t25/2dt=0.E143

Exercise 2

For which numbersnN, the following function

fnt=w02+x212n12Pnw0w02+x21/2E144

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

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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.

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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