Open access peer-reviewed chapter

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

By Kayupe Kikodio Patrick

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

DOI: 10.5772/intechopen.94865

Downloaded: 52

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 ξRand 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 L21121dxof 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 11form 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 Pnxand the Bessel function of order 0are 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 PWΩand 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=0is the mother wavelet where tis 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 Xμbe a measure space and let N2L2Xμbe a sub-closed space of infinite dimension. Let Cnn=0be 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 Kxyis a reproducing kernel, N2is 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 Nxis 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=0being an orthonormal basis of eigenstates of the quantum harmonic oscillator. Then, the space N2is the Fock spaceFCand Nz=π1ezz¯,zC.

3. The Paley-wiener space PWΩand 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 Fsatisfy Definition 3, we call Ωthe type of Fwhere

Ω=limr+suplogMrrE16

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

  1. For all ε>0there exists Cεsuch that

    FzCεeΩ+εz;

  • There exists C>0such that

    FzCeΩz;

  • as z+

    Fz=oeΩz.

  • Then cleary, 321Fis 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 PWΩpis the image via the Fourier transformof the Lp-function that are supported in ΩΩ. We will be interested in the case p=2, in which PWΩto denote the Paley-Wiener space PWω2. From the Plancherel formulawe have

    fPWΩ2=ĝPWΩ2=2πgL2=f̂L2=fL2.E19

    Hence, by polarization, for f,φPWΩ,

    fφPWΩ=fφL2.E20

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

    • FRL2Rand

    Fz=oeΩzasz+,E21

    • there exists fL2Rwith suppf̂ΩΩsuch that

    Fz=12πRf̂ξeizξ.E22

    The function fPWΩif and only if fL2Rand f=FR(that is, fis the restriction to the real line of a function F), where Fis an entire function of exponential type such that Fz=oeΩzfor z+.

    Theorem 1.2 The Paley-Wiener space PWΩis 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 Pnxand the Legendre function ψnxare 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 Xand Pdenotes respectively the position and momentum operators, a+and aare the creation and annihilation operators. The eigenvalues of operators Hare 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=0in 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 Hconnected with the Legendre polynomials Pnxis an operator of multiplication by argument Xψn=xψn.Taking into account of the relation (28), then

    Xψnx=bnψn+1x+bn1ψn1x,E30

    whee bnare coefficients defined by Eq. (29). Because n=01/bn=+, Xis a self-adjoint operator on the Hilbert space H(see [30, 31, 32]). The momentum operator Pby 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 Xand Pon 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.

    4. Wavelet theory on Rand 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 ψsatisfying the so-called admissibility condition

    Cψψ̂ξ2ξ<,E33

    where ψ̂being the Hankel transform of ψ. Not every vector in Nsatisfies the above condition. A vector ψsatisfying (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 abof the real numbers, with a>0, from translations and dilatations of the function ψ, we obtain a family of wavelets ψa,bNas

    ψa,bx=1aψxba,ψ1,0=ψ.E35

    Here ais the parameter of dilation (or scale) and bis 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 INis the identity operator on N. The continuous wavelet transformof an arbitrary vector (signal) fNat the scale aand the position bis given by

    Sfab=0ftψa,btdt.E38

    The wavelet transform Sfabhas several properties [34]:

    • It is linear in the sense that:

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

    • It is translation invariant:

    Sτbfab=Sfabb

    where τbrefers to the translation of the function fby bgiven

    τbfx=fxb.

    • It is dilatation-invariant, in the sense that, if fsatisfies the invariance dilatation property fx=λfrxfor some λ,r>0fixed 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 fin the L2- sense from its wavelet transform as

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

    The function Sfis the continuous wavelet transform of the signal f. The parameter 1/arepresents the signal frequency of fand bits 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 Sfis 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 ψand ψare 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 ψis called the analyzing wavelet and ψthe 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 Nψand 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 Onξin (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 σξis 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 σξsuch 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 ϑξform an overcomplete basis in the Hilbert space H(Figure 1).

    Figure 1.

    Plots of the probability distributionPnξversusξfor various values ofn.

    When the GCS (47) describes a quantum system, the probability of finding the state ψnin some normalized state ϑξof the state Hilbert space His 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 Φnξand the positive sequences ρnby 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=xand 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=L21121dxspanned by eigenstates ψnin (27) onto PW1L2Ras.

    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 ψnunder the transform Wshould exactly be

    Wψnξ=inπ2n+12ξJn+12ξ.E79

    Now, by writing (75) as

    Wψnξ=11eixξψnxdx2,

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

    Wψnξ=2n+1211eixξPnxdx,

    the integral 78can 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 ψnL21121dxhas the integral representation

    ψnx=hnξKξx¯ξE81

    where the function Φnξand the positive sequences ρnare given in (48) and (50) respectively, the measure ξis 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 1pand μ>0, denote

    LσpR+ψsuchasψp,σp=0ψxpx<

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

    x=x2μ2μ+12Γμ+32dx.E85

    Now, let us consider the function

    jx=2μ12Γμ+12x12μJμ12x,E86

    where Jμ12xis the Bessel function of order lμ1/2given by

    Jlx=x2lk=01kk!Γk+l+1x22k.E87

    For μ=1, the function jx=O0xcoincides with equation 2discussed in the introduction. For each function ϕL1,σ0, the Hankel transform of order μis defined by

    ϕ̂x0jxtϕtt,0x<.E88

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

    ϕx=0jxtϕ̂tt.E89

    From ([45], p. 127) if ϕxand Φxare 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 yand for each fixed x, and

    ψx.2,σψ2,σ.E93

    The map yτyψis continuous from 0into 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=0and μ1/2=0, we arrive at

    0Dxyzz=1.E95

    The Bessel Wavelet copy ψa,bare 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 aand the position bby

    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 a2μ1aand 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 ψ̂is 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 J0xthe Bessel’s function of the first kind. Also note that

    02w02t222w02+t25t<.E121

    The Bessel wavelet transform of ftis 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=2and 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>0and μ+2ν>0, while ζis written in place of q2+r22qrcosϕ1/2, where 2F1denotes the hypergeometric function. For parameters p=w0/a,q=b/a,r=z,μ=3and 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 waveletftdefined in6.34versust, for various values of the parametersw0.

    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 ftLσ20such 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 ftis 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.

    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 11form 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.

    © 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Kayupe Kikodio Patrick (February 24th 2021). Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory, Wavelet Theory, Somayeh Mohammady, IntechOpen, DOI: 10.5772/intechopen.94865. Available from:

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