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.

Highlights

  • 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 introduced by Klauder [4, 5]

  • 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 this chapter we are interested in the construction of the generalized coherent state (GCS) and the theory of wavelets

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Summary

Introduction

Coherent state (CS) was originally introduced by Schrödinger in 1926 as a Gaussian wavepacket to describe the evolution of a harmonic oscillator [1]. Wavelet Theory introduced a new family of CS as a suitable superposition of the associated Bessel functions and in [13–15] the authors 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). This tool permits the representation of L2functions in a basis well localized in time and in freqency.

Generalized coherent states formalism
The Paley-wiener space PWΩ
The Legendre Hamiltonian
Wavelet theory on and the reproduction of kernels
GCS for the Legendre Hamiltonian pffiffiffiffi
Coherent state transform
Application 2: continuous Bessel wavelet transform
Example
Conclusions
Full Text
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