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

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.