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.

Electromagnetic Scattering of Inhomogeneous Plane Wave by Ensemble of Cylinders

The interaction between an ensemble of cylinders and an inhomogeneous plane wave is introduced and is determined, in the present paper, through a rigorous theoretical approach. Scattered electromagnetic field generated by an indefinite number of infinite circular cylinders is analyzed by the application of the generalized vector cylinder harmonics (VCH) expansion. The exact mathematical model relied upon to represent this scenario considers the so-called complex-angle formalism reaching a superposition of vectorial cylindrical-harmonics and Foldy-Lax Multiple scattering equations (FLMSE) to account for the multiscattering process between the cylinders. The method was validated by comparing the numerical results obtained with the use of the finite element method and a homemade Matlab code

Electromagnetic Scattering by a Cylinder in a Lossy Medium of an Inhomogeneous Elliptically Polarized Plane Wave

In this paper, a rigorous theoretical approach, adopted in order to generalize the Vectorial CylindricalHarmonics (VCH) expansion of an inhomogeneous elliptically polarized plane wave, is presented. An application of the VCH expansion to analyze electromagnetic field scattered by an infinite circular cylinder is presented. The results are obtained using the so-called complex-angle formalism reaching a superposition of Vectorial Cylindrical-Harmonics. To validate the method, a Matlab code was implemented. Also, the validity of the methodology was confirmed through some comparisons between the proposed method and the numerical results obtained based on the Finite Element Method (FEM) in the canonical scenario with a single cylinder.

Fourier components of cylindrical illuminance

Fourier analysis is applied to the profile of illuminance around the curved surface of a small cylinder, due to one, two and three point sources of light and due to an extended source. This reveals only one odd harmonic, whose amplitude is equal to half that of the illumination vector. Lighting can also generate a diminishing series of even harmonics. The lighting at any point in open space can be expressed in terms of the harmonics of illuminance around three mutually perpendicular cylinders. Visual discomfort is often associated with the presence of even harmonics. The photometry of cylindrical and semi-cylindrical illuminance, and of cylindrical harmonics, is discussed.

Slow viscous flow of two porous spherical particles translating along the axis of a cylinder

We describe the motion of two freely moving porous spherical particles located along the axis of a cylindrical tube with background Poiseuille flow at low Reynolds number. The stream function and a framework based on cylindrical harmonics are adopted to solve the flow field around the particles and the flow within the tube, respectively. The two solutions are employed in an iterated framework using the method of reflections. We first consider the case of two identical particles, followed by two particles with different dimensions. In both cases, the drag force coefficients of the particles are solved as functions of the separation distance between the particles and the permeability of the particles. The detailed flow field in the vicinity of the two particles is investigated by plotting the streamlines and velocity contours. We find that the particle–particle interaction is dependent on the separation distance, particle sizes and permeability of the particles. Our analysis reveals that when the permeability of the particles is large, the streamlines are more parallel and the particle–particle interaction has less effect on the particle motion. We further show that a smaller permeability and bigger particle size generally tend to squeeze the streamlines and velocity contour towards the wall.