Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: An overall theoretical discussion of the FC-AD approach is provided and the methodology is extended to linear hyperbolic PDEs and an example concerning the Wave Equation shows a complete absence of the debilitating ''dispersion error'', that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.
100 citations
TL;DR: The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures and the theory is compactly presented with an emphasis on the analogy to the normal Fourier transform.
Abstract: In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.
100 citations
TL;DR: In this article, a Walsh operational matrix of PWM waveforms is introduced as a means of obtaining the Walsh spectral equations of waveforms, and a noniterative algorithm that produces piecewise-linear, global solutions between angles and for the angles is proposed.
Abstract: A method for selective harmonic elimination in pulse-width-modulated (PWM) inverter waveforms by the use of Walsh functions is presented. The Walsh operational matrix of PWM is introduced as a means of obtaining the Walsh spectral equations of PWM waveforms. The slope and intercept Fourier operational matrices of PWM are also introduced as a means of obtaining Fourier spectral equations of PWM waveforms. A noniterative algorithm that produces piecewise-linear, global solutions between angles and for the angles is proposed. The algorithm also produces the full range of variation of fundamental voltage for given harmonic elimination constraints. The set of systems of linear equations obtained replaces the system of nonlinear transcendental equations used in the Fourier series harmonic elimination approach. In general, the algorithm makes possible the synthesis of two-state PWM inverter waveforms with specified old harmonic content. >
100 citations
TL;DR: In this article, a theoretical formulation for the determination of the displacements, strains, and stresses in a three-dimensional transversely isotropic linearly elastic medium is presented, which can be used in a variety of elastodynamic as well as elastostatic problems.
Abstract: A theoretical formulation is presented for the determination of the displacements, strains, and stresses in a three-dimensional transversely isotropic linearly elastic medium. By means of a complete representation using two displacement potentials, it is shown that the governing equations of motion for this class of problems can be uncoupled into a fourth-order and a second-order partial differential equation in terms of the spatial and time coordinate under general conditions. Compatible with Fourier expansions and Hankel transforms in a cylindrical coordinate system, the formulation includes a complete set of transformed displacement-potential, strain-potential, and stress-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. As an illustration of the application of the method, the solution for a half-space under the action of arbitrarily distributed, time-harmonic surface traction is derived, including its specialization to uniform patch loads and point forces. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are also included for cases of different degree of the material anisotropy, frequency of excitation, and compared with existing solutions.
99 citations
TL;DR: In this paper, the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers' equation is analyzed, and it is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth.
Abstract: The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.
99 citations