scholarly journals Notes on Fourier analysis, XXXII. On the summability $\left( {C,1} \right)$ of the Fourier series

1950 ◽  
Vol 26 (7) ◽  
pp. 5-9
Author(s):  
Noboru Matsuyama
Keyword(s):  
Fourier Series ◽  
Fourier Analysis

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

A Bandpass Active Filter for Fourier Analysis Laboratory

1995 ◽  
Vol 32 (4) ◽  
pp. 350-353
Author(s):  
Marcelo Basilio Joaquim
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Signal Analysis ◽  
Active Filter ◽  
Experimental Demonstration ◽  
High Q ◽  
State Variable ◽  
Analysis Laboratory ◽  
Tuning Frequency

A bandpass active filter for Fourier analysis laboratory In this note is presented a modified bandpass state-variable active filter which exhibits constant gain, high-Q, where its tuning frequency can easily be varied. This circuit proves to be useful for experimental demonstration of the Fourier series in communications or signal analysis laboratories.

scholarly journals On Fourier Series of Fuzzy-Valued Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Uğur Kadak ◽  
Feyzi Başar
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Uniform Convergence ◽  
Level Sets ◽  
Closed Interval ◽  
Complex Form ◽  
Dirichlet Kernel ◽  
Extension Principle ◽  
Science And Engineering ◽  
Interest In Science

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally, by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functions at each point of discontinuity, where one-sided limits exist.

scholarly journals Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 72 ◽  
Author(s):  
Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Water Waves ◽  
Ocean Waves ◽  
Airborne Lidar ◽  
Ocean Dynamics ◽  
Theoretical Interpretation ◽  
Maximum Height ◽  
Nonlinear Phenomena ◽  
Nonlinear Schrödinger

I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements.

scholarly journals Signatures of New Physics Versus the Ridge Phenomenon in Hadron–Hadron Collisions at the LHC

Physics ◽  
2019 ◽  
Vol 1 (1) ◽  
pp. 84-91 ◽  
Author(s):  
Miguel-Angel Sanchis-Lozano ◽  
Edward K. Sarkisyan-Grinbaum
Keyword(s):  
Fourier Series ◽  
Large Hadron Collider ◽  
Fourier Analysis ◽  
Parton Shower ◽  
Hadron Collider ◽  
New Physics ◽  
Beyond The Standard Model ◽  
Final State ◽  
Azimuthal Correlations ◽  
The Standard Model

In this paper, we consider the possibility that a new stage of matter stemming from hidden/dark sectors beyond the Standard Model, to be formed in p p collisions at the LHC (Large Hadron Collider), can significantly modify the correlations among final-state particles. In particular, two-particle azimuthal correlations are studied by means of a Fourier series sensitive to the near-side ridge effect while assuming that hidden/dark particles decay on top of the conventional parton shower. Then, new (fractional) harmonic terms should be included in the Fourier analysis of the azimuthal anisotropies, encoding the hypothetical new physics contribution and enabling its detection in a complementary way to other signatures.

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