Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

A historical perspective of spectrum estimation

Abstract: The prehistory of spectral estimation has its roots in ancient times with the development of the calendar and the clock. The work of Pythagoras in 600 B.C. on the laws of musical harmony found mathematical expression in the eighteenth century in terms of the wave equation. The struggle to understand the solution of the wave equation was finally resolved by Jean Baptiste Joseph de Fourier in 1807 with his introduction of the Fourier series. The Fourier theory was extended to the case of arbitrary orthogonal functions by Sturm and Liouville in 1836. The Sturm-Liouville theory led to the greatest empirical success of spectral analysis yet obtained, namely the formulation of quantum mechanics as given by Heisenberg and Schrodinger in 1925 and 1926. In 1929 John von Neumann put the spectral theory of the atom on a firm mathematical foundation in his spectral representation theorm in Hilbert space. Meanwhile, Wiener developed the mathematical theory of Brownian movement in 1923, and in 1930 he introduced generalized harmonic analysis, that is, the spectral representation of a stationary random process. The common ground of the spectral representations of von Neumann and Wiener is the Hilbert space; the von Neumann result is for a Hermitian operator, whereas the Wiener result is for a unitary operator. Thus these two spectral representations are related by the Cayley-Mobius transformation. In 1942 Wiener applied his methods to problems of prediction and filtering, and his work was interpreted and extended by Norman Levinson. Wiener in his empirical work put more emphasis on the autocorrelation function than on the power spectrum. The modern history of spectral estimation begins with the breakthrough of J. W. Tukey in 1949, which is the statistical counterpart of the breakthrough of Fourier 142 years earlier. This result made possible an active development of empirical spectral analysis by research workers in all scientific disciplines. However, spectral analysis was computationally expensive. A major computational breakthrough occurred with the publication in 1965 of the fast Fourier transform algorithm by J. S. Cooley and J. W. Tukey. The Cooley-Tukey method made it practical to do signal processing on waveforms in either the time or the frequency domain, something never practical with continuous systems. The Fourier transform became not just a theoretical description, but a tool. With the development of the fast Fourier transform the field of empirical spectral analysis grew from obscurity to importance, and is now a major discipline. Further important contributions were the introduction of maximum entropy spectral analysis by John Burg in 1967, the development of spectral windows by Emmanuel Parzen and others starting in the 1950's, the statistical work of Maurice Priestley and his school, hypothesis testing in time series analysis by Peter Whittle starting in 1951, the Box-Jenkins approach by George Box and G. M. Jenkins in 1970, and autoregressive spectral estimation and order-determining criteria by E. Parzen and H. Akaike starting in the 1960's. To these statistical contributions must be added the equally important engineering contributions to empirical spectrum analysis, which are not treated at all in this paper, but form the subject matter of the other papers in this special issue.

mathematical-and-experimental-modeling-of-physical-and-biological-processes

Abstract: Introduction: The Iterative Modeling Process Modeling and Inverse Problems Mechanical Vibrations Inverse Problems Mathematical and Statistical Aspects of Inverse Problems Probability and Statistics Overview Parameter Estimation or Inverse Problems Computation of sigman, Standard Errors, and Confidence Intervals Investigation of Statistical Assumptions Statistically Based Model Comparison Techniques Mass Balance and Mass Transport Introduction Compartmental Concepts Compartment Modeling General Mass Transport Equations Heat Conduction Motivating Problems Mathematical Modeling of Heat Transfer Experimental Modeling of Heat Transfer Structural Modeling: Force/Moments Balance Motivation: Control of Acoustics/Structural Interactions Introduction to Mechanics of Elastic Solids Deformations of Beams Separation of Variables: Modes and Mode Shapes Numerical Approximations: Galerkin's Method Energy Functional Formulation The Finite Element Method Experimental Beam Vibration Analysis Beam Vibrational Control and Real-Time Implementation Introduction Controllability and Observability of Linear Systems Design of State Feedback Control Systems and State Estimators Pole Placement (Relocation) Problem Linear Quadratic Regulator Theory Beam Vibrational Control: Real-Time Feedback Control Implementation Wave Propagation Fluid Dynamics Fluid Waves Experimental Modeling of the Wave Equation Size-Structured Population Models Introduction: A Motivating Application A Single Species Model (Malthusian Law) The Logistic Model A Predator/Prey Model A Size-Structured Population Model The Sinko-Streifer Model and Inverse Problems Size Structure and Mosquitofish Populations Appendix A: An Introduction to Fourier Techniques Fourier Series Fourier Transforms Appendix B: Review of Vector Calculus References appear at the end of each chapter.

Wave modulation and breakdown

Abstract: Time series of amplitude, frequency, wavenumber and phase speed are measured in an unstable deep-water wavetrain using a Hilbert-transform technique. The modulation variables evolve from sinusoidal perturbations that are well described as slowly varying Stokes waves, through increasingly asymmetric modulations that finally result in very rapid jumps or ‘phase reversals’. These anomalies appear to correspond to the ‘crest pairing’ described by Ramamonjiarisoa & Mollo-Christensen (1979). The measurements offer a novel local description of the instability of deep-water waves which contrasts markedly with the description afforded by conventional Fourier decomposition. The measurements display very large local modulations in the phase speed, modulations that may not be anticipated from measurements of the phase speeds of individual Fourier components travelling (to leading order) at the linear phase speed (Lake & Yuen 1978).

The curvature perturbation in a box

Abstract: The stochastic properties of cosmological perturbations are best defined through the Fourier expansion in a finite box. I discuss the reasons for that with reference to the curvature perturbation, and explore some issues arising from it.