Fourier series

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

Approximation of infinite-dimensional systems

Abstract: A Fourier series-based method for approximation of stable infinite-dimensional linear time-invariant system models is discussed. The basic idea is to compute the Fourier series coefficients of the associated transfer function T/sub d/(Z) and then take a high-order partial sum. Two results on H/sup infinity / convergence and associated error bounds of the partial sum approximation are established. It is shown that the Fourier coefficients can be replaced by the discrete Fourier transform coefficients while maintaining H/sup infinity / convergence. Thus, a fast Fourier transform algorithm can be used to compute the high-order approximation. This high-order finite-dimensional approximation can then be reduced using balanced truncation or optimal Hankel approximation leading to the final finite-dimensional approximation to the original infinite-dimensional model. This model has been tested on several transfer functions of the time-delay type with promising results. >

Wavelets and other orthogonal systems

Abstract: ORTHOGONAL SERIES General Theory Examples A PRIMER ON TEMPERED DISTRIBUTIONS Intuitive Introduction Test Functions Tempered Distribution Fourier Transforms Periodic Distributions Analytic Representations Sobolev Spaces AN INTRODUCTION TO ORTHOGONAL WAVELET THEORY Multiresolution Analysis Mother Wavelet Reproducing Kernels and a Moment Condition Regularity of Wavelets as a Moment Condition Mallat's Decomposition and Reconstruction Algorithm Filters CONVERGENCE AND SUMMABILITY OF FOURIER SERIES Pointwise Convergence Summability Gibbs' Phenomenon Periodic Distributions WAVELETS AND TEMPERED DISTRIBUTIONS Multiresolution Analysis of Tempered Distributions Wavelets Based on Distributions Distributions with Point Support Approximation with Impulse Trains ORTHOGONAL POLYNOMIALS General Theory Classical Orthogonal Polynomials Problems OTHER ORTHOGONAL SYSTEMS Self-Adjoint Eigenvalue Problems on Finite Intervals Hilber-Schmnidt Integral Operators An Anomaly: The Prolate Spheroidal Functions A Lucky Accident? Rademacher Functions Walsh Function Periodic Wavelets Local Sine or Cosine Base Biorthogonal Wavelets POINTWISE CONVERGENCE OF WAVELET EXPANSIONS Reproducing Kernel Delta Sequences Positive and Quasi-Positive Delta Sequences Local Convergence of Distribution Expansions Convergence Almost Everywhere Rate of Convergence of the Delta Sequence Other Partial Sums of the Wavelet Expansion Gibbs' Phenomenon Positive Scaling Functions A SHANNON SAMPLING THEOREM IN WAVELET SUBSPACES A Riesz Basis of Vm The Ampling Sequence in Vm Examples of Sampling theorems The Sampling Sequence in Tm Shifted Sampling Gibbs' Phenomenon for Sampling Series Irregular Sampling in Wavelet Subspaces EXTENSIONS OF WAVELET SAMPLING THEOREMS Oversampling with Scaling Functions Hybrid Sampling Series Positive Hybrid Sampling The Convergence of the Positive Hybrid Series Cardinal Scaling Functions Interpolating Multiwavelets Orthogonal Finite Element Multiwavelets TRANSLATION AND DILATION INVARIANCE IN ORTHOGONAL SYSTEMS Trigonometric System Orthogonal Polynomials An Example Where Everything Works An Example Where Nothing Works Weak Translation Invariance Dilations and Other Operations ANALYTIC REPRESENTATIONS VIA ORTHOGONAL SERIES Trigonometric Series Hermite Series Legendre Polynomial Series Analytic and Harmonic Wavelets Analytic Solutions to Dilation Equations Analytic Representation of Distributions by Wavelets Wavelets Analytic in the Entire Complex Plane ORTHOGONAL SERIES IN STATISTICS Fourier Series Density Estimators Hermite Series Density Estimators The Histogram as a Wavelet Estimator Smooth Wavelet Estimators of Density Local Convergence Positive Density Estimators Based on Characteristic Functions Positive Estimators Based on Positive Wavelets Density Estimation with Noisy Data Other Estimation with Wavelets Threshold Methods ORTHOGONAL SYSTEMS AND STOCHASTIC PROCESSES K-L Expansions Stationary Processes and Wavelets A Series with Uncorrelated Coefficients Wavelets Based on Band Limited Processes Nonstationary Processes Each chapter also contains a Problems section

Combinatorial Sublinear-Time Fourier Algorithms

Abstract: We study the problem of estimating the best k term Fourier representation for a given frequency sparse signal (i.e., vector) A of length N≫k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of $\hat{\mathbf{A}}$, and estimate their coefficients, in polynomial(k,log N) time. Randomized sublinear-time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem (Gilbert et al. in ACM STOC, pp. 152–161, 2002; Proceedings of SPIE Wavelets XI, 2005). In this paper we develop the first known deterministic sublinear-time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method (Gilbert et al. in Proceedings of SPIE Wavelets XI, 2005). Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in (Iwen in Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA’08), 2008).

Fourier series expansion of the transfer equation in the atmosphere-ocean system

Abstract: We consider radiative transfer in a plane-parallel atmosphere bounded by a rough ocean surface. The problem is solved by using a Fourier series decomposition of the radiation field. For the case of a Lambertian surface as a boundary condition, this decomposition is classically achieved by developing the scattering phase matrix in a series of Legendre functions. For the case of a rough ocean surface, we obtain the decomposition by developing both the Fresnel reflection matrix and the wave facet distribution function in Fourier series. This procedure allows us to derive the radiance field for the case of the ruffled ocean surface, with a computation time only a few percent larger than for the case of a Lambertian surface.