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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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28 Feb 1991
TL;DR: In this article, the authors define the Walsh functions on the interval [0, 1] and the Walsh Fourier series as a generalization of the Fourier-Stieltjes series.
Abstract: 1 Walsh Functions and Their Generalizations.- 1.1 The Walsh functions on the interval [0, 1).- 1.2 The Walsh system on the group.- 1.3 Other definitions of the Walsh system. Its connection with the Haar system.- 1.4 Walsh series. The Dirichlet kernel.- 1.5 Multiplicative systems and their continual analogues.- 2 Walsh-Fourier Series Basic Properties.- 2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums.- 2.2 The Lebesgue constants.- 2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series.- 2.4 Other tests for uniform convergence.- 2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point.- 2.6 The Walsh system as a complete, closed system.- 2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series.- 2.8 Fourier series in multiplicative systems.- 3 General Walsh Series and Fourier-Stieltjes Series Questions on Uniqueness of Representation of Functions by Walsh Series.- 3.1 General Walsh series as a generalized Stieltjcs series.- 3.2 Uniqueness theorems for representation of functions by pointwise convergent Walsh series.- 3.3 A localization theorem for general Walsh series.- 3.4 Examples of null series in the Walsh system. The concept of U-sets and M-sets.- 4 Summation of Walsh Series by the Method of Arithmetic Mean.- 4.1 Linear methods of summation. Regularity of the arithmetic means.- 4.2 The kernel for the method of arithmetic means for Walsh- Fourier series.- 4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions.- 4.4 (C, 1) summability of Fourier-Stieltjes series.- 5 Operators in the Theory of Walsh-Fourier Series.- 5.1 Some information from the theory of operators on spaces of measurable functions.- 5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.- 5.3 Partial sums of Walsh-Fourier series as operators.- 5.4 Convergence of Walsh-Fourier series in Lp[0, 1).- 6 Generalized Multiplicative Transforms.- 6.1 Existence and properties of generalized multiplicative transforms.- 6.2 Representation of functions in L1(0, ?) by their multiplicative transforms.- 6.3 Representation of functions in Lp(0, ?), 1 < p ? 2, by their multiplicative transforms.- 7 Walsh Series with Monotone Decreasing Coefficient.- 7.1 Convergence and integrability.- 7.2 Series with quasiconvex coefficients.- 7.3 Fourier series of functions in Lp.- 8 Lacunary Subsystems of the Walsh System.- 8.1 The Rademacher system.- 8.2 Other lacunary subsystems.- 8.3 The Central Limit Theorem for lacunary Walsh series.- 9 Divergent Walsh-Fourier Series Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions.- 9.1 Everywhere divergent Walsh-Fourier series.- 9.2 Almost everywhere convergence of Walsh-Fourier series of L2[0, 1) functions.- 10 Approximations by Walsh and Haar Polynomials.- 10.1 Approximation in uniform norm.- 10.2 Approximation in the Lp norm.- 10.3 Connections between best approximations and integrability conditions.- 10.4 Connections between best approximations and integrability conditions (continued).- 10.5 Best approximations by means of multiplicative and step functions.- 11 Applications of Multiplicative Series and Transforms to Digital Information Processing.- 11.1 Discrete multiplicative transforms.- 11.2 Computation of the discrete multiplicative transform.- 11.3 Applications of discrete multiplicative transforms to information compression.- 11.4 Peculiarities of processing two-dimensional numerical problems with discrete multiplicative transforms.- 11.5 A description of classes of discrete transforms which allow fast algorithms.- 12 Other Applications of Multiplicative Functions and Transforms.- 12.1 Construction of digital filters based on multiplicative transforms.- 12.2 Multiplicative holographic transformations for image processing.- 12.3 Solutions to certain optimization problems.- Appendices.- Appendix 1 Abelian groups.- Appendix 2 Metric spaces. Metric groups.- Appendix 3 Measure spaces.- Appendix 4 Measurable functions. The Lebesgue integral.- Appendix 5 Normed linear spaces. Hilbert spaces.- Commentary.- References.

193 citations

Journal ArticleDOI
TL;DR: Methods for obtaining a parsimonious sinusoidal series representation or model of biological time-series data are described and illustrated, capable of higher resolution than a conventional Fourier series analysis and used to identify nonlinear systems with unknown structure.
Abstract: We describe and illustrate methods for obtaining a parsimonious sinusoidal series representation or model of biological time-series data. The methods are also used to identify nonlinear systems with unknown structure. A key aspect is a rapid search for significant terms to include in the model for the system or the time-series. For example, the methods use fast and robust orthogonal searches for significant frequencies in the time-series, and differ from conventional Fourier series analysis in several important respects. In particular, the frequencies in our resulting sinusoidal series need not be commensurate, nor integral multiples of the fundamental frequency corresponding to the record length. Freed of these restrictions, the methods produce a more economical sinusoidal series representation (than a Fourier series), finding the most significant frequencies first, and automatically determine model order. The methods are also capable of higher resolution than a conventional Fourier series analysis. In addition, the methods can cope with unequally-spaced or missing data, and are applicable to time-series corrupted by noise. Fially, we compare one of our methods with a wellknown technique for resolving sinusoidal signals in noise using published data for the test time-series.

193 citations

Journal ArticleDOI
TL;DR: In this article, a unified formulation is used to compare 40 theories for multilayered composites and sandwich plates which are loaded by transverse pressure with various inplane distributions (harmonic, constant, triangular and tent-like).

192 citations

Journal ArticleDOI
TL;DR: In this paper, the influence of echoes on the sensitivity of Fourier transform spectroscopy is considered, and it is concluded that the achievable gain does not normally outweigh the complications arising from retention or enhancement of echoes.
Abstract: In NMR Fourier transform spectroscopy, rf pulses are periodically applied to a spin system to generate a sequence of free induction decays which form the basis of spectroscopic measurements. In inhomogeneous magnetic fields, distortions of the free induction decay arise from superimposed spin‐echo effects. They are strongly dependent on molecular diffusion which is also responsible for the nonexponential envelope of the echo train observed in a stopped pulse experiment. These effects are investigated by means of two independent methods, the Fourier expansion method and the partition method. The results are discussed and used to explain measurements made on a one‐spin system. The influence of echoes on the sensitivity of Fourier transform spectroscopy is considered, and it is concluded that the achievable gain does not normally outweigh the complications arising from retention or enhancement of echoes. Experimental parameters are derived for efficient suppression of echoes by means of a periodically pulsed...

192 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023270
2022702
2021511
2020510
2019589
2018580