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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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TL;DR: In this article, the authors derived a perturbation series for general κ which converges even at the peakon limit, and they also gave three analytical representations for the spatially periodic generalization of peakon, the coshoidal wave.
85 citations
TL;DR: In this paper, the Fourier series method was used to predict the onset of radial ejection in a cubic ion trap with respect to a single-frequency on-resonance excitation.
85 citations
TL;DR: In this paper, a domain decomposition technique for solving vibration problems of uniform and stepped cylindrical shells with arbitrary boundary conditions is presented, where the displacement components of each shell domain are expanded in the form of a double mixed series: Fourier series for the circumferential variable and polynomials/series for the axial variable.
85 citations
TL;DR: The results show that the property of a Boolean function having a concise Fourier representation is locally testable and an “implicit learning” algorithm is given that lets us test any subproperty of Fourier concision.
Abstract: We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\mathbb{F}_2^n$. In both cases we also prove lower bounds showing that any testing algorithm—even an adaptive one—must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Building on these results, we give an “implicit learning” algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].
85 citations
TL;DR: Six mathematical functions to describe the chronobiology of cortisol concentrations were assessed and the single cosine function, which predicts symmetrical cortisol concentrations, was satisfactory in describing the baseline and suppressed cortisol concentrations.
Abstract: Six mathematical functions to describe the chronobiology of cortisol concentrations were assessed. Mean data from a dose-proportionality study of inhaled fluticasone propionate were fitted with an indirect response model using various biorhythmic functions (single cosine, dual ramps, dual zero-order, dual cosines, and Fourier series with 2 and n-harmonics) for production rate. Data with known parameters and random variation were also generated and fitted using the ADAPT II program. Fitted parameters, model estimation criteria, and runs tests were compared. Models with preassigned functions: the dual ramps, the dual zero-order and the dual cosines provide maximum and minimum times for cortisol release rate, were suitable for describing asymmetric circadian patterns and yielding IC50values. Fourier analysis differs from the other methods in that it uses the placebo data to recover equations for cortisol secretion rate rather than by postulation. Nonlinear regression for Fourier analysis, instead of the L2-norm method, was useful to characterize the baseline cortisol data but was restricted to a maximum of two harmonics. Apart from the single cosine function, which predicts symmetrical cortisol concentrations, all methods were satisfactory in describing the baseline and suppressed cortisol concentrations. On the other hand, Fourier series with L2-norm produced the best unbiased estimate for baseline patterns. The Fourier method is flexible, accurate, and can be extended to other drug-induced changes in normal periodic rhythms.
84 citations