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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 selection of an optimal parametric angle θ describing a closed magnetic flux surface is considered with regard to accelerating the convergence rate of the Fourier series for the Cartesian coordinates x(θ,φ)≡R−R0 and y(φ, φ) ≥Z−Z0.
Abstract: The selection of an optimal parametric angle θ describing a closed magnetic flux surface is considered with regard to accelerating the convergence rate of the Fourier series for the Cartesian coordinates x(θ,φ)≡R−R0 and y(θ,φ)≡Z−Z0. A system of algebraic equations, which are quadratic in the Fourier amplitudes of x and y, is derived by minimizing the width of the surface power spectrum. The solution of these nonlinear equations, together with the prescribed surface geometry, determines a unique optimal angle. A variational principle is used to solve these constraint equations numerically. Application to the representation of three‐dimensional magnetic flux surfaces is considered.
79 citations
TL;DR: In this article, a modification of the classical Fourier expansio n, whereby in ( 1, 1) the sinnx functions are replaced by sin�(n 1 )x, n � 1.
Abstract: In this paper we consider a modification of the classical Fourier expansio n, whereby in ( 1, 1) the sinnx functions are replaced by sin�(n 1 )x, n � 1. This has a number of important advantages in the approximation of analytic, nonperiodic functions. In particular, expansion coefficients decay like O n 2 � , rather than like O n 1 � . We explore theoretical features of these modified Fourier expansions, prove suitable versions of Fej´
79 citations
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TL;DR: In this article, a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples is proposed, where the edge set of the image is localized to the zero set of a trigonometric polynomial and the Fourier coefficients of the partial derivatives satisfy a linear annihilation relation.
Abstract: We introduce a method to recover a continuous domain representation of a piecewise constant two-dimensional image from few low-pass Fourier samples. Assuming the edge set of the image is localized to the zero set of a trigonometric polynomial, we show the Fourier coefficients of the partial derivatives of the image satisfy a linear annihilation relation. We present necessary and sufficient conditions for unique recovery of the image from finite low-pass Fourier samples using the annihilation relation. We also propose a practical two-stage recovery algorithm which is robust to model-mismatch and noise. In the first stage we estimate a continuous domain representation of the edge set of the image. In the second stage we perform an extrapolation in Fourier domain by a least squares two-dimensional linear prediction, which recovers the exact Fourier coefficients of the underlying image. We demonstrate our algorithm on the super-resolution recovery of MRI phantoms and real MRI data from low-pass Fourier samples, which shows benefits over standard approaches for single-image super-resolution MRI.
79 citations
TL;DR: In this article, a general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ, it is shown that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded.
Abstract: A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2
78 citations
TL;DR: In this article, the variational statement, governing equations and corresponding Ritz approximations are derived in Cartesian, cylindrical and spherical coordinates for the evaluation of the natural frequencies of free vibrations of elastic cylinders and spheres.
78 citations