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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: The problem of the maximization of the energy produced by a self reacting point absorber subject to motion restriction is addressed and the optimal control problem is reformulated as a non linear program where the properties of the cost function and of the constraint are affected by the choice of the basis functions.
72 citations
TL;DR: In this paper, it is shown that with polymer fibres this causes serious errors in the normalization, and in the values of those low harmonics used in the size and disorder determination, and prevents reliable values being obtained.
Abstract: Methods which determine the number and disorder of lattice planes in a crystal from the Fourier cosine coefficients of the intensity profile of an X-ray reflection use only the low harmonics and require that the coefficients be normalized so that the zero harmonic is unity. Experimentally, the profiles can only be recorded over a smaller range of scattering angle than required by the theory, and it is necessary to subtract background, which is likely to be estimated with considerable error, before determining the coefficients. It is shown that with polymer fibres this causes serious errors in the normalization, and in the values of those low harmonics used in the size and disorder determination, and prevents reliable values being obtained. Methods which avoid normalization and use only high harmonics are needed. It is shown that disorder may be obtained in such a way, but not size, for which low-order normalized coefficients are essential. A method of extrapolation is described and tested which enables the accurate high harmonics to be used to improve the estimates of the low ones. Whilst this will yield more reliable values of crystal size than are obtainable from existing methods, the accuracy depends entirely on the validity of the extrapolation, which cannot be tested in many cases of interest.
71 citations
TL;DR: The intrinsic ability of the method of spherical cap harmonic analysis to separate external and internal sources allows the calculation of equivalent ionospheric and induced currents that are able to explain variations of the geomagnetic field over a portion of the earth's surface as discussed by the authors.
Abstract: SUMMARY
The intrinsic ability of the method of spherical cap harmonic analysis to separate external and internal sources allows the calculation of equivalent ionospheric and induced currents that are able to explain variations of the geomagnetic field over a portion of the earth's surface. Formulations for current densities and current functions are derived and found to be analogous to those derived for the global case from conventional spherical harmonic analysis. Although spherical cap formulations for current density have been given by another worker, they were incorrect because of an error in defining the equivalent current. An example of the use of current functions is given by modelling variations from hourly mean values recorded at 40 geomagnetic observatories over Europe during a very quiet day in 1978. The modelling can be done spatially for each of the 24 hours separately, or spatially and temporally either by expressing each spatial coefficient as a Fourier series or by smoothing the spatial coefficients obtained from the separate hourly models.
71 citations
TL;DR: A new method of instrument calibration and data analysis is presented for single-etalon interferometric measurements of winds, temperatures, and emission line intensities for the multichannel Fabry-Perot interferometer on the Dynamics Explorer spacecraft.
Abstract: A new method of instrument calibration and data analysis is presented for single-etalon interferometric measurements of winds, temperatures, and emission line intensities. The technique has been developed for the multichannel Fabry-Perot interferometer on the Dynamics Explorer spacecraft. A numerical representation of the instrumental transfer function is used based on a truncated Fourier series with empirically determined coefficients. The numerical form is compared with the conventional analytic form. The Fourier coefficients describing the instrument function are generated at the wavelength of a stable He–Ne laser and are translated to other wavelengths using an interpolation technique for both phase and power. A quasi-linear least-squares fitting process involving matrices provides for a rapid and accurate data reduction.
71 citations
Book•
30 Mar 2000
TL;DR: In this article, the authors present density theorems, linear Chebyshev approximation, degree of approximation, interpolation, Fourier series, Spline Functions, Orthogonal Polynomials and best approximation in normed linear spaces.
Abstract: Density Theorems / Linear Chebyshev Approximation / Degree of Approximation / Interpolation / Fourier Series / Spline Functions / Orthogonal Polynomials / Best Approximation in Normed Linear Spaces / Bibliography / Symbols and Notation / Index.
71 citations