The Fourier series of Gegenbauer's function.
Fourier series of Gegenbauer function, examining convergence characteristics
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The fourier series of gegenbauer's function
Theoretical analysis of Fourier series of Gegenbauer function - methods for integration of Gegenbauer function and Fourier coefficients
FOURIER SERIES↗
Exponential Approximations Using Fourier Series Partial Sums
The problem of accurately reconstructing a piece-wise smooth, 2(pi)-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are approximated to within O(N(sup -M-2), and the associated jump of the k(sup th) derivative of f is approximated to within O(N(sup -M-l+k), as N approaches infinity, and the method is robust. These estimates are then used with a class of singular basis functions, which have certain 'built-in' singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M approaches infinity. The technique is illustrated with several examples.
Banerjee, Nana S.↗
Rational trigonometric approximations using Fourier series partial sums
A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.
Geer, James F.↗
Simplification of multiple Fourier series - An example of algorithmic approach
This paper describes one example of multiple Fourier series which originate from a problem of spectral analysis of time series data. The example is exercised here with an algorithmic approach which can be generalized for other series manipulation on a computer. The generalized approach is presently pursued towards applications to a variety of multiple series and towards a general purpose algorithm for computer algebra implementation.
Ng, E. W.↗
Products of multiple Fourier series with application to the multiblade transformation
A relatively simple and systematic method for forming the products of multiple Fourier series using tensor like operations is demonstrated. This symbolic multiplication can be performed for any arbitrary number of series, and the coefficients of a set of linear differential equations with periodic coefficients from a rotating coordinate system to a nonrotating system is also demonstrated. It is shown that using Fourier operations to perform this transformation make it easily understood, simple to apply, and generally applicable.
Kunz, D. L.↗
Fourier Series Operating Package
This report presents a computer program for multiplying, adding, differentiating, integrating, "barring" and scalarly multiplying "literal" Fourier series as such, and for extracting the coefficients of specified terms.
Charnow, Milton L.↗
The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations
A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.
Hannah, S. R.↗
Equivalence of Green's function and the Fourier series representation of composites with periodic microstructure
A computer program which is being developed to analyze the heterogeneous stress and strain history variation at the 'damage critical' locations of a composite structure operating at elevated temperatures is described. The theoretical foundations behind this program are described. The relationship between Fourier series and Green's function approaches is elucidated.
Walker, Kevin P.↗
Locating the Discontinuities of a Bounded Function by the Partial Sums of its Fourier Series I: Periodical Case
A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on identities which determine the jumps of a 2(pi)-periodic bounded not-too-highly oscillating function by the partial sums of its differentiated Fourier series. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function. We study the accuracy of approximation and establish asymptotic expansions for the approximations of a 27(pi)-periodic piecewise smooth function with one discontinuity. By an appropriate linear combination, obtained via derivatives of different order, we significantly improve the accuracy. Next, we use Richardson's extrapolation method to enhance the accuracy even more. For a function with multiple discontinuities we establish simple formulae which "eliminate" all discontinuities of the function but one. Then we treat the function as if it had one singularity following the method described above.
Kvernadze, George↗
Comparison of various methods for mathematical analysis of the Foucault knife edge test pattern to determine optical imperfections
The linearized integral equation for the Foucault test of a solid mirror was solved by various methods: power series, Fourier series, collocation, iteration, and inversion integral. The case of the Cassegrain mirror was solved by a particular power series method, collocation, and inversion integral. The inversion integral method appears to be the best overall method for both the solid and Cassegrain mirrors. Certain particular types of power series and Fourier series are satisfactory for the Cassegrain mirror. Numerical integration of the nonlinear equation for selected surface imperfections showed that results start to deviate from those given by the linearized equation at a surface deviation of about 3 percent of the wavelength of light. Several possible procedures for calibrating and scaling the input data for the integral equation are described.
Gatewood, B. E.↗
A new approach to the solution of large, full matrix equations: A two-dimensional potential flow feasibility study
An approach to the solution of matrix problems resulting from integral equations of mathematical physics is presented. Based on the inherent smoothness in such equations, the problem is reformulated using a set of orthogonal basis vectors, leading to an equivalent coefficient problem which can be of lower order without significantly impairing the accuracy of the solution. This approach was evaluated using a two-dimensional Neumann problem describing the inviscid, incompressible flow over an airfoil. Two different kinds of mode functions were investigated, namely eigenfunction series and Fourier series. The method using Fourier series was found preferable. It uses all of the coefficients from a Fast Fourier Transform algorithm in an approximate method which exploits the known structure of the transformed coefficient matrix and very promising results for the flow over a realistic airfoil are obtained. On the basis of the results presented here, an order of magnitude reduction in this computer time can be expected for such problems as compared with the time for a direct matrix solution.
James, R. M.↗
Steady-state phase error for a phase-locked loop subjected to periodic Doppler inputs
The performance of a carrier phase locked loop (PLL) driven by a periodic Doppler input is studied. By expanding the Doppler input into a Fourier series and applying the linearized PLL approximations, it is easy to show that, for periodic frequency disturbances, the resulting steady state phase error is also periodic. Compared to the method of expanding frequency excursion into a power series, the Fourier expansion method can be used to predict the maximum phase error excursion for a periodic Doppler input. For systems with a large Doppler rate fluctuation, such as an optical transponder aboard an Earth orbiting spacecraft, the method can be applied to test whether a lower order tracking loop can provide satisfactory tracking and thereby save the effect of a higher order loop design.
Chen, C.-C.↗
A spectral boundary integral equation method for the 2-D Helmholtz equation
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.
Hu, Fang Q.↗
Sigma Filter
The GLA Fourth-Order model is needed to smooth the topography. This is to remove the Gibbs phenomenon. The Gibbs phenomenon occurs whenever we truncate a Fourier Series. The Sigma factors were introduced to reduce the Gibbs phenomenon. It is found that the smooth Fourier series is nothing but the original Fourier series with its coefficients multiplied by corresponding sigma factors. This operator can be applied many times to obtain high order sigma filtered field and is easily applicable using FFT. It is found that this filter is beneficial in deriving the topography.
Balgovind, R. C.↗
Comparison of circular orbit and Fourier power series ephemeris representations for backup use by the upper atmosphere research satellite onboard computer
The Upper Atmosphere Research Satellite (UARS) is a three-axis stabilized Earth-pointing spacecraft in a low-Earth orbit. The UARS onboard computer (OBC) uses a Fourier Power Series (FPS) ephemeris representation that includes 42 position and 42 velocity coefficients per axis, with position residuals at 10-minute intervals. New coefficients and 32 hours of residuals are uploaded daily. This study evaluated two backup methods that permit the OBC to compute an approximate spacecraft ephemeris in the event that new ephemeris data cannot be uplinked for several days: (1) extending the use of the FPS coefficients previously uplinked, and (2) switching to a simple circular orbit approximation designed and tested (but not implemented) for LANDSAT-D. The FPS method provides greater accuracy during the backup period and does not require additional ground operational procedures for generating and uplinking an additional ephemeris table. The tradeoff is that the high accuracy of the FPS will be degraded slightly by adopting the longer fit period necessary to obtain backup accuracy for an extended period of time. The results for UARS show that extended use of the FPS is superior to the circular orbit approximation for short-term ephemeris backup.
Kast, J. R.↗
Computer assisted expansion of 1/Delta series with Newton iteration
The Newton iteration method has been widely applied to the solution of various equations such as Kepler's equation. In this study it is used in planetary and satellite theory as a general procedure for Fourier series inversion. The method is used for the construction of the 1/Delta series either in literal form or in numerical form with small eccentricities and inclinations substituted in advance. This usually results in very compact series. With the Newton iteration procedure and a computerized series manipulation technique, the Fourier series of 1/Delta of the mutual perturbations among most natural satellites can be easily constructed.
Chao, C. C.↗
Optimized laser turrets for minimum phase distortion
An analysis and computer program which optimizes laser turret geometry to obtain minimum phase distortion is described. Phase distortion due to compressible, inviscid flow over small perturbation laser turrets in subsonic or supersonic flow is calculated. The turret shape is determined by a two dimensional Fourier series; in a similar manner, the flow properties are given by a Fourier series. Phase distortion is calcualted for propagation at serveral combinations of elevation and azimuth angles. A sum is formed from the set of values, and this sum becomes the objective function for an optimization computer program. The shape of the turret is varied to provide minimum phase distortion.
Vanderplaats, G. N.↗