Showing papers on "Fourier series published in 1977"

Numerical analysis of spectral methods : theory and applications

Abstract: Spectral Methods Survey of Approximation Theory Review of Convergence Theory Algebraic Stability Spectral Methods Using Fourier Series Applications of Algebraic Stability Analysis Constant Coefficient Hyperbolic Equations Time Differencing Efficient Implementation of Spectral Methods Numerical Results for Hyperbolic Problems Advection-Diffusion Equation Models of Incompressible Fluid Dynamics Miscellaneous Applications of Spectral Methods Survey of Spectral Methods and Applications Properties of Chebyshev and Legendre Polynomial Expansions.

Digital Filters

Abstract: sequency as a generalized frequency is introduced, and the frequency is used as a parameter to distinguish individual functions that belong to sets of nonsinusoidal functions. The sixth chapter is devoted to the study of the Walsh-Hadamard transform (WHT) and algorithms to compute it. The concept of the Walsh spectra and their properties are presented with physical significance. Special attention is given to the analogy between the Walsh-Hadamard and the discrete Fourier transforms. In Chapter 7, a study is made of the generalized Haar, Slant, and discrete cosine transforms. Fast algorithms to compute these transforms

Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

Abstract: A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman [3], and in this case the answer is known (see [l I]). Interest in Problem B was recently signalled by 1. Segal [6] who studied the special case S {(x, y) yZ xz I} and gave the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic-invariant norm. In this paper we give a complete solution when S is a quadratic surface given by

Fourier Analysis of Time Series: An Introduction

Abstract: Fitting Sinusoids. The Search for Periodicity. Harmonic Analysis. The Fast Fourier Transform. Examples of Harmonic Analysis. Complex Demodulation. The Spectrum. Some Stationary Time Series Theory. Analysis of Multiple Series. Further Topics. References. Indexes.

Fourier coefficients of the resolvent for a Fuchsian group.

Abstract: *~T + ~T~2~)~2ifc.y-r— acting on a Hubert space §k of automorphic forms dx dy) dx of weight k e IR. In this paper, we present the basic eigenfunction expansions of Gs k(z, z') and discuss applications to conditionally convergent Poincare series and series of Dirichlet type for Fuchsian groups of the first kind, and to the spectral decomposition of §* for groups of the second kind. The outline of the paper is äs follows: in § l we set up the basic eigenfunctions of Dk and the spectral decomposition of f>fc for the trivial group, used in the construction of an automorphic \"prime-form\" and automorphic functions with prescribed automorphic eigenfunctions are introduced in § 2 and are used to give summation methods for the classical kernel functions on a compact Riemann surface, äs well äs for the construction of an automorphic \"prime-form\" and automorphic function with prescribed singularities. The Fourier coefficients of the resolvent at a parabolic cusp are worked out in § 3 and include many special cases of historical interest; the use of the resolvent here explicates certain multiplicative relations of Hecke type and \"expansions of zero\" associated to analytic forms of positive dimension for the modular group. Finally, in § 4 we consider Fourier developments at the hyperbolic fixed-points, including a summation method for the period matrix of a compact Riemann surface; for a Fuchsian group of the second kind with a single free side, the continuous spectral measure for §k is given in terms of a Poisson kernel and Fourier coefficients analogous to the Eisenstein series for horocyclic groups.

Analysis of periodically switched linear circuits

Abstract: A new approach, to the exact analysis of linear circuits containing periodically operated switches is presented. After reformulation of the state equations conventional Fourier analysis is used to determine the response to arbitrary deterministic or stochastic inputs. The analysis is applicable also to improper circuits and circuits causing discontinuities in the state variables at the switching instants. The switches may operate in an arbitrary fashion with a common switching period. Explicit closed form solutions for both the (Fourier coefficients of the) periodically time-varying transfer function \hat{H}(f, t) and the impulse response h(t,\tau) are derived. These results are most suitable for computer aided design. Applications to switched filters and modulators are given.

On the Nonlinear Theory for Gravity Waves on the Ocean's Surface. Part I: Derivations

Abstract: A general hydrodynamic solution is derived for arbitrary gravity-wave fields on the ocean surface by extending Stokes' (1847) original perturbational analysis. The solution to the nonlinear equations of motion is made possible by assuming that the surface height is periodic in both space and time and thus can be described by a Fourier series. The assumption of periodicity does not limit the generality of the result because the series can be made to approach an integral representation by taking arbitrarily large fundamental periods with respect to periods of the dominant ocean waves actually present on the surface. The observation areas and times over which this analysis applies are assumed small, however, compared to the periods required for energy exchange processes; hence an “energy balance” (or steady-state) condition is assumed to exist within the observed space-time intervals. This in turn implies the condition of statistical stationarity of the Fourier height coefficients when one generaliz...

Stability of the Fourier method

Abstract: In this paper we develop a stability theory for the Fourier (or pseudo-spectral) method for linear hyperbolic and parabolic partial differential equations with variable coefficients.

Applications of linear algebra

Abstract: Constructing curves and surfaces through specified points plane geometry equilibrium of rigid bodies graphy theory theory of games the assignment problem Markov chains Leontief economic models forest management computer graphics equilibrium temperature distributions - some applications in genetics age-specific population growth harvesting of animal populations least squares fitting to data a least squares model for human hearing Fourier series cubic spline interpolation cryptography computer tomography linear programming

Fourier Biometrics: Harmonic Amplitudes as Multivariate Shape Descriptors

Abstract: Younker, J. L. and R. Ehrlich (Department of Geology, Indiana-Purdue University, Indianapolis, Indiana 46202) 1977. Fourier biometrics: Harmonic amplitudes as multivariate shape descriptors. Syst. Zool. 26:336-342.-Fourier analysis in closed form provides a highly efficient method for measuring overall morphological similarity and identifying specific types of morphological variation. In shape analysis of ostracode outlines, continuous variation representing phylogenetically induced changes as well as shape families reflecting taxonomic categories can be recognized. Two-dimensional shape is described by a series of terms in a Fourier expansion of periphery radius as a function of polar angle for members of the ostracode genus Rabilimis and for six genera of Hemicytheridae. Raw data, consisting of Cartesian coordinates of the two-dimensional outline are converted to polar coordinates using the center of gravity of the form as a reference point. Coefficients of each term in the expansion represent the contribution of a fundamental harmonic form to the total shape description. When morphological variability is partitioned across the harmonic amplitude spectrum so that individual coefficients represent separate sources of morphological variation, removal of specific harmonics allows comparison of variation resulting from genetic vs. nongenetic factors. [Morphological analysis; Fourier biometrics.] Taxonomic criteria used in the classification of organisms are often the result of artificial dissection of overall form into characters that can be quantitatively or qualitatively expressed. Variation in morphology results from both genetic and nongenetic factors, with the strength of the nongenetic factors determining the ease of recognition of true taxonomic differences between taxa. The use of multivariate techniques such as principal components permits isolation of underlying axes of variation, but clearly the results obtained are only as good as the parameters chosen to represent morphological variation. Thompson (1942) suggested that morphological variation could be described by the use of radial coordinates and polar equations, expressing changes in growth directions and rates of growth by altering the coefficients of polar trigonometric equations. Sneath (1967) attempted to look at variation in overall form by means of trend surface analysis, basically an ex1 Present address: Department of Geological Sciences, Box 4348, University of Illinois-Chicago Circle, Chicago, Illinois 60680. tension of coordinate transformation as proposed by Thompson. The closed-form Fourier shape analysis technique, described by Ehrlich and Weinberg (1970), is a method capable of greatly increasing the efficacy of form analysis. Their initial application involved shape analysis of sand grains as a means for discrimination of distinct populations, and for monitoring gradual changes in grain shape. Applied to biological form, this method permits the study of morphological variation in the manner envisioned by Thompson. Closed-form Fourier shape analysis treats the total two-dimensional cross sectional view, producing a series of orthogonal terms which describe the outline as precisely as required. Each term represents the contribution of a known shape component (fundamental harmonic order) to the overall shape description. Individual terms can be used as multivariate data for taxonomic studies, and the overall shape spectrum provides a continuous variable which can be monitored temporally and spatially. Fourier analysis in Cartesian coordinates has been used in seismic data processing as a means whereby certain

An ILLIAC program for the numerical simulation of homogeneous incompressible turbulence

Abstract: An algorithm and ILLIAC computer program, developed for the simulation of homogeneous incompressible turbulence in the presence of an applied mean strain, are described. The turbulence field is represented spatially by a truncated triple Fourier series (spectral method) and followed in time using a fourth-order Runge-Kutta algorithm. These include: (1) transformation of variables suggested by Taylor's sudden-distortion theory; (2) implicit viscous diffusion by use of an integrating factor; (3) implicit pressure calculation suggested by Taylor's sudden-distortion theory, and (4) inexpensive control of aliasing by random and phased coordinate shifts.

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

Abstract: A theoretical study is presented of the spatial stability of flow in a circular pipe to small but finite axisymmetric disturbances. The disturbance is represented by a Fourier series with respect to time, and the truncated system of equations for the components up to the second-harmonic wave is derived under a rational assumption concerning the magnitudes of the Fourier components. The solution provides a relation between the damping rate and the amplitude of disturbance. Numerical calculations are carried out for Reynolds numbers R between 500 and 4000 and βR [les ] 5000, β being the non-dimensional frequency. The results indicate that the flow is stable to finite disturbances as well as to infinitesimal disturbances for all values of R and βR concerned.

Large deflections of rectangular membranes under uniform pressure

Abstract: The Foppl large deflection equations for laterally loaded membranes are solved for uniform load and for edges which are fixed normal to the edge but are free to move parallel to the edge. Both the deflection function and the Airy stress function are expanded in Fourier series. The resulting coupled non-linear cubic equations for the deflection function coefficients are truncated and solved by means of an iterative procedure. Results for the center normal deflection and the stress resultants at selected points are calculated with the use of 100 or more equations and are found to differ significantly from the previously accepted approximate results.

Scattering by periodic metal surfaces with sinusoidal height profiles--A theoretical approach

Abstract: A theory of scattering by periodic metal surfaces is presented that utilizes the physical optics approximation to determine the current distribution in the metal surface to first order, but modifies this approximate distribution by multiplication with a Fourier series whose fundamental period is that of the surface profile (Floquet's theorem). The coefficients of the Fourier series are determined from the condition that the field radiated by the current distribution into the lower (shielded) half-space must cancel the primary plane wave in this space range. The theory reduces the scatter problem to the familiar task of solving a linear system. For certain basic types of surface profiles, including the sinusoidal profile considered here, the coefficients of the linear system are obtained as closed form expressions in well-known functions (Bessel functions for sinusoidal profiles and exponential functions for piecewise linear profiles). The theory is thus amenable to efficient computer evaluation. Comparison of numerical results based on this theory with data obtained by recent numerical schemes shows that for depths of surface grooves less than a wavelength and for unrestricted groove widths, reliable and comparable, if not more accurate, data is obtained, in many cases at considerably cheaper computational cost.

Exponential Fourier densities and optimal estimation and detection on the circle

Abstract: A new representation, called an exponential Fourier density, of a probability density on a circle, S^{1} is introduced. It is shown that a density of bounded variation on S^{1} can be approximated as closely as desired by such a representation in the space of square-integrable functions on S^{1} . The exponential Fourier densities have the desirable feature of being closed under the operation of taking conditional distributions. Facilitated by the use of these densities, finite-dimensional, recursive, and optimal estimation and detection schemes are derived for some simple models including a PSK communication system. A deficiency of the exponential Fourier densities is that they are not closed under convolution. How to circumvent this deficiency is still an open question.

ON $ (H,k)$-SUMMABILITY OF MULTIPLE TRIGONOMETRIC FOURIER SERIES

Abstract: A theorem is proved from which, in particular, it follows that if on , then the multiple trigonometric Fourier series of and all conjugate series are -summable almost everywhere on for every . In the case where this result was obtained by Marcinkiewicz (Collected papers, PWN, Warsaw, 1964). That it is unimprovable, in a certain sense, follows from a result of Saks (On the strong derivatives of functions of intervals, Fund. Math. 25 (1935), 235-252).Bibliography: 15 titles.

A Fourier series algorithm for the analysis of reflectance data

Abstract: A simple Fourier series algorithm is presented for the determination of the phase, and hence the other optical constants from measurements of the reflectance. The advantages of the proposed procedure and its mathematical equivalence with the Kramers-Kronig relations are discussed.

Spherical Harmonics Solutions of Multi-Dimensional Neutron Transport Equation by Finite Fourier Transformation

Abstract: A method of solution of a monoenergetic neutron transport equation in PL approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.

On the Iterative Solution of a Stochastic Mapping

Abstract: M. Henon has demonstrated [1] that the simple deterministic mapping, xk+1=axk+bx2 k-xk-1, possessing a conservation theorem similar to Liouville’s Theorem, can exhibit the same “stochastic” behavior which, in Hamiltonian dynamics, is considered necessary for establishing an “approach to equiLibrium” or “transport laws”. While this mapping is so simple that it could be studied on a pocket-calculator its analytic solution is not available. This is probably due to the fact that it is so successful in imitating the behavior of a “non- integrable” Hamiltonian system of differential equations.

On the Construction of Poincaré–Lindstedt Solutions: The Nonlinear Oscillator Equation

Abstract: The equation of motion in a potential energy well $Y'' = F(Y)$ is formally integrated by a combined power and Fourier series. A new notation is used to reduce the integration problem to the algebraic problem of the solution of two recursion relations in a finite form. Two general algorithms are obtained from the recursion relations for motions in asymmetrical and symmetrical, parabolic wells. A third algorithm is presented for the periodic solutions of a form of the Emden–Fowler equation. The computer versions of the algorithms for parabolic wells are checked against independent analytical solutions of the equation for time dependent radial motion in the Newtonian two-body problem and the equation of Blasius. The limitations of the solutions to small to moderate amplitudes are found by the analytical-computer solution of the radial part of the orbital and scattering notions in a Lennard–Jones six-twelve potential.