Showing papers on "Fourier series published in 1982"

An Improved Method for Numerical Inversion of Laplace Transforms

Abstract: An improved procedure for numerical inversion of Laplace transforms is proposed based on accelerating the convergence of the Fourier series obtained from the inversion integral using the trapezoidal rule. When the full complex series is used, at each time-value the epsilon-algorithm computes a .(trigonometric) Pade approximation which gives better results than existing acceleration methods. The quotient-difference algorithm is used to compute the coefficients of the corresponding continued fraction, which is evaluated at each time-value, greatly improving efficiency. The convergence of the continued fraction can in turn be accelerated, leading to a further improvement in accuracy.

Forward modeling by a Fourier method

Abstract: A Fourier or pseudospectral forward-modeling algorithm for solving the two-dimensional acoustic wave equation is presented. The method utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform. time derivatives which appear in the wave equation are calculated by second-order differcncing. The scheme requires fewer grid points than finite-diffcrcnce methods to achieve the same accuracy. It is therefore believed that the Fourier method will prove more efficient than finitedifference methods. especially when dealing with threedimensional models. The Fourier forward-modeling method was tested against two problems, a single-layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling. The numerical results agreed with both the analytic and physical model results. Furthermore, the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account.

Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis

Abstract: Methods for the determination of crystallite size and lattice strain from X-ray diffraction line broadening are discussed. The subsequent steps of measurement, data correction and evaluation are elucidated; alternatives are indicated. Emphasis is laid on the rigorous analysis of line profiles in terms of Fourier coefficients. For the analysis in terms of integral breadth and full width at half maximum a powerful method exists which adopts a Voigt function for describing the shape of the profiles. Size broadening, strain broadening and single-line methods are commented. A practical example is given of the influence of a non-ideal standard line profile and of different background estimates when a Fourier deconvolution and a Warren-Averbach size-strain analysis are performed. It is expected that line profile analysis will become an automated routine-like analytical method soon, since the tools are available: non-expensive computers, error calculations and commercially available software.

Random Fourier Series with Applications to Harmonic Analysis. (AM-101), Volume 101

Abstract: In this book the authors give the first necessary and sufficient conditions for the uniform convergence a.s. of random Fourier series on locally compact Abelian groups and on compact non-Abelian groups. They also obtain many related results. For example, whenever a random Fourier series converges uniformly a.s. it also satisfies the central limit theorem. The methods developed are used to study some questions in harmonic analysis that are not intrinsically random. For example, a new characterization of Sidon sets is derived. The major results depend heavily on the Dudley-Fernique necessary and sufficient condition for the continuity of stationary Gaussian processes and on recent work on sums of independent Banach space valued random variables. It is noteworthy that the proofs for the Abelian case immediately extend to the non-Abelian case once the proper definition of random Fourier series is made. In doing this the authors obtain new results on sums of independent random matrices with elements in a Banach space. The final chapter of the book suggests several directions for further research.

Fast polarization modulated ellipsometer using a microprocessor system for digital Fourier analysis

Abstract: A new type of spectroscopic automatic ellipsometer using a piezobirefringent element for polarization modulation at 50 kHz is described. Instead of lock‐in amplifiers the data‐acquisition system consists of a 12.8‐MHz digital sampling of the detected signal with a high word rate 8‐bit ADC, followed by on line Fourier transformation of the accumulated data with a short instruction cycle (∼200 ns) microprocessor, driven by a commercial microcomputer. The absolute minimum time required for measuring one set of Fourier coefficients is thus reduced to the modulation period of 20 μs. For digital error reduction purposes and signal‐to‐noise ratio improvement a basic 5 ms sequence of 256 accumulated periods per point is chosen. At this data‐acquisition rate a precision of 5×10−4 is obtained. Further accumulation over 10 s leads to 10−5 precision capability. A detailed analysis of various sources of inaccuracy leads to an estimate of 0.5° maximum systematic error on the ellipsometric angles and ψ and Δ. Applicatio...

A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions

Abstract: A numerical method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface and variable bed topography. It is applied to the unsteady motion of non-breaking water waves of arbitrary magnitude over a horizontal bed. All horizontal variation is approximated by truncated Fourier series. This and finite-difference representation of the time variation are the only necessary approximations. Although the method loses accuracy if the waves become sharp-crested at any stage, when applied to non-breaking waves the method is capable of high accuracy.The interaction of one solitary wave overtaking another was studied using the Fourier method. Results support experimental evidence for the applicability of the Korteweg-de Vries equation to this problem since the waves during interaction are long and low. However, some deviations from the theoretical predictions were observed - the overtaking high wave grew significantly at the expense of the low wave, and the predicted phase shift was found to be only roughly described by theory. A mechanism is suggested for all such solitary-wave interactions during which the high and fast rear wave passes fluid forward to the front wave, exchanging identities while the two waves have only partly coalesced; this explains the observed forward phase shift of the high wave.For solitary waves travelling in opposite directions, the interaction is quite different in that the amplitude of motion during interaction is large. A number of such interactions were studied using the Fourier method, and the waves after interaction were also found to be significantly modified - they were not steady waves of translation. There was a change of wave height and propagation speed, shown by the present results to be proportional to the cube of the initial wave height but not contained in third-order theoretical results. When the interaction is interpreted as a solitary wave being reflected by a wall, third-order theory is shown to provide excellent results for the maximum run-up at the wall, but to be in error in the phase change of the wave after reflection. In fact, it is shown that the spatial phase change depends strongly on the place at which it is measured because the reflected wave travels with a different speed. In view of this, it is suggested that the apparent time phase shift at the wall is the least-ambiguous measure of the change.

A historical perspective of spectrum estimation

Abstract: The prehistory of spectral estimation has its roots in ancient times with the development of the calendar and the clock. The work of Pythagoras in 600 B.C. on the laws of musical harmony found mathematical expression in the eighteenth century in terms of the wave equation. The struggle to understand the solution of the wave equation was finally resolved by Jean Baptiste Joseph de Fourier in 1807 with his introduction of the Fourier series. The Fourier theory was extended to the case of arbitrary orthogonal functions by Sturm and Liouville in 1836. The Sturm-Liouville theory led to the greatest empirical success of spectral analysis yet obtained, namely the formulation of quantum mechanics as given by Heisenberg and Schrodinger in 1925 and 1926. In 1929 John von Neumann put the spectral theory of the atom on a firm mathematical foundation in his spectral representation theorm in Hilbert space. Meanwhile, Wiener developed the mathematical theory of Brownian movement in 1923, and in 1930 he introduced generalized harmonic analysis, that is, the spectral representation of a stationary random process. The common ground of the spectral representations of von Neumann and Wiener is the Hilbert space; the von Neumann result is for a Hermitian operator, whereas the Wiener result is for a unitary operator. Thus these two spectral representations are related by the Cayley-Mobius transformation. In 1942 Wiener applied his methods to problems of prediction and filtering, and his work was interpreted and extended by Norman Levinson. Wiener in his empirical work put more emphasis on the autocorrelation function than on the power spectrum. The modern history of spectral estimation begins with the breakthrough of J. W. Tukey in 1949, which is the statistical counterpart of the breakthrough of Fourier 142 years earlier. This result made possible an active development of empirical spectral analysis by research workers in all scientific disciplines. However, spectral analysis was computationally expensive. A major computational breakthrough occurred with the publication in 1965 of the fast Fourier transform algorithm by J. S. Cooley and J. W. Tukey. The Cooley-Tukey method made it practical to do signal processing on waveforms in either the time or the frequency domain, something never practical with continuous systems. The Fourier transform became not just a theoretical description, but a tool. With the development of the fast Fourier transform the field of empirical spectral analysis grew from obscurity to importance, and is now a major discipline. Further important contributions were the introduction of maximum entropy spectral analysis by John Burg in 1967, the development of spectral windows by Emmanuel Parzen and others starting in the 1950's, the statistical work of Maurice Priestley and his school, hypothesis testing in time series analysis by Peter Whittle starting in 1951, the Box-Jenkins approach by George Box and G. M. Jenkins in 1970, and autoregressive spectral estimation and order-determining criteria by E. Parzen and H. Akaike starting in the 1960's. To these statistical contributions must be added the equally important engineering contributions to empirical spectrum analysis, which are not treated at all in this paper, but form the subject matter of the other papers in this special issue.

A Complete Set of Fourier Descriptors for Two-Dimensional Shapes

Abstract: A set of Fourier descriptors for two-dimensional shapes is defined which is complete in the sense that two objects have the same shape if and only if they have the same set of Fourier descriptors. It also is shown that the moduli of the Fourier coefficients of the parameterizing function of the boundary of an object do not contain enough information to characterize the shape of an object. Further a relationship is established between rotational symmetries of an object and the set of integers for which the corresponding Fourier coefficients of the parameterizing function are nonzero.

Numerical evaluation of the dielectric polarization distribution from thermal‐pulse data

Abstract: A method for numerically carrying out the Fourier analysis for the thermal‐pulse experiment is given. It is shown that it is possible to obtain the polarization distribution across the thickness of a thin film (25 μm) to within the limits set by the experimental data. For such films, resolution of the distribution to within 0.1 of the film thickness is possible. Results are given for the experiment by using a charge measurement rather than a voltage measurement. The effect of a finite‐width pulse is shown to cut off the Fourier coefficients in such a way as to smooth any distribution. Pulsing the sample alternately on both sides is shown to greatly increase the resolution of the experiment. Results for a PVF2 film and a P(VF2‐TFE) copolymer film show that interesting details can be found by the experiment.

Modal coupling in the vibration of fluid‐loaded cylindrical shells

Abstract: An approach is presented to evaluate the pressure field and vibratory response of a finite fluid‐loaded cylindrical shell with infinite rigid extensions which are connected to the shell. The approach combines a generalized Fourier series or in vacuo eigenfunction expansion of the velocity field of the shell with a Green’s function and integral equation representation of the acoustic loading on the shell. Although modes with different circumferential wavenumbers are decoupled, all modes with identical circumferential wavenumbers are coupled via the fluid. The algebraic equations which account for this coupling include both shell impedances and acoustic impedances which are mode dependent. General integral expressions are presented for the acoustic impedances which include both self radiation and interaction impedances. In order to illustrate the general characteristics of these acoustic impedances, the impedances are investigated for a particular set of eigenfunctions. Low‐frequency and asymptotic expressi...

Biorthogonal expansions of functions in series of exponents on intervals of the real axis

Abstract: CONTENTSIntroductionChapter I. Non-harmonic Fourier series (behaviour on the initial interval) ??1.1. Minimal systems of exponentials ??1.2. Expansions of functions in , ??1.3. Expansions of functions in Comments and supplements Chapter II. Non-harmonic Fourier series (the behaviour on the real axis) ??2.1. Extension of convergence of quasi-polynomials ??2.2. Continuation of functions from the initial interval ??2.3. Convergence and summability of non-harmonic Fourier series in the -norm () on every segmentComments and supplements Chapter III. Properties of the system ??3.1. Basis properties ??3.2. Angles between subspaces of exponentials Comments and supplements References

Applications of fourier series technique to transient heat transfer problem

Abstract: For transient heat transfer problem with uniform initial temperature, the Laplace transformation method is considerably powerful. However, it is very difficult and complicated to solve the inverse transform of the subsidiary equation of the given differential equation. The technique of Fourier series can be used in order to obtain the inverse transform. The analysis of transient heat transfer problem of two-dimensional and one-dimensional straight fins is considered here to testify the merit of this method.

Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem

Abstract: On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group SL(3,ℝ), which is automorphic with respect to the discrete group SL(3,ℤ) The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions σs(n)=∑d¦n,d>od5 make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as Χ → ∞ of the sum ∑n⩽Χτ3(n)τ3(n+κ), where τ3(n) is the number of solutions of the equation d1d2d3=n in natural numbers Part II of the present work will be devoted to this binary problem At the end of the paper properties of special functions used in Theorem 1 are discussed

The Carleson-Hunt theorem on fourier series

Abstract: Fourier SeriesSystems, Approximation, Singular Integral Operators, and Related TopicsTopics in Analysis and Its ApplicationsAmerican journal of mathematicsModern Fourier AnalysisTransactions of the American Mathematical SocietyThe Abel PrizeThe Carleson-Hunt Theorem on Fourier SeriesClassical and Modern Fourier AnalysisAn Introduction to Non-Harmonic Fourier Series, Revised Edition, 93A Course in Functional AnalysisModern Fourier AnalysisHarmonic AnalysisCommutative Harmonic Analysis ICarleson Curves, Muckenhoupt Weights, and Toeplitz OperatorsPointwise Convergence of Fourier SeriesBrownian MotionDifferentiation of Integrals in RnDirichlet SeriesExplorations in Harmonic AnalysisChaos in Classical and Quantum MechanicsMartingales in Banach SpacesFourier AnalysisTrigonometric SeriesClassical Fourier AnalysisThe Geometry of Fractal SetsPerspectives in AnalysisCommutative Harmonic Analysis IVFourier Analysis on Local Fields. (MN-15)Fourier Restriction, Decoupling and ApplicationsWave Packet AnalysisMeasure and IntegralRevue Roumaine de Mathématiques Pures Et AppliquéesFractals in Probability and AnalysisA Panorama of Harmonic AnalysisInterpolation of OperatorsA Course in Abstract Harmonic AnalysisThe Carleson-Hunt theorem on Fourier seriesOrlicz Spaces and Generalized Orlicz SpacesDiophantine Approximation and Dirichlet Series

Envelopes of narrow-band signals

Abstract: In some cases, a rapidly oscillating transient term appears in the expression for the envelope of a narrow-band signal. Such terms depend upon the particular definition used to define the envelope. Although these terms are usually so small that they can be neglected, they are of theoretical interest. Here we discuss three commonly used envelope definitions, and derive expressions for the difference between the corresponding envelopes.

Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg–de Vries equation

Abstract: It has been shown by Novikov [Funct. Anal. Appl. 8, 236 (1974)], Dubrovin et al. [Russian Math. Surveys 31, 59 (1976)], Lax [Commun. Pure Appl. Math. 28, 141 (1975)], McKean and van Moerbeke [Inv. Math. 30, 217 (1975)], and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions (’’polycnoidal waves’’) which can be expressed in terms of multidimensional Riemann theta functions. Here, it is shown that via Poisson summation, the Fourier series that define the theta functions can be transformed into an infinite series of Gaussian functions. Because the lowest terms of the Gaussian series generate the usual solitary waves, it is possible to intimately explore the relationship between solitary waves and these spatially periodic ’’polycnoidal’’ waves. Also, by using the Gaussian series, one can perturbatively calculate phase velocities and wave structure for the ’’polycnoidal’’ wave even in the strongly nonlinear regime for which the soliton (or mult...

Interpretation of multigated Fourier functional images.

Abstract: First-harmonic Fourier analysis is currently used to aid in the interpretation of multigated cardiac studies. Its intrinsic inaccuracies are not generally appreciated. This study investigates the characteristics and magnitudes of the errors of this technique. The study analyzes computer-generated phantoms that isolate the various motions of the ventricles (contraction, translation, and rotation) with the first-harmonic approach. The first-harmonic output is compared with a more accurate fitting scheme using multiple terms of the Fourier expansion. Significant artifacts of the inaccuracy of the first harmonic appear in the phantom studies and are observed in patient examples. We conclude that caution is needed in interpreting first-harmonic phase and amplitude images, and particularly in associating them with parameters like the onset of contraction and the stroke volume.

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Abstract: Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ℝ2 can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f1(y), f2y)], -1≦y≦1, f(y)e C1[-1, 1], f″e L2[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f1(±1)=0 and, in any case, to [0, f2(±1)-f1(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f1(±1) ≠ 0.

Scattering from wires and open circular cylinders of finite length using entire domain Galerkin expansions

Abstract: Electromagnetic (EM) scattering by finite-length, perfectly conducting open cylinders (i.e., ducts or tubes) with circular cross sections is considered. The case of a straight wire, viewed as a thin cylinder, is examined in this context. The salient features of this study are a) use of the electric field integral equation (EFIE) as a starting point, b) solution of this equation by the Galerkin method, and c) representation of the axial variation of the currents on the scatterer by an entire domain (Fourier series) expansion. Edge modes are considered in the expansion set and their effect is examined. The open cylinder backscatter cross section is computed as a function of aspect angle for various radii and lengths and is compared with measured data.

Prelude to Hopf bifurcation in an epidemic model: analysis of a characteristic equation associated with a nonlinear Volterra integral equation.

Abstract: We discuss a simple deterministic model for the spread, in a closed population, of an infectious disease which confers only temporary immunity The model leads to a nonlinear Volterra integral equation of convolution type We are interested in the bifurcation of periodic solutions from a constant solution (the endemic state) as a certain parameter (the population size) is varied Thus we are led to study a characteristic equation Our main result gives a fairly detailed description (in terms of Fourier coefficients of the kernel) of the traffic of roots across the imaginary axis As a corollary we obtain the following: if the period of immunity is longer than the preceding period of incubation and infectivity, then the endemic state is unstable for large population sizes and at least one periodic solution will originate

Some approaches for location of centroids of quartz grain outlines to increase homology between Fourier amplitude spectra

Abstract: The ability to test for similarities and differences among families of shapes by closed-form Fourier expansion is greatly enhanced by the concept of homology. Underlying this concept is the assumption that each term of a Fourier series, when compared to the same term in another series, represents the “same thing”. A method that ensures homology is one which minimizes the “centering error,” as reflected in the first harmonic term of the Fourier expansion. The problem is to chose a set of edge points derived from a much larger, but variable, number of edge points such that a valid homologous Fourier series can be calculated. Methods are reviewed and criteria given to define a “proper” solution. An algorithm is presented which takes advantage of the fact that minimization of the “error term” can be accomplished by minimizing the distance between the origin of the polar coordinate system in the calculation of the Fourier series and the shape centroid. The use of this algorithm has produced higher quality solutions for quartz grain provenance studies.

A note on the Fourier series model for analysing line transect data.

Abstract: The Fourier series model offers a powerful procedure for the estimation of animal population density from line transect data. The estimate is reliable over a wide range of detection functions. In contrast, analytic confidence intervals yield, at best, 90% confidence for nominal 95% intervals. Three solutions, one using Monte Carlo techniques, another making direct use of replicate lines and the third based on the jackknife method, are discussed and compared.