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In the article it is proved that a specific generalized Cauchy-Riemann system with singularity on a circle has only the trivial solution in the class of infinitely differentiable functions. [ABSTRACT FROM AUTHOR]

In this paper the stability of a multifrequency model of a PWM converter is investigated. A multifrequency model is a model based on Fourier series that contains as a special case the so-called state space average model. In contrast to a state space average model a multifrequency model may also include so-called higher order harmonics, where the zeroth order harmonic corresponds to the (moving) average. This paper focuses on a specific PWM converter, namely a Ćuk converter, and it is proved that a multifrequency model of a Ćuk converter with fixed duty ratio is asymptotically stable. This result generalizes the known corresponding result for a state space average model of a Ćuk converter with fixed duty ratio. Taking all the harmonics into account the result also illustrates the well-known fact that a Ćuk converter with a fixed duty ratio and a finite switching frequency is asymptotically stable in the following sense. If the signals in a Ćuk converter do not correspond with a periodic behaviour, they will however do so in the limit, i.e. as time goes to infinity the signals will become periodic, and this limiting periodic behaviour is unique. Although the paper mainly deals with the stability issues for a Ćuk converter, it is possible to use the ideas of the paper to derive similar results for other types of PWM converters. [ABSTRACT FROM AUTHOR]

A full-wave electromagnetic model of a symmetrical star junction of N rectangular waveguides coupled in E-plane is presented. The model is based on a simple trigonometric-series expansion that is constructed for the field in the connecting region using the domain product technique and symmetry properties of the geometry. Before solving the final system of algebraic equations, all mathematical manipulations are carried out analytically without any numerical procedure. The analysis is applied to scattering characteristics of the junction. The validity and efficiency of the solution are verified by computational and comparison tests. The obtained frequency dependencies of S-matrices are presented for N = 3 to N = 7. [ABSTRACT FROM AUTHOR]

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫[sub 0][sup π] (sin x/x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function f(x) is 'very nearly' equal to the convolution integral 1/π ∫[sub -∞][sup +∞] f(x - t)(sin nt/t) dt. [ABSTRACT FROM AUTHOR]

A new approach has been presented to characterise phase spectra for simulating realistic nonstationary characteristics in synthetic accelerograms. The phase characteristics of the recorded earthquake accelerograms have been studied for this purpose and it has been found that the phase curve/unwrapped phases exhibit a monotonic downward trend which allows the problem of phase characterisation to be cast as a constrained nonlinear programming problem. The phase spectrum is first characterised by matching mean and variance of the generated distribution of relative phases with those obtained from recorded motions. As a practical application, it is shown how phase spectra can be characterised for an ensemble of synthetic accelerograms so as to maximise the severity of sample realisations. [ABSTRACT FROM AUTHOR]

This article examines the coupled thermoelasticity of an infinitely long, triple-layer annular cylinder subjected to a constant or time-dependent change in boundary temperatures. The governing equations, taking into account the thermomechanical coupling term, are expressed in terms of temperature increment and displacement. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. The transient distributions of temperature increments and stresses in the real domain are presented numerically. Using a Fourier series technique, the inversion to the real domain is obtained, and no thermoelastic potentials are presented in the solution process. The results indicate that the coupling parameter can lead to a lagging effect in both the temperature and the stress distributions. The present method is also suitable for cases with different boundary conditions, such as time-dependent changes in surrounding temperature. [ABSTRACT FROM AUTHOR]

Presents information on a study which analyzed fourier integrals. Details on the Riemann-Lebesgue Lemma; Analysis of various fourier integrals.

A technique is proposed that permits solving to within 15 or more decimal places, in explicit form, initial value problems for nonlinear ordinary differential equations having oscillatory periodic solutions. The technique is elementary and relies on employing a numerical solver to generate a solution having a high degree of precision, estimating the period of the numerical solution, and then estimating the Fourier series coefficients of the numerical solution. Using the computer algebra system Mathematica, its routine NDSolve is employed with working precision set to 34, to obtain a (truncated) Fourier series solution which, when substituted back into the equation, yields a residual of less than 4 × 10[sup -16] (in absolute value) and agrees within 1.5 × 10[sup -15] with the numerically generated solution over the first period. The technique is quite suitable for discussion in a second semester beginning course with computer laboratory component and can be applied to any equation expected to have a periodic solution. It is considered that this procedure shows the 'correct use' of mathematical technology, blending strong computing and visualization capability with theoretical considerations that permits one to do new and deeper investigations. [ABSTRACT FROM AUTHOR]

The sliding controller is very effective in dealing with system uncertainties defined in compact sets. If the bounds of the uncertainties are not available, the adaptive sliding controller might be designed. One restriction for the adaptive sliding scheme is that the unknown parameter should be constant, which is not always satisfied in practice. For a non-linear system with general uncertainties (i.e. time varying with unknown bounds), both the traditional sliding control and adaptive sliding control do not work properly. This paper proposes a new sliding control scheme for non-linear systems containing time-varying uncertainties with unknown bounds. The uncertainties are assumed to be piecewise continuous functions of time and satisfy the Dirichlet conditions. By representing these uncertainties in finite-term Fourier series, they can be estimated by updating the Fourier coefficients. Since the coefficients are time-invariant, update laws are easily obtained from the Lyapunov approach to guarantee output error convergence. Computer simulations are performed to show efficacy of the proposed schemes. [ABSTRACT FROM AUTHOR]

The well-known physicist A. A. Michelson started quite an interesting correspondence in the journal Nature in 1898. He complained about the convergence of continuous Fourier series approximations to a discontinuous function as being 'utterly at variance with the physicist's notions of quantity'. J. W. Gibbs essentially settled matters in 1899 and this situation has become to be called the Gibbs' phenomenon. It is discussed in many texts but appears to be always focused on the discontinuity of a simple step function. The details for an arbitrary jump discontinuity are illuminating and provide an interesting example to discuss the difference between pointwise convergence and uniform convergence. Two proofs are given that the Gibbs' phenomenon only depends on the size of the jump and is a multiple of the integral ƒ[sup π]0(sin x/x) dx. The demonstration and calculations are suitable for an advanced calculus class and provide very nice applications of Riemann sums and uniform convergence. [ABSTRACT FROM AUTHOR]

The Karplus equation has been investigated by ab initio computation of the spin-spin coupling constants for a series of rotated ethane geometries. The couplings have been calculated at the self-consistent field (SCF) level as well as using the second-order polarization propagator approximation (SOPPA) and the second-order polarization propagator approximation with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)) and have been compared with results of previous calculations. The four principal components of the coupling constants rather than just the Fermi-contact have been calculated, and the common supposition that the Fermi-contact term is totally dominant has been confirmed. The derivatives of the orbital paramagnetic and orbital diamagnetic terms are significant but opposite in sign for the case of this rotation in ethane. It is found that the coefficients in the Karplus equation are largely overestimated at the SCF level, whereas the SOPPA(CCSD) results are in good agreement with coefficients derived from experimental coupling constant data or the results of multiconfigurational self-consistent field (MCSCF) calculations. It is further observed that extending the Fourier series in the Karplus equation to include cos(3θ) and cos(4θ) terms neither significantly improves the quality of the fit nor significantly changes the values of the other coefficients. In order to simulate the Abraham and Pachler equation, calculations varying the nuclear charge on hydrogen have been performed. These will allow an abstract but flexible prediction of the effect of electronegative substituents. [ABSTRACT FROM AUTHOR]

This paper discusses the large signal performance of low-voltage analogue functional elements built around MOS field-effect transistors (MOSFETs). Fourier-series approximations are obtained for the transfer characteristics of the basic building blocks; such as the balanced source-coupled pair, the unbalanced source-coupled pair with different transistor sizes, the source-coupled pair with bias offset and the squaring circuits. Using the Fourier-series approximations of these basic building blocks, closed-form expressions are obtained for the amplitudes of the harmonics and intermodulation products at the output of analogue functional elements; such as the basic differential pairs, half-wave rectifiers, full-wave rectifiers and multipliers excited by a multisinusoidal input signals. Using these expressions comparison between the large signal performance of analogue functional elements can be performed and the parameters required for a predetermined performance can be determined. [ABSTRACT FROM AUTHOR]

The paper investigates the question 'Is the Gibbs' phenomenon evident in Fourier series solutions to the harmonic oscillator equation with periodic piecewise continuous forcing?' In investigating this, an alternative to the Laplace transform technique for solving such equations is promoted which is thought to be simpler (although equivalent). Error analysis, comparing truncated Fourier series solutions with exact solutions obtained with the Laplace transform, shows a trend in the residuals. This trend is accounted for and an improved form for the (truncated) Fourier series solution is obtained which permits global error to be controlled and ease of computation preserved. [ABSTRACT FROM AUTHOR]

Analytic Fourier series admit a uniform exponential truncation bound. Applied to the fast multipole method this leads to pertinent truncation bounds for the multipole potential expansion. [ABSTRACT FROM AUTHOR]

Presents several articles that contain insights on topics taught in the first years of undergraduate mathematics. Collapsed matrices with almost the same eigenstuff; Applications of fourier series in classical guitar technique; Problem for the study of the center of gravity in calculus classes.

Harmonic oscillator equations of the form y + lambda[sup 2]y = h(t) where lambda is a real constant and h(t) is a continuous, piecewise smooth, periodic 'forcing' function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving 'term-by-term' by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated 'term-by-term'. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research. [ABSTRACT FROM AUTHOR]

By means of the Direct Simulation Monte Carlo (DSMC) method, the Boltzmann equation is numerically solved for a gas of hard spheres enclosed between two parallel plates kept at different temperatures and subject to the action of a gravity field normal to the plates. The profiles of pressure, density, temperature and heat flux are seen to be quite sensitive to the value of the gravity acceleration g. If the gravity field and the heat flux are parallel (g > 0), the magnitudes of both the temperature gradient and the heat flux are smaller than in the opposite case (g < 0). When considering the actual heat flux relative to the value predicted by the Fourier law, it is seen that, if g > 0, the ratio increases as the reduced local field strength increases, while the opposite happens if g < 0. The simulation results are compared with theoretical predictions for Maxwell molecules. [ABSTRACT FROM AUTHOR]

The dissociation of a dislocation in the {111} plane of a fcc crystal into two Shockley partials is studied in the framework of the generalized Peierls model. The interplanar atomic misfit energy (the gamma-surface) is represented by a twodimensional Fourier series in which the 'stacking fault energy' gamma g and the maximum stacking energy gamma m can be varied independently. Whereas for Volterra dislocations the separation d0 of the Shockley partials depends only on gamma g , it turns out that in the more general treatment the equilibrium separation d depends also on gamma m. Hence previous experimental determinations of gamma g from TEM observations have to be re-evaluated. The energy EP to recombine the two Shockley partials also depends on the value of gamma m. Agreement with the dissociation energy EV for a Volterra dislocation in screw orientation can only be obtained when the 'recombination radius' is chosen rc 0.15b. [ABSTRACT FROM AUTHOR]

Considers microwave imaging of parallel perfectly conducting cylinders in free space using real-coded genetic algorithm. Shape functions approximated by trigonometric series; Relative error between the measured scattered field and the simulated one; Advantages and disadvantages of the method.

By using a different, but equivalent, representation for the general Fourier-series representation, it is shown that the available computer programs, originally designed for intermodulation analysis of memoryless handpass non-linear amplifiers, are adaptable to wideband amplifiers after very minor modifications. [ABSTRACT FROM AUTHOR]

Analyzes the Fourier series development for a periodical signal relative to the orthonormal trigonometric function set. Differential equation solution; Electric circuit analysis; Models for multipliers and operational amplifiers.

This paper describes a method by which the transfer function of a minimum-phase digital filter can be determined from a knowledge of either the gain function or the phase function. The method makes use of the development of the real and imaginary parts of the transfer function in Fourier series whose coefficients arc determined if either the gain or the phase is specified as a function-of frequency. Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Secrecy coding schemes based on certain pruned linear transforms are suggested. They exploit the structural redundancy in the kernels of the transforms.
Many transform kernels have a large amount of structural redundancy. Fourier, Walsh and Hadamard transforms are well-known examples. There are other transforms with similar properties, for example, Chebyshev (Ahmed and Fisher 1970) and generalized Hadamard Fourier transforms (Andrews and Kane 1970, Andrews and Caspari 1970, Caspari 1970). The kernels of the generalized Hadamard-Fourier transforms are obtained by the Kronecker product of the Fourier and Hadamard kernels. In all these cases the structural redundancy of the transform kernel is such that certain lower-order transform kernels are embedded in the higher-order transform kernels, provided certain elementary relations hold between these orders. The consequence of this structure is that (Moharir 1971, 1973, 1974) the transform coefficients can be pruned to retain only the properly defined subsequence, truncated sequence or truncated subsequence of the transform coefficients. Also the knowledge of the pruned transform domain is equivalent to the knowledge of certain mutually exclusive and exhaustive subsums (in general, weighted) over the signal samples. Such a pruning of the transform domain will be called Fourier-Kronecker pruning because the consequence of the redundant structure of the transform kernel mentioned above is derived either from the fact that the transform kernel is a Kronecker product of the lower-order transform kernels or that the transform kernel, or one of the components of a transform kernel, is a Fourier kernel of composite order.
This paper exploits the property of the transform kernels referred to above, for the transform domain secrecy coding schemes. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to design a low-pass filter by the standard Fourier series method and to investigate analytically the effect of varying the sampling frequency for a given number of taps and cut-off frequency. It is shown that to achieve the sharpest, cut-off one requires a sampling frequency as dose to the Nyquist frequency as is permissible by aliasing and the filtering of higher harmonics of the transfer function. [ABSTRACT FROM AUTHOR]

A Dyadic Green's function appropriate to radiation of sources in a uniaxially anisotropic medium is derived in terms of Fourier integrals. The integrals are then evaluated asymptotically using a method recently developed by Lighthill which has the advantage of showing clearly how the far field depends on the shape of the dispersion surfaces. As an illustration, the electrical field due to an oscillating dipole is worked out. [ABSTRACT FROM AUTHOR]

We study a nonparametric deconvolution density estimation problem. The estimator is obtained by an EM algorithm for a smoothed maximum likelihood estimation problem, which has a unique continuous solution. We present an implementation of the procedure incorporating a data-driven discrepancy principle for selecting the smoothing parameter. Simulations illustrate the good properties of the resulting estimator when the unknown distribution is smooth and has regularly varying thin tails. Comparisons with a Fourier kernel deconvolution method are made for the case of normal noise. We show that under mild smoothness conditions, the estimator based on the data-driven smoothing parameter is strongly consistent. [ABSTRACT FROM AUTHOR]

This article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model. The large-sample properties of parametric goodness-of-fit tests for linearity are considered. The inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression. Large-sample theory is derived for two variants of this type of statistic. The results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n-1/2. Simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives. [ABSTRACT FROM AUTHOR]

Aliasing is a signal-confounding problem that arises when a continuous-time signal is sampled at a rate slower than twice the highest frequency component of a Fourier series representation of the signal. Aliasing can be especially serious for social-science time series applications, since the sampling designs used to construct most social-science data bases are fixed by considerations other than the nature of the underlying continuous-time mechanisms. After collecting sampled data, it is of value to test the observations for the presence of aliasing. It is shown that the nature of the support set of a sampled band-limited stationary signal can be used to motivate an amended version of the Hinich bispectrum test for Gaussianity (Hinich 1982) as a test for aliasing. [ABSTRACT FROM AUTHOR]

We discuss the use of standard logistic regression techniques to estimate hazard rates and survival curves from censored data. These techniques allow the statistician to use parametric regression modeling on censored data in a flexible way that provides both estimates and standard errors. An example is given that demonstrates the increased structure that can be seen in a parametric analysis, as compared with the nonparametric Kaplan-Meier survival curves. In fact, the logistic regression estimates are closely related to Kaplan-Meier curves, and approach the Kaplan-Meier estimate as the number of parameters grows large. [ABSTRACT FROM AUTHOR]

We show that Kronmal and Tarter's well-known role for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings. [ABSTRACT FROM AUTHOR]

Stratified sampling is considered, where (a) the mean integrated squared error (MISE) metric is used in place of the mean squared error (MSE) metric; (b) the entire distribution [i.e., f(x)], rather than a property of the distribution [e.g., E(x)], is used as a target of the procedure; (c) the distribution f(x) is estimated by a truncated series f(x) (to counterbalance model complexity with sample size availability); and finally, (d) samples are taken both with and without replacement. In the last regard, series term inclusion rules are generalized to deal with samples taken without replacement from a finite population. Two special cases are treated in detail. The first shows that even for sample sizes as small as three and for data forms as elementary as the bivariate binary vectors, the use of orthogonal series representation can lead to smaller expected error than would be achievable through use of conventional representation and methods. The second demonstrates that in the continuous case, MISE-based and conventional sample size selection rules can differ substantially. [ABSTRACT FROM AUTHOR]

The problem of estimating the probability density function of a strictly stationary process is considered. To study the effect of a dependence structure on the efficiency of a kernel density estimator, the mean integrated squared error (MISE) of the Fourier integral estimator (FIE) is derived on the assumption that the observed data are generated by a first-order autoregressive process. Numerical results for the normal and Cauchy densities show that even moderate departures from independence can lead to a considerable loss in efficiency of the FIE. In addition to efficiency considerations, the issue of determining an optimal smoothing parameter for the FIE under the autoregressive model is addressed. [ABSTRACT FROM AUTHOR]

Estimation in time series with an unknown number of deterministic trigonometric components with unknown amplitudes and frequencies is considered. A stepwise procedure is used. At each step the frequency of the largest term in the periodogram of the residual series is used as a starting value for finding the best frequency in the least squares sense. The procedure is stopped when there are no further significant harmonic components, When tested by a multiple-test procedure. The fitting procedure is tried on various time series, including the sunspot series. For long-term prediction the deterministic model does better for the sunspot series than, for example, autoregressive models. [ABSTRACT FROM AUTHOR]

In this article, the author gives an algorithm for estimation of the scaling parameter &b.lambda; in the kernel-type density estimate derived from the Fourier integral. In order to show the effect of estimating the scaling parameter, a Monte Carlo study compares the integrated square error (ISE) for the Fourier integral estimate (FIE) with the estimated parameter to the ISE for the FIE with scaling parameter chosen to be optimal for a given sample size and distribution. The corresponding ISE for an estimate using the normal kernel is also computed to show how the FIE compares with a nonnegative kernel estimate. The mean integrated squared error (MISE) is a measure of the global performance of an estimate. The scaling parameter that minimizes the MISE will be called MISE optimal. One hundred samples of size 25, 100, and 250 were generated for the normal, Cauchy, exponential with parameter one, chi-square with four degrees of freedom and chi-square with 10 degrees of freedom. These densities were chosen to give examples of several degrees of smoothness since the MISE for the FIE is known to decrease as the number of derivatives of the density increases.

A Cramer-von Mises-type statistic for testing symmetry, proposed independently by Orlov (1972) and Rothman and Woodroofe (1972), is decomposed into components in the manner of Durbin and Knott (1972); the components are found to be related to Fourier series expansions of the underlying distribution function. Asymptotic power properties of the statistic and its components in testing symmetry about the origin against location shift alternatives when observations have double exponential or normal distributions are described. From these power considerations, it is suggested that the first component is a more useful statistic for the testing problem than the overall Cramer-von Mises-type statistic. The components are shown to be asymptotically equivalent to linear rank statistics. On the basis of computational convenience, a new linear rank statistic for testing symmetry, the analog of the first component, is proposed. [ABSTRACT FROM AUTHOR]

Four new methods for classification of multivariate binary data are presented, based on an orthogonal expansion of the density in terms of discrete Fourier series.. The performance of these methods in 11 populations of various structures was measured in terms of mean error of misclassification and was compared to three well-known methods. Also, performance in density estimation was measured for the appropriate methods. In general, the new methods seem to be superior for classification as well as for density estimation. [ABSTRACT FROM AUTHOR]

Looks at the connection which demonstrates more similarities between power series and Fourier series, and the explanation of some of the differences between the two series. Cooefficients of both series; Ultraspherical polynomials; Introduction of a set of orthogonal polynomials...now called Legendre polynomials; How the connection is given between Fourier and power series.

Discusses an application of the Fourier series to the most significant digit problem. Concept of Benford's logarithmic law of first significant digits; Definition of the Fourier coefficients; Parvesal's formula; Invariance of the log distribution for the first significant digits of products.

Illustrates the Fourier series of a regular polygon in the complex plane. Examples testing the efficiency of the illustration.

Shows that the Fourier series expansion of the function x2k and its derivative lead directly to recursive formulas for zeta(2k) and L(2K-1) that do not require acquaintance with the Bernoulli or Euler numbers.

The standard approach to finding antiderivatives of trigonometric expressions such as sin(ax) cos(bx) is to make use of certain trigonometric identities. The disadvantage of this technique is that it gives no insight into the problem, but relies on students using a memorized formula. This note considers a technique for finding antiderivatives of such trigonometric expressions that is based on the product rule for derivatives and the inverse relation between the derivative and the indefinite integral. [ABSTRACT FROM AUTHOR]

We use a dual convolution to the classical convolution in L¹(R+) to find other expressions to Fourier trigonometric convolutions. These convolutions are used to get trigonometric equalities and formulae with special functions. [ABSTRACT FROM AUTHOR]