Showing papers on "Fourier series published in 1979"

The Spontaneous Appearance of a Singularity in the Shape of an Evolving Vortex Sheet

Abstract: The evolution of a small amplitude initial disturbance to a straight uniform vortex sheet is described by the Fourier coefficients of the disturbance. An approximation to the exact evolution equation for these coefficients is proposed and it is shown, by an asymptotic analysis valid at large times, that the solution of the approximate equations develops a singularity at a critical time. The critical time is proportional to ln $(\epsilon ^{-1})$, where $\epsilon $ is the initial amplitude of the disturbance and the singularity itself is such that the nth Fourier coefficient decays like $n^{-2.5}$ instead of exponentially. Evidence-not conclusive, however-is present to show that the approximation used is adequate. It is concluded that the class of vortex layer motions correctly modelled by replacing the vortex layer by a vortex sheet is very restricted; the vortex sheet is an inadequate approximation unless it is everywhere undergoing rapid stretching.

Linear Systems, Fourier Transforms and Optics

Abstract: (1979). Linear Systems, Fourier Transforms and Optics. Optica Acta: International Journal of Optics: Vol. 26, No. 7, pp. 836-836.

Fitting of Mathematical Functions to Biomechanical Data

Abstract: A method is presented to decide the order a polynomial or Fourier series best represents a set of biomechanical data points, with regard to the calculation of the first and second derivatives of the data.

Comparison of WKB calculations and experimental results for three-dimensional cochlear models.

Abstract: The WKB asymptotic method is applied to the calculation of cochlear models with square scala cross section, for which the fluid motion is fully three dimensional. The analysis begins with the exact solution for wave propagation in a duct with constant properties. This solution is somewhat tedious but straightforward, since it requires a Fourier series expansion across the duct. Then with the formulation of Whitham [Linear and Nonlinear Waves (Wiley, New York, 1974)], the approximate solution is readily generated for the duct with properties which vary slowly along the length. Numerical calculations are carried out for the experimental models of Cannel [Ph.D. thesis, Univ. of Warwick (1969)] and Helle [Dr.-Ing. disser., Technische Univ., Muchen (1974)] who furnish quantitative details of both "basilar membrane" response and model parameters. Without any free parameters for adjusting, the present WKB solution shows quite satisfactory agreement with the experimental model results. Computer time is reasonable; the calculation of displacement envelope and phase at a number of stations along the cochlea for a given frequency requires only one second of CPU time. Thus the credibility and practically of the approach is established for the investigation of yet more realistic and more elaborate cochlear models.

Maximum Likelihood Estimation of Fourier Coefficients to Describe Seasonal Variations of Parameters in Stochastic Daily Precipitation Models

Abstract: Fourier series are convenient expressions for the seasonally fluctuating values of parameters in stochastic models of precipitation. Least-squares methods are often used to estimate the Fourier series coefficients, but this method has two important disadvantages. First the “data” points are in fact estimates of parameters, and because of varying sample size, they may have unequal variances and should not be given equal weight. Second, there is no statistically sound procedure to test the significance of individual harmonics. In this paper we investigate methods to obtain maximum likelihood estimates of the Fourier coefficients to describe the seasonal variability in the parameters for a stochastic rainfall model. Parameters are obtained from a two-state Markov chain model for wet and dry day occurrence, and from a mixed exponential model for distribution of depth on wet days. The procedure is demonstrated on four sample stations scattered across the continental United States. A constrained multiv...

Inversion of satellite magnetic anomaly data

Abstract: A method of finding a first approximation to a crustal magnetization distribution from inversion of satellite magnetic anomaly data is described. Magnetization is expressed as a Fourier series in a segment of spherical shell. Input to this procedure is an equivalent source representation of the observed anomaly field. Instability of the inversion occurs when high frequency noise is present in the input data, or when the series is carried to an excessively high wave number. Preliminary results are given for the United States and adjacent areas.

The Rayleigh hypothesis in the theory of reflection by a grating

Abstract: In this paper, the Rayleigh hypothesis in the theory of reflection by a grating is investigated analytically. Conditions are derived under which the Rayleigh hypothesis is rigorously valid. A procedure is presented that enables the validity of the Rayleigh hypothesis to be checked for a grating whose profile can be described by an analytic function. As examples, we consider some grating profiles described by a finite Fourier series. Numerical results are then presented.

Possible inverse cascade behavior for drift-wave turbulence

Abstract: The turbulent spectral properties of the dynamical equation of Hasegawa and Mima (1978) governing the evolution of the electrostatic potential in drift-wave turbulence is investigated for two formulations of the problem: (1) as a nondissipative initial value problem, with the potential represented by a truncated Fourier series with large number of terms, and (2) as a dissipative problem with a small viscous dissipation at very short spatial scales, and a long wavelength forcing term at longer wavelengths It is found that Hasegawa and Mima's prediction for the nondissipative, truncated initial value modal problem is accurate, but substantial differences exist for the forced dissipative case between computer results and analytical predictions based on a wave kinetic equation of Kadomtsev Much better agreement is found with a simple dual-cascade model based on Kraichnan's generalization of Kolmogorov's cascade arguments

Stochastic functional fourier series, Volterra series, and nonlinear systems analysis

Abstract: A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.

Synthesis of discrete-interval binary signals with specified Fourier amplitude spectra

Abstract: A closed-form solution is presented for the discrete-interval, binary, periodic signal the complex Fourier coefficient spectrum of which optimally approximates in the least squares sense a desired complex Fourier coefficient spectrum. From this solution an efficient numerical procedure is derived for synthesis of discrete-interval, binary, periodic signals the Fourier amplitude spectrum of which is optimal in the same sense. Numerical examples show the practical feasibility of the procedure.

Simplifications in the x‐ray line‐shape analysis

Abstract: It is shown that a Fourier series associated with the Warren‐Averbach line‐shape analysis can be fitted with only five parameters to a pair of peaks. These interrelate the Fourier coefficients and thereby provide a simplified series which has been applied to the study of a Mo film on a Si crystal. The parameters include the average particle size, the first neighbor rms strain, a term which gives the variation in rms strain with cell separation, and two instrumental broadening coefficients. Although considerable simplification is possible, equivalent information can be obtained as compared with the original analysis and the ’’hook effect’’ is eliminated in the fitted coefficients.

FUNCTIONAL EQUATIONS OF WHITTAKER FUNCTIONS ON p-ADIC GROUPS

Abstract: Introduction. The Fourier coefficients of Eisenstein series on a rational tube domain have Euler product expansions, and one can view the Euler factors as functions of the weight of the Eisenstein series. Our main result is the existence of a functional equation for these functions, the local singular series. To prove this we observe that the local singular series is the value of a Whittaker function W attached to a class 1 principal series representation PS (X) induced from a character X on P, a maximal parabolic subgroup of G. That is, W transforms under left translation by elements of the unipotent radical, N, of P according to a character 4. The crucial point is to show that for suitable 4 there is only one operator, up to scalar multiple, that intertwines PS(x) with the space of Whittaker functions for 4. Functional equations then follow from the existence of intertwining operators corresponding to certain elements of the Weyl group. In contrast to [9] and [12], in dealing with Whittaker models we work with a horocycle N that is not maximal. This is necessary because we are dealing with degenerate principal series representations. In a separate paper, we apply the functional equations to the computation of local singular series for Baily's exceptional modular group. We now summarize the contents of the present paper. In Section 1 we introduce the principal series representations PS(X) on G and generic characters. By means of the Frobenius Reciprocity Law and a filtration on the space PS(x) deriving from the cellular decomposition of G with respect to a pair of parabolic subgroups, we reduce the uniqueness theorem to a study of certain integrals. In Section 2 we prove a technical lemma about p-adic fiber spaces and use it to show that the integrals mentioned above must all vanish. In Section 3 we prove the uniqueness theorem (3.2). Then we show how to obtain Whittaker maps as Cauchy principal value integrals. This gives rise to analytic families of distributions on G. As a consequence of

Analysis of Economic Time Series: A Synthesis

Abstract: A History of the Idea of Unobserved Components in the Analysis of Economic Time Series. Introduction to the Theory of Stationary Time Series. The Spectral Representation and Its Estimation. Formulation and Analysis of Unobserved-Components Models. Elements of the Theory of Prediction and Extraction. Formulation of Unobserved-Components Models and Canonical Forms. Estimation of Unobserved-Components and Canonical Models. Appraisal of Seasonal Adjustment Techniques. On the Comparative Structure of Serial Dependence in Some U.S. Price Series. Formulation and Estimation of Mixed Moving-Average Autoregressive Models for Single Time Series: Examples. Formulation and Estimation of Multivariate Mixed Moving-Average Autoregressive Time-Series Models. Formulation and Estimation of Unobserved-Components Models: Examples. Application to the Formulation of Distributed-Lag Models. A Time-Series Model of the U.S. Cattle Industry. Appendices: The Work of Buys Ballot. Some Requisite Theory of Functions of a Complex Variable. Fourier Series and Analysis. Whittle's Theorem. Inversion of Tridiagonal Matrices and a Method for Inverting Toeplitz Matrices. Spectral Densities, Actual and Theoretical, Eight Series. Derivation of a Distributed-Lag Relation between Sales and Production: A Simple Example. References. Author Index. Subject Index.

Simulation and Design of Microwave Class-C Amplifiers through Harmonic Analysis

Abstract: A method for analyzing microwave class-C amplifiers is proposed which satisfies the requirements of a wide application field, and, at the same time, operates with a fast runing time and without convergence problems. It is based on the partitioning of the circuit into linear and nonlinear subnetworks for which, respectively, frequency-domain and time-domain equations are written. Then, taking into account that the time-domain and frequency-domain representations are related by the Fourier series, the circuit behavior is described by means of a system of nonlinear equations whose unknowns are the harmonic components of the incident waves at all the connections. To overcome the numerical problems arising in the search for the solution of this system when strong nonlinearities are involved, a special step-by-step procedure is adopted. The problem is transformed into the search for the solution of a sequence of well-conditioned systems of equations corresponding to a sequence of well-chosen circuits obtained from the original one through progressive changes of the input signal starting from 0 up to the nominal value. The program which implements the method is also described and the results of the analysis relative to a class-C amplifier are compared with measured values.

Algorithms for Linear Interpolator and Interpolation Error for Minimal Stationary Stochastic Processes

Abstract: Algorithms for linear interpolator and interpolation error for a minimal univariate weakly stationary stochastic process with discrete multiparameter are derived. The Fourier coefficients of the inverse of the spectral density play an important role in the determination of these algorithms.

Fourier series of functions of Λ-bounded variation

Abstract: It is shown that the Fourier coefficients of functions of Abounded variation, A = fAn} are O(\/n). This was known for An = n+ 1, -1 < 8 < 0. The classes L and HBV are shown to be complementary, but L and ABV are not complementary if ABV is not contained in HBV. The partial sums of the Fourier series of a function of harmonic bounded variation are shown to be uniformly bounded and a theorem analogous to that of Dirichlet is shown for this class of functions without recourse to the Lebesgue test. We have shown elsewhere that functions of harmonic bounded variation (HBV) satisfy the Lebesgue test for convergence of their Fourier series, but if a class of functions of A-bounded variation (ABV) is not properly contained in HBV, it contains functions whose Fourier series diverge [1]. We have also shown that Fourier series of functions of class { n 13+1) BV, 1 < ,B < 0, are (C, /3) bounded, implying that the Fourier coefficients are 0(n13) [2]. Here we shall estimate the Fourier coefficients of functions in ABV. Without recourse to the Lebesgue test, we shall prove a theorem for functions of HBV analogous to that of Dirichlet and also show that the partial sums of the Fourier series of an HBV function are uniformly bounded. From this one can conclude that L and HBV are complementary classes, i.e., Parseval's formula holds (with ordinary convergence) forf E L and g E HBV. We shall see that L and ABV are not complementary if ABV is not a subclass of HBV. 1. Definitions and results. Let f be a real function on an interval [a, b], A = {) n} a nondecreasing sequence of positive numbers such that E l/Xn diverges, and { In) a sequence of nonoverlapping intervals In = [an, bn] C [a, b]. The function f is said to be of A-bounded variation (ABV) if E If(an) f(bn)I/;A converges for every choice of {I). The supremum of these sums is called the A-variation of f, denoted by VA(f; [a, b]). When A = {n), the class is referred to as the functions of harmonic bounded variation (HBV). We shall suppose that [a, b] = [0, 27T] and our functions have period 27T. Two classes of functions, K and K1, are said to be complementary [4, p. 157] if f E K and g E K1 implies 1 2w 1 0 fg dx = aoaf + 2 (a a' + bkbk) T0 2 1 Received by the editors April 14, 1978. AMS (MOS) subject classifications (1970). Primary 42A28; Secondary 26A45. ISupported in part by NSF grant MCS77-00840. ? 1979 American Mathematical Society 0002-9939/79/0000-0171 /$02.25 119 This content downloaded from 157.55.39.104 on Sun, 19 Jun 2016 06:34:03 UTC All use subject to http://about.jstor.org/terms

Simulation of space‐random fields for solution of stochastic boundary‐value problems

Abstract: The technique of digital simulation of Gaussian random fields (which are nonhomogeneous in space or are part of homogeneous random fields in space) is presented to a solution of stochastic boundary‐value problems. The method consists of expanding the simulated field, with known mean and autocorrelation function, in series in terms of the structural ’’natural’’ mode shapes, and the Fourier coefficients of the truncated series are then simulated as random normal vectors. The method is applicable to static or dynamic stochastic two‐point boundary‐value problems in mechanics of solids.

Scattering of an offset two‐dimensional Gaussian beam wave by a cylinder

Abstract: The scattering of a two‐dimensional Gaussian beam wave by a perfectly conducting or dielectric cylinder centered off both the beam axis and the beam waist is treated by the Fourier‐series‐expansion method. The incident‐beam fields are expanded in a Fourier series with respect to the polar angle in a cylindrical coordinate, and the expansion coefficients are numerically evaluated. Several scattering patterns are obtained by the use of these expansion coefficients, and the dependence of the scattering characteristics on the offset of the incident‐beam wave is examined.

On the Inversion of an Integral Equation Relating Two Wavefunctions in Planes of an Optical System Suffering from an Arbitrary Number of Aberrations

Abstract: If the wavefunction in the (not necessarily gaussian) image plane of an optical instrument is distorted by an arbitrary number of aberrations, the wavefunction in planes situated between the image plane and the plane of the specimen holder cannot be reconstructed by a Fourier series or a Fourier integral. This paper shows that the wavefunctions in those planes are uniquely determined by the values of the wavefunction in the image plane, and that an inversion formula, which in its structure is very similar to the well-known Fourier series, can be derived. Mathematically the problem concerns the inversion of the integral equation h (x 1, x 2) = Z † exp iS (x 1, x 2, y 1, y 2) ‚ (y 1, y 1) dy 1 dy 2 if the eikonal S is a multinomial.

On a theorem of Paley and the Littlewood conjecture

Abstract: (1) Z~~ lf(hk)j 2 <= 8(l[/llx) 2, for all functions f in L~. Corollary 1. I f F is a finite subset o f the integers having N elements, then (2) Ilflll --> [(log, N)/8] 1/2 mip If(n)l, for all functions f in L~. A similar estimate was proved under added assumptions about the size of the coefficients f ( n ) by S. K. Pichorides [25]. We shall discuss other work on the Littlewood conjecture at the end of this section and at the end of the paper.

Nonparametric Estimation of Population Density for Line Transect Sampling Using FOURIER Series

Abstract: A nonparametric, robust density estimation method is explored for the analysis of right-angle distances from a transect line to the objects sighted. The method is based on the FOURIER series expansion of a probability density function over an interval. With only mild assumptions, a general population density estimator of wide applicability is obtained.

Distribution functions for the Brownian motion of particles in a periodic potential driven by an external force

Abstract: The stationary distribution functions for the Brownian motion of particles driven by an external force are calculated by expanding the velocity part into Hermite functions and the space part into a Fourier series. Insertion into the Fokker-Planck equation leads to a matrix continued fraction for the lowest two coefficients of the Hermite functions. Higher order terms are found by reverse iteration. Results are shown for a cosine potential. The good convergence allows the calculation in the full range of damping constants. For small friction the distribution function is in good agreement with previous results and the maxima are given by the solutions without noise.