Showing papers on "Fourier series published in 1974"

Almost Periodic Differential Equations

Abstract: Almost periodic functions.- Uniformly almost periodic families.- The fourier series theory.- Modules and exponents.- Linear constant coefficient equations.- Linear almost periodic equations.- Exponential dichotomy and kinematic similarity.- Fixed point methods.- Asymptotic almost periodic functions and other weaker conditions.- Separated solutions.- Stable solutions.- First order equations.- Second order equations.- Averaging.

Diffusion and field-gradient effects in NMR Fourier spectroscopy

Abstract: In NMR Fourier transform spectroscopy, rf pulses are periodically applied to a spin system to generate a sequence of free induction decays which form the basis of spectroscopic measurements. In inhomogeneous magnetic fields, distortions of the free induction decay arise from superimposed spin‐echo effects. They are strongly dependent on molecular diffusion which is also responsible for the nonexponential envelope of the echo train observed in a stopped pulse experiment. These effects are investigated by means of two independent methods, the Fourier expansion method and the partition method. The results are discussed and used to explain measurements made on a one‐spin system. The influence of echoes on the sensitivity of Fourier transform spectroscopy is considered, and it is concluded that the achievable gain does not normally outweigh the complications arising from retention or enhancement of echoes. Experimental parameters are derived for efficient suppression of echoes by means of a periodically pulsed...

Fourier Series on Spheres

Abstract: Spectral methods for the numerical solution of problems in spherical (and spheroidal) geometries are discussed. A new class of expansions based on special Fourier series are shown to lead to efficient and accurate simulations. A detailed exposition is given of the properties of surface harmonic series and transform methods and their relation to the new Fourier series on spheres. With resolution 1/N in both longitude and latitude, spectral methods using surface harmonics require 0(N) arithmetic operations per retained mode per time step while those based on Fourier series on spheres require only 0(logN) operations per retained mode per time step.

Identification of Three-Dimensional Objects Using Fourier Descriptors of the Boundary Curve

Abstract: The feasibility of a method for the identification of a three-dimensional object from information contained in the boundary of its silhouettes is demonstrated. A silhouette is characterized by parametric representation of its boundary curve in the complex plane. After normalization and transformation, a set of Fourier descriptors is derived for every silhouette. A minimum distance classifier uses the descriptors to identify the three-dimensional object and to estimate its position and attitude with respect to a known reference coordinate system. The method was tested for identification of four aircraft representing complex and nonconvex objects. Simulation results, quantitative and statistical, are presented.

Determination of the single-scattering probability distribution from plural-scattering data

Abstract: A technique is presented which enables the recovery of the probability distribution for single scattering from plural-scattering electron energy loss data. Neither the scattering parameter t/? nor details of the component processes need be known. The computational method uses Fourier series in order to overcome a number of practical problems in the application of convolution series methods, to include instrumental effects and to permit the processing of data with large values of the scattering parameter. The effects of noise, specimen oxidation and the accuracy of the technique are considered.

On fourier’s analysis of linear inequality systems

Abstract: Fourier treated a system of linear inequalities by a method of elimination of variables. This method can be used to derive the duality theory of linear programming. Perhaps this furnishes the quickest proof both for finite and infinite linear programs. For numerical evaluation of a linear program, Fourier’s procedure is very cumbersome because a variable is eliminated by adding each pair of inequalities having coefficients of opposite sign. This introduces many redundant inequalities. However, modifications are possible which reduce the number of redundant inequalities generated. With these modifications the method of Fourier becomes a practical computational algorithm for a class of parametric linear programs.

Fourier analysis of stationary processes

Abstract: This paper begins with a description of some of the important procedures of the Fourier analysis of real-valued stationary discrete time series. These procedures include the estimation of the power spectrum, the fitting of finite parameter models, and the identification of linear time invariant systems. Among the results emphasized is the one that the large sample statistical properties of the Fourier transform are simpler than those of the series itself. The procedures are next generalized to apply to the cases of vector-valued series, multidimensional time series or spatial series, point processes, random measures, and finally to stationary random Schwartz distributions. It is seen that the relevant Fourier transforms are evaluated by different formulas in these further cases, but that the same constructions are carried out after their evaluation and the same statistical results hold. Such generalizations are of interest because of current work in the fields of picture processing and pulse-code modulation.

Convergence in the mean and almost everywhere of fourier series in polynomials orthogonal on an interval

Abstract: Let be the system of polynomials orthonormal on with weight where , , on and ( is the modulus of continuity in ). Consider the class of functions , where Let denote the partial sums of the Fourier series of a function with respect to the system .In the paper, conditions are obtained on the exponents of the functions and and the exponent that are necessary and sufficient for the boundedness in of each of the operators and . Sufficient conditions for the convergence of the partial sums to in the mean and almost everywhere in are revealed as a consequence. It is proved that these conditions are best possible on the class (for in the case of convergence almost everywhere). Estimates of the polynomials and necessary and sufficient conditions for their boundedness in the mean are also obtained.Bibliography: 26 items.

Sinusoids versus Walsh functions

Abstract: In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.

Fourier series and estimation on the circle with applications to synchronous communication--I: Analysis

Abstract: A wide variety of continuous- and discrete-time estimation problems on the circle S^1 are considered with the aid of Fourier series analysis. Measurement and diffusion update equations are derived for the conditional expectation of certain functions of the parameter to be estimated, and we investigate the use of Fourier series to obtain easily implemented optimal estimation equations. A variety of important examples--phase tracking, frequency demodulation, and phase demodulation in the presence of oscillator instabilities, additive noise, Rayleigh fading, or any combination of these--are considered.

Fourier series as a quantitative descriptor of miospore shape

Abstract: Fourier series is a mathematical expression of sine and cosine waves which can be used to describe completely the outline of a miospore. The method of obtaining the Fourier series for a single miospore is presented and an illustration is included to show the application of Fourier series to the solution of palynologic problems. Investigation suggests that definable differences in the outlines of various species of pine pollen can be used to differentiate among them, but the technique used to prepare pine pollen for microscopic examination affects pollen shape. Furthermore, the changes in shape brought about by different preparation techniques are not constant from species to species. These differences have been quantitatively evaluated by means of a Fourier series and appropriate statistical tests. Further investigation suggests that under certain conditions, composite pollen shapes can be generated from the Fourier series of several grains and can be used to locate the areas on the periphery of a grain which are most variable from species to species.

On the a.e. convergence of Walsh-Kaczmarz-Fourier series

Abstract: It is shown that partial sums of Walsh-KaczmarzFourier series of functions in the Orlicz class L(log+ L)2 converge a.e. The proof utilizes an estimate of P. Sjolin on the partial sums of the usual Walsh-Fourier series. The Walsh-Kaczmarz system is a reordering of the usual Walsh system within dyadic blocks of indices 2N to 2N+1, N=O, 1, * f f. The a.e. convergence properties of Fourier series with respect to this system have been investigated by L. A. Balashov [1] and K. H. Moon [7]. Balashov showed that there exist functions in the Orlicz class L(log+ L)1", 0 ...read moreread less

Construction of automorphic forms and integrals

Abstract: It is well known that modular forms of positive dimension have Fourier coefficients given by certain infinite series involving Kloostermann sums and the modified Bessel function of the first kind. In this paper a functional equation which characterizes all such Fourier series is found. It is also shown that these Fourier series have a construction similar to that of Poincare' series of negative dimension.

Computational techniques based on the Lanczos representation

Abstract: In his book Discourse on Fourier Series, Lanczos deals in some detail with representations of f(x) of the type f(x) = h,-i(x) + gp(x) where h,-I(x) is a polynomial of degree p - 1 and gp(x) has the property that its full range Fourier coefficients converge at the rate r-P In Part I, some properties of h,(x) and of the series Ih,(x) l0 are described These prop- erties are used here to provide criteria for the convergence or divergence of the Euler-Mac- laurin series, in the case when f(x) is an analytic function The similarities and differences between this series and the Lidstone and other two-point series are briefly mentioned In Part II, the Lanczos representation is employed to derive an approximate representation F(x) for an analytic function f(x) on the interval (0, 1) is derived This has the form P-i m12 F(x) = E Xqi1Bq(x)/q! +2 E (Iu cos 2irrx + V, sin 2irrx) toq1 rhO and requires for its determination the values of the derivatives f (a-l )(1) - f (q-l )(O) (q = 1, 2, * * p - 1) and the regularly spaced function values f(j/m) (j = 0, 1, * * *, m) It involves replacing gp(x) by a discrete Fourier expansion based on trapezoidal rule approximations to its Fourier coefficients This representation is a powerful one The drawback is that it requires derivatives Most of Part II is devoted to the effect of using only approximate derivatives It is shown that when these are successively less accurate with increasing order (the sort of behaviour encountered using finite difference formula), then the representation is still powerful and reliable In a computational context the only penalty for using inaccurate derivatives is that a larger value of m may-or may not-be required to attain a specific accuracy PART I** PROPERTIES OF THE SEQUENCE hp(x) 1 The Lanczos Representation In this section, we outline a derivation of what we term Lanczos' representation for a function f(x) We suppose that f(x) is an analytic function of x and is real valued when x is real For convenience, we suppose that f(x) is analytic in a region of the complex plane which contains the unit interval (0, 1), a restriction which we denote by

Fourier series and estimation on the circle with applications to synchronous communication--II: Implementation

Abstract: The practical implementation of the infinite-dimensional optimal estimation results presented in Part I of this series is considered. Several techniques are described in detail. Included among these is the so-called "assumed density" approximation technique. Finite-dimensional suboptimal filtering equations based on this method are derived for several of the phase-tracking/demodulation problems studied in Part I. Finally, these techniques are applied to a phase tracking problem of importance in navigation systems such as Omega, and simulation results are reported that favorably compare a system designed using these techniques to an optimal phase-lock loop and an optimal linear system.

Schaum's outline of theory and problems of Fourier analysis : with applications to boundary value problems

Abstract: Boundary Value Problems Fourier Series and Applications Orthogonal Functions Gamma, Beta and Other Special Functions Fourier Integrals and Applications Bessel Functions and Applications Legendre Functions and Applications Hermite, Laguerre and Other Orthogonal Functions Appendices A: Uniqueness of Solutions Appendices B: Special Fourier Series Appendices C: Special Fourier Transforms Appendices D: Tables of Values for J0(x) and J1(x) Appendices E: Zeros of Bessel Functions

The Budan-Fourier theorem for splines

Abstract: The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.

On piecewise-linear basis functions and piecewise-linear signal expansions

Abstract: A set of piecewise-linear (PL) basis functions for signal or functional decomposition is introduced. These basis functions provide a PL approximation to the signal and an a priori determination of the required number for a finite term expansion to achieve a certain pointwise approximation error. Moreover, the expansion coefficients are linear combinations of samples of the function to be expanded and are virtually trivial to determine.

Backscattered Short Pulse Response of Surface Waves from Dielectric Spheres

Abstract: A short pulse response for surface waves involved in the backscattering by dielectric spheres is considered in the time domain. Since the surface-wave contributions for the cw backscattering are known, pulse returns for the surface waves are obtained from the cw solutions by Fourier synthesis. Large savings in computation time for a Fourier series were realized with the use of the fast Fourier transform algorithm. The return positions in the short pulse response can be approximately estimated from a scattering model for surface waves depicted by Van de Hulst. With the propagation constant in the dielectric sphere rather than in free space for a wave traveling along the sphere surface, a better agreement between the predicted and calculated positions for the short pulse returns was obtained. Significant returns of surface waves for the backscattering come from the surface waves that have made the maximum number of shortcuts possible through the sphere.

Enhancement of directional wave spectrum estimates

Abstract: Determination of the directional distribution of ocean surface waves is of practical importance and analytical schemes for it are developed and discussed here. Based on a generalized representation of wave properties such as surface elevation, subsurface pressure or horizontal components of water particle velocity, acceleration or wave force, two general schemes of analysis are developed. In one scheme the predictive equations for the directional distribution of both the amplitude and phase of waves are derived. Distribution of energy as a function of direction for random waves is obtained in the other scheme. Fourier series parameterization is used to represent directional spectrum. The truncation of the series dictated by data limitations introduce directional spread and negative side lobes for the estimated directional spectrum. A procedure to remove these undesirable side lobes by a non-negative smoothing function is described. The smoothing causes further directional spread. Methods for obtaining better directional resolution are discussed. Data adaptive spectral analysis techniques such as Maximum Likelihood Method and Maximum Entropy Method are suggested.