Showing papers on "Fourier series published in 1970"
IBM1
TL;DR: A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described and some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.
Abstract: A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described. In this method, the normalized phase function for scattering is represented as a Fourier series whose maximum required number of terms depends upon the zenith angles of the directions of incident and of scattered radiation. Some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.
159 citations
TL;DR: In this article, it is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated.
Abstract: The topics of orthogonality and Fourier series occupy a central position in analysis. Nevertheless, there is surprisingly little statistical literature, with the exception of that of time series and regression, which involves Fourier analysis. In the last decade however, several papers have appeared which deal with the estimation of orthogonal expansions of distribution densities and cumulatives. Cencov [1] and Van Ryzin [6] considered general properties of orthogonal expansion based density estimators and the latter applied these properties to obtain classification procedures. Schwartz [3] and the authors [2] and [4] investigated respectively the Hermite and Trigonometric special cases. The authors also obtained certain general results which apply not only to estimators of the population density but also to estimators of the population cumulative [2], [4] and [5]. In this paper several results derived for the univariate case are extended to the multivariate case. Also a new relationship is obtained which involves general Fourier expansions and estimators. Although there is some reason for calling the Gram-Charlier estimation of distribution densities a Fourier method, one fundamental aspect of Fourier methods is not shared by Gram-Charlier estimation. Gram-Charlier techniques make no use of Parseval's Formula or related error relationships of Fourier analysis. The ease with which the mean integrated square error (MISE) is evaluated, when Fourier methods are applied, accounts for most of the recent interest in this area. Section 1 of this paper deals with an investigation of two general MISE relationships for multivariate estimates of Fourier expansions. The relationship given in Theorem 2 is particularly simple and yet includes the four MISE's which are involved in the estimation problem. In Section 2 the choice of orthogonal functions is restricted to the trigonometric polynomials. It is shown that the MISE of multidimensional trigonometric polynomial estimators are related in a simple way to the Fourier coefficients of the distribution which is being estimated. This result is of considerable utility since it yields a rule for deciding which terms should be included in the estimate of the multivariate density.
73 citations
69 citations
TL;DR: In this paper, sets of coefficients for four finite difference methods of numerical integration are presented that will integrate without truncation error products of fourier and ordinary polynomials, and these sets are formulated such that they are free from computational difficulties.
Abstract: Sets of coefficients for four finite difference methods of numerical integration are presented that will integrate without truncation error products of fourier and ordinary polynomials. These sets are formulated such that they are free from computational difficulties.
68 citations
01 Jan 1970
TL;DR: In this paper, nonlinear panel flutter analysis and response under random excitation or nonlinear aerodynamic loading, using Rayleigh-Ritz approximation to Hamilton variational principle.
Abstract: Nonlinear panel flutter analysis and response under random excitation or nonlinear aerodynamic loading, using Rayleigh-Ritz approximation to Hamilton variational principle
63 citations
IBM1
TL;DR: It is shown that a section of a scattering function vs scattering angle curve can be adequately represented by a fourier series with less than 2x + 10 terms.
Abstract: Results of computations are presented to show the variations of coefficients of four different Legendre series, one for each of the four scattering functions needed in describing directional dependence of the radiation scattered by a sphere. Values of the size parameter (x) covered for this purpose vary from 0.01 to 100.0. An adequate representation of the entire scattering function vs scattering angle curve is obtained after making use of about 2x + 10 terms of the series. It is shown that a section of a scattering function vs scattering angle curve can be adequately represented by a fourier series with less than 2x + 10 terms. The exact number of terms required for this purpose depends upon values of the size parameter and refractive index, as well as upon the values of the scattering angles defining the section under study. Necessary expressions for coefficients of such fourier series are derived with the help of the addition theorem of spherical harmonics.
62 citations
TL;DR: In this article, the authors apply the theory of interpolation spaces to different parts of approximation theory and study the rate of convergence of summation processes of Fourier series and Fourier integrals.
Abstract: In this paper we apply the theory of interpolation spaces to different parts of Approximation theory. We study the rate of convergence of summation processes of Fourier series and Fourier integrals. The main body of the paper is devoted to a study of the rate of convergence of solutions of difference schemes for parabolic initialvalue problems with constant coefficients and to related problems.
36 citations
TL;DR: In this paper, the M6bius inversion technique is applied to the Poisson summation formula, which results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series.
Abstract: The M6bius inversion technique is applied to the Poisson summation formula. This results in expressions for the remainder term in the Fourier coefficient asymptotic expansion as an infinite series. Each element of this series is a remainder term in the corresponding Euler-Maclaurin summation formula, and the series has specified convergence properties. These expressions may be used as the basis for the numerical evaluation of sets of Fourier coefficients. The organization of such a calculation is described, and discussed in the context of a broad comparison between this approach and various other standard methods.
25 citations
TL;DR: A matrix series is derived that converges to the inverse of a covariance matrix that can be inverted by taking finite Fourier transforms.
Abstract: A matrix series is derived that converges to the inverse of a covariance matrix. The members of the series are derived from a circular matrix that can be inverted by taking finite Fourier transforms. An example of the method is presented.
24 citations
22 citations
TL;DR: In this article, a method of analysis for circular cylindrical shells under non-uniform external loads is presented for moderately large displacements and take secondary creep into account, where thermal effects and initial imperfections are included.
01 Jan 1970
01 Jan 1970
TL;DR: In this article, the authors take up a number of special topics in harmonic analysis on compact groups, including the algebra of absolutely convergent Fourier series, an entity about which a good deal although far from everything is known.
Abstract: In this chapter, we take up a number of special topics in harmonic analysis on compact groups. Section 34 deals with the algebra of absolutely convergent Fourier series, an entity about which a good deal although far from everything is known. Sections 35 and 36 are a detailed treatment of multipliers for various classes of Fourier transforms on compact groups; some facts about locally compact Abelian groups are obtained as well. In § 37, we study lacunary Fourier transforms, again on compact groups, and in § 38, ideal theory for certain convolution algebras of functions on compact groups.
TL;DR: The standard method of computing the mutual information between two stochastic processes with finite energy replaces the processes with their Fourier coefficients, and this procedure is mathematically justified here for random signals w,(ω) square-integrable in the product space t × ω where t ∊ [O, T] and w is an element of a probability space.
Abstract: The standard method of computing the mutual information between two stochastic processes with finite energy replaces the processes with their Fourier coefficients. This procedure is mathematically justified here for random signals w,(ω) square-integrable in the product space t × ω where t ∊ [O, T] and w is an element of a probability space. A natural notion of the sigma field generated by w, (ω) is presented and it is shown to coincide with the sigma field generated by the random Fourier coefficients of w,(ω) in any complete orthonormal system in L 2 [O, T]. This justifies the use of Fourier coefficients in mutual information computations. Capacity is calculated for finite and infinite-dimensional channels, where the output signal consists of a filter (general Hilbert-Schmidt operator) operating on the input signal with additive Gaussian noise. The finite-dimensional optimal signal is obtained. In the infinite-dimensional case capacity can be approached arbitrarily closely with finite-dimensional inputs. The question of the existence of an infinite-dimensional signal which achieves capacity is considered. There are channels for which no signal achieves capacity. Some results are obtained when the noise coordinates are independent in the eigensystem of the filter.
TL;DR: The American Mathematical Monthly: Vol. 77, No. 2, No 2, pp 119-133 as mentioned in this paper, is a collection of articles about Fourier series and Fourier diagrams.
Abstract: (1970). Some Aspects of Fourier Series. The American Mathematical Monthly: Vol. 77, No. 2, pp. 119-133.
TL;DR: In this paper, the negative powers of the mutual distance between two bodies are developed into series converging at any moment but that of collision, and on the base of these expansions the series have been constructed representing Δ−γ in the perturbation theory of celestial mechanics.
Abstract: The negative powers of the mutual distanceΔ−γ between two bodies are developed into series converging at any moment but that of collision. On the base of these expansions the series have been constructed representingΔ−γ in the perturbation theory of celestial mechanics. In the general case, including intersecting orbits, the terms are quasi-periodic functions of the time. In the case of non-intersecting orbits the expansion is a double Fourier series in the mean anomalies. All the expansions have a literal form with respect to osculating elements.
TL;DR: The name of Joseph Fourier (1768-1830) is largely associated with the mathematical analysis of heat diffusion and methods of solving partial differential equations by means of Fourier series, Fourier integrals and the calculus of differential operators.
Abstract: The name of Joseph Fourier (1768-1830) is largely associated with the mathematical analysis of heat diffusion and methods of solving partial differential equations by means of Fourier series, Fourier integrals and the calculus of differential operators. But his interests in mathematics encompassed other fields also, and one of his achievements was to create singlehanded a basic theory of linear programming.
TL;DR: In this article, classical solutions of the heat flow and diffusion equations expressed in terms of Fourier series and error functions are outlined, and three practical applications of the mathematical solutions are described: they are the kiln drying of fish, measurement of diffusion rates in polymers and the burning of wood.
Abstract: Summary Classical solutions of the heat‐flow and diffusion equations expressed in terms of Fourier series and error functions are outlined. Numerical methods essential for solving non‐linear problems are discussed. Three practical applications of the mathematical solutions are described: they are the kiln‐drying of fish, the measurement of diffusion rates in polymers and the burning of wood.
TL;DR: Dduce an average purine-pyrimidine pair for DNA using the International Tables for Crystallography to reach the same conclusions as when B = 6 A2.
Abstract: duce an average purine-pyrimidine pair for DNA. The atomic form factors are from International Tables for Crystallography; they are multiplied by weighting factors of 1⁄2/2 or 1⁄44 where appropriate. The temperature factor has B =6A2 since this is more appropriate for DNA. (Those familiar with the diffraction from DNA models are aware that much of the sparseness of DNA data with periodicities less than 3 A is not due only to attenuation factors that increase continuously with diffraction angle but to the fact that the molecular transform itself is small between 3 and 2 A.) However we have repeated the experiments with B = 15 A and 30 A and reach the same conclusions as when B = 6 A2. 9. M. H. F. Wilkins, in Biological Structure and Function (Academic Press, New York, 1960), vol. 1, pp. 13-32. 10. D. A. Marvin, M. H. F. Wilkins, L. D. Hamilton, Acta Cryst. 20, 663 (1966).
Patent•
28 Jan 1970
TL;DR: In this article, an improved method for generating the Fourier series coefficients corresponding to a waveform segment was proposed, which is identical to knowledge of the complexamplitude frequency spectrum of the waveform.
Abstract: Any time segment of a time-varying waveform may be expressed in terms of a Fourier series. That is knowledge of the complexamplitude frequency spectrum of the waveform, which is identical to knowledge of the Fourier series coefficients for the waveform segment, enable one to reconstruct the waveform over the time segment in question. This invention is an improved method for generating the Fourier series coefficients corresponding to such a waveform segment.
TL;DR: In this paper, a procedure is developed for locating singularities of two-dimensional exterior harmonic functions, which is capable of extension to more general differential equations and, in principle at least, to higher dimensions.
Abstract: A procedure is developed for locating singularities of two-dimensional, exterior harmonic functions $\Phi $. The method is capable of extension to more general differential equations and, in principle at least, to higher dimensions. It utilizes the fact that $\Phi $ may be expanded in Fourier series about a point $P_0 $ external to the boundary C. On the circle of convergence lies at least one singularity of $\Phi $. The envelope formed by the circles of convergence as $P_0 $ describes a closed curve about C will bound the singularities. The radius of the circle of convergence is obtained by determining the asymptotic behavior of the Fourier coefficients. For the Dirichlet problem, this is made possible by expressing as the potential of a double layer, of density $\mu $, on C. The asymptotic behavior of the coefficients is governed by the singularities of $\mu $ in the plane of the complex arclength parameter s. Properties of $\mu $ are found by using the Fredholm integral equation of the second kind whic...
TL;DR: In this paper, it was shown that the eigenfunction expansion with respect to {un} of a function φ on (0, π) has properties similar to those of the Fourier cosine expansion of φ.
TL;DR: In this paper, the Fourier series of a 2π-periodic function f(x, y) is shown to converge to a value bounded above the limit superior and below the limit inferior of f (x+u, y+v), u, v → 0, depending on the manner in which the series is summed.
Abstract: Let a2π-periodic function f(x, y) be continuous in some neighbourhood of the point (x, y) except possibly along finitely many lines l1, l2, ..., lk terminating at (x, y). The problem of convergence of the Fourier series of f(x, y) at the point (x, y) is examined in some detail. It is established that under certain restrictions on the variation of f(x, y), and also on the lines l1, l2, ..., lk, the fourier series converges to a value bounded above by the limit superior, and below by the limit inferior of f(x+u, y+v), u, v →0, this value depending on the manner in which the series is summed.
TL;DR: The analysis of arbitrary time samples of signals of interest in terms of a Fourier series in effect forces the signal to be periodic with a fundamental period equal to the sample length.
Abstract: The analysis of arbitrary time samples of signals of interest in terms of a Fourier series in effect forces the signal to be periodic with a fundamental period equal to the sample length. This causes sinusoidal components in the signal that are not harmonic in the sample interval to appear to be discontinuous at the ends of the periods; each such component leads to a complete set of the harmonic terms determined by the analysis. The determination of the inharmonic sinusoidal components can be improved by taking suitable combinations of the coefficients determined by the analysis, or by a weighting of the input data to remove the discontinuity. It is shown that improvements of the convergence are accompanied by a corresponding broadening of the principal response.
TL;DR: In this paper, the authors derived some recurrence formulae which can be used to calculate the Fourier expansions of the functions n cosmv and n sinmv in terms of the eccentric anomaly E or the mean anomaly M. These basic series were given in ex-plicit form in the classical literature.
Abstract: In this paper we derive some recurrence formulae which can be used to calculate the Fourier expansions of the functions (r/a) n cosmv and (r/a) n sinmv in terms of the eccentric anomaly E or the mean anomaly M. We also establish a recurrence process for computing the series expansions for aI1 n and m when the expansions of two basic series are known. These basic series were given in ex- plicit form in the classical literature. The recurrence formulae are linear in the functions involved and thus make very simple the computation of the series.
TL;DR: For any given set E ⊂, of measure zero, a functionf(t) ε C (0, 2π), is constructed whose Fourier series is unboundedly divergent on E.
Abstract: For any given set E ⊂ [0, 2π), of measure zero, a functionf(t) e C (0, 2π), is constructed whose Fourier series is unboundedly divergent on E. If E is closed, there is a functionϕ(t) e C (0, 2π), whose Fourier series diverges unboundedly on E and converges on [0, 2π)E.