Showing papers on "Fourier series published in 1988"

Hyperbolic distribution problems and half-integral weigth Maass forms.

Abstract: (Actually n ~ is replaced by d(n)log ~ 2n where d(n) is the divisor function.) A striking application of(1.2) is to give the uniform distribution of certain lattice points in Z 3 on a sphere centered at the origin with increasing radius, without imposing Linnik's condition and in a quantitative sense. Motivated by the corresponding problem in negative curvature, which is the distribution of Heegner points and closed geodesics on PSL2(7Z)\\H, one is led to establish the analogue of

There exists a neural network that does not make avoidable mistakes

Abstract: The authors show that a multiple-input, single-output, single-hidden-layer feedforward network with (known) hardwired connections from input to hidden layer, monotone squashing at the hidden layer and no squashing at the output embeds as a special case a so-called Fourier network, which yields a Fourier series approximation properties of Fourier series representations. In particular, approximation to any desired accuracy of any square integrable function can be achieved by such a network, using sufficiently many hidden units. In this sense, such networks do not make avoidable mistakes. >

Tauberian theorems for generalized functions

Abstract: Notation and Definitions.- 1: Some Facts on the Theory of Distributions.- 1. Distributions and their properties.- 1. Spaces of test functions.- 2. The space of distributions D?(O).- 3. The space of distributions S?(F).- 4. Linear operations on distributions.- 5. Change of variables.- 6. L -invariant distributions.- 7. Direct product of distributions.- 8. Convolution of distributions.- 9. Convolution algebras of distributions.- 2. Integral transformations of distributions.- 1. The Fourier transform of tempered distributions.- 2. Fourier series of periodic distributions.- 3. The B -transform of distributions.- 4. Fractional derivatives (primitives).- 5. The Laplace transform of tempered distributions.- 6. The Cauchy kernel of the tube domain TC.- 7. Regular cones.- 8. Fractional derivatives (primitives) with respect to a cone.- 9. The Radon transform of distributions with compact support in an odd-dimensional space.- 3. Quasi-asymptotics of distributions.- 1. General definitions and basic properties.- 2. Automodel (regularly varying) functions.- 3. Quasi-asymptotics over one-parameter groups of transformations.- 4. The one-dimensional case. Quasi-asymptotics at infinity and at zero.- 5. The one-dimensional case. Asymptotics by translations.- 6. Quasi-asymptotics by selected variable.- 2: Many-Dimensional Tauberian Theorems.- 4. The General Tauberian theorem and its consequences.- 1. The Tauberian theorem for a family of linear transformations.- 2. The general Tauberian theorem for the dilatation group.- 3. Tauberian theorems for nonnegative measures.- 4. Tauberian theorems for holomorphic functions of bounded argument.- 5. Admissible and strictly admissible functions.- 1. Families of linear transformations under which a cone is invariant.- 2. Strictly admissible functions for a family of linear transformations.- 3. Admissible functions of a cone.- 4. Some examples of admissible functions of a cone.- 6. Comparison Tauberian theorems.- 1. Preliminary theorems.- 2. The comparison Tauberian theorems for measures and for holomorphic functions with nonnegative imaginary part.- Comments on Chapter 2.- 3: One-Dimensional Tauberian Theorems.- 7. The general Tauberian theorem and its consequences.- 1. The general Tauberian theorem and its particular cases.- 2. Quasi-asymptotics of a distribution f from S+? and a function arg f?.- 3. Tauberian theorem for distributions from the class .- 4. The decomposition theorem.- 8. Quasi-asymptotic properties of distributions at the origin.- 1. The general case.- 2. Quasi-asymptotics of distributions from H and asymptotic properties of the reproducting functions of measures.- 9. Asymptotic properties of the Fourier transform of distributions from M+.- 1. Asymptotic properties of the Fourier transform of finite measures.- 2. Asymptotic properties of the Fourier transform of distributions from M+.- 3. The Abel and Cezaro series summation with respect to an automodel weight.- 10. Quasi-asymptotic expansions.- 1. Open and closed quasi-asymptotic expansions.- 2. Quasi-asymptotic expansions and convolutions.- 4: Asymptotic Properties of Solutions of Convolutions Equations.- 11. Quasi-asymptotics of the fundamental solutions of convolution equations.- 1. Quasi-asymptotics and convolution.- 2. Quasi-asymptotics of the fundamental solutions of hyperbolic operators with constant coefficients.- 3. Quasi-asymptotics of the solutions of the Cauchy problem for the heat equation.- 12. Quasi-asymptotics of passive operators.- 1. The translationally-invariant passive operators.- 2. The fundamental solution and the Cauchy problem.- 3. Quasi-asymptotics of passive operators and their fundamental solutions.- 4. Differential operators of the passive type.- 5. Examples.- Comments on Chapter 4.- 5: Tauberian Theorems for Causal Functions.- 13. The Jost-Lehmann-Dyson representation.- 1. The Jost-Lehmann-Dyson representation in the symmetric case.- 2. Inversion of the Jost-Lehmann-Dyson representation in the symmetric case.- 3. The Jost-Lehmann-Dyson representation in the general case.- 14. Automodel asymptotics for the causal functions and singularities of their Fourier transforms on the light cone.- 1. Some preliminary results and definitions.- 2. The main theorems.- 3. On forbidden asymptotics in the Bjorken domain.- 4. Asymptotic properties of the two-point Wightman function.- Comments on Chapter 5.

Fourier series direct method for variational problems

Abstract: A direct method for solving variational problems using Fourier series is presented. An operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations. Illustrative examples are also given.

Multiphase Averaging for Classical Systems: With Applications to Adiabatic Theorems

Abstract: 1 Introduction and Notation.- 1.1 Introduction.- 1.2 Notation.- 2 Ergodicity.- 2.1 Anosov's result.- 2.2 Method of proof.- 2.3 Proof of Lemma 1.- 2.4 Proof of Lemma 2.- 3 On Frequency Systems and First Result for Two Frequency Systems.- 3.1 One frequency introduction and first order estimates.- 3.2 Increasing the precision higher order results.- 3.3 Extending the time-scale geometry enters.- 3.4 Resonance a first encounter.- 3.5 Two frequency systems Arnold's result.- 3.6 Preliminary lemmas.- 3.7 Proof of Arnold's theorem.- 4 Two Frequency Systems Neistadt's Results.- 4.1 Outline of the problem and results.- 4.2 Decomposition of the domain and resonant normal forms.- 4.3 Passage through resonance: the pendulum model.- 4.4 Excluded initial conditions, maximal separation, average separation.- 4.5 Optimality of the results.- 4.6 The case of a one-dimensional base.- 5 N Frequency Systems Neistadt's Result Based on Anosov's Method.- 5.1 Introduction and results.- 5.2 Proof of the theorem.- 5.3 Proof for the differentiable case.- 6 N Frequency Systems Neistadt's Results Based on Kasuga's Method.- 6.1 Statement of the theorems.- 6.2 Proof of Theorem 1.- 6.3 Optimality of the results of Theorem 1.- 6.4 Optimality of the results of Theorem 2.- 7 Hamiltonian Systems.- 7.1 General introduction.- 7.2 The KAM theorem.- 7.3 Nekhoroshev's theorem introduction and statement of the theorem.- 7.4 Analytic part of the proof.- 7.5 Geometric part and end of the proof.- 8 Adiabatic Theorems in One Dimension.- 8.1 Adiabatic invariance definition and examples.- 8.2 Adiabatic series.- 8.3 The harmonic oscillator adiabatic invariance and parametric resonance.- 8.4 The harmonic oscillator drift of the action.- 8.5 Drift of the action for general systems.- 8.6 Perpetual stability of nonlinear periodic systems.- 9 The Classical Adiabatic Theorems in Many Dimensions.- 9.1 Invariance of action, invariance of volume.- 9.2 An adiabatic theorem for integrable systems.- 9.3 The behavior of the angle variables.- 9.4 The ergodic adiabatic theorem.- 10 The Quantum Adiabatic Theorem.- 10.1 Statement and proof of the theorem.- 10.2 The analogy between classical and quantum theorems.- 10.3 Adiabatic behavior of the quantum phase.- 10.4 Classical angles and quantum phase.- 10.5 Non-communtativity of adiabatic and semiclassical limits.- Appendix 1 Fourier Series.- Appendix 2 Ergodicity.- Appendix 3 Resonance.- Appendix 4 Diophantine Approximations.- Appendix 5 Normal Forms.- Appendix 6 Generating Functions.- Appendix 7 Lie Series.- Appendix 8 Hamiltonian Normal Forms.- Appendix 9 Steepness.- Bibliographical Notes.

Radiative Transfer through Arbitrarily Shaped Optical Media. Part I: A General Method of Solution

Abstract: A general transform method is presented for studying problems of radiative transfer through absorbing, emitting and anisotropically scattering media exposed to arbitrary radiation conditions on its boundaries. The method permits quite arbitrary horizontal and vertical variability in the scattering and extinction properties of the medium bounded by a surface whose albedo and bidirectional reflection function varies from point to point. The technique developed incorporates a two-dimensional Fourier transform of the radiative transfer equation and a full Fourier expansion in azimuth. The general solution is based on the use of invariant imbedding principles in the form of doubling and adding algorithms. In developing these algorithms the principles of invariance are derived for three-dimensional geometry. Differences and similarities to the one-dimensional transfer problem are highlighted throughout. The method is applied to two special problems, namely the reflection by an atmosphere overlying or s...

Lectures on Integral Transforms

Abstract: Averaging operators and the Bochner theorem The Fourier transform in $L^1$ The inversion theorem in $L^1$. The Poisson integral Harmonic functions. The Dirichlet problem for a ball and a half-space The Fourier transform in $L^2$ Hermite functions Spherical functions Positive definite functions The Hankel transform Orthogonal polynomials and the moment problem The class $H^2$. The Paley-Wiener theorem Boundary properties of functions analytic in the upper half-plane and the Hilbert transform The Poisson summation formula and some of its applications Applications of the Laplace and Fourier transforms to the solution of boundary value problems in mathematical physics Fourier transforms of increasing functions. The Wiener-Hopf technique.

Surface waves of large amplitude beneath an elastic sheet. part 2. galerkin solution

Abstract: This study continues the work of Forbes (1986) on periodic waves beneath an elastic sheet floating on the surface of an infinitely deep fluid. The solution is sought as a Fourier series with coefficients that are computed numerically. Waves of extremely large amplitude are found to exist, and results are presented for waves belonging to several different nonlinear solution branches, characterized by different numbers of inflexion points in the wave profiles. The existence of multiple solutions, conjectured in the previous paper (Forbes 1986), is confirmed here by direct numerical computation.

Current Source Density Estimation and Interpolation Based on the Spherical Harmonic Fourier Expansion

Abstract: A method for the spatial analysis of EEG and EP data, based on the spherical harmonic Fourier expansion (SHE) of scalp potential measurements, is described. This model provides efficient and accurate formulas for: (1) the computation of the surface Laplacian and (2) the interpolation of electrical potentials, current source densities, test statistics and other derived variables. Physiologically based simulation experiments show that the SHE method gives better estimates of the surface Laplacian than the commonly used finite difference method. Cross-validation studies for the objective comparison of different interpolation methods demonstrate the superiority of the SHE over the commonly used methods based on the weighted (inverse distance) average of the nearest three and four neighbor values.

Classical Fourier Transforms

Abstract: I. Fourier transforms on L1 (-?,?).- 1. Basic properties and examples.- 2. The L1 -algebra.- 3. Differentiability properties.- 4. Localization, Mellin transforms.- 5. Fourier series and Poisson's summation formula.- 6. The uniqueness theorem.- 7. Pointwise summability.- 8. The inversion formula.- 9. Summability in the L1-norm.- 10. The central limit theorem.- 11. Analytic functions of Fourier transforms.- 12. The closure of translations.- 13. A general tauberian theorem.- 14. Two differential equations.- 15. Several variables.- II. Fourier transforms on L2(-?,?).- 1. Introduction.- 2. Plancherel's theorem.- 3. Convergence and summability.- 4. The closure of translations.- 5. Heisenberg's inequality.- 6. Hardy's theorem.- 7. The theorem of Paley and Wiener.- 8. Fourier series in L2(a,b).- 9. Hardy's interpolation formula.- 10. Two inequalities of S. Bernstein.- 11. Several variables.- III. Fourier-Stieltjes transforms (one variable).- 1. Basic properties.- 2. Distribution functions, and characteristic functions.- 3. Positive-definite functions.- 4. A uniqueness theorem.- Notes.- References.

Identification of Nonlinear Multi-Degree-of-Freedom Systems : Presentation of an Identification Technique

Abstract: A new technique of identifying nonlinear multi-degree-of-freedom systems has been presented The basic procedures of this technique are : (1) obtaining the data of a periodic external force applied to the system and of the periodic steady-state response induced by it ; (2) expressing the nonlinear terms of the system in the form of polynomials with unknown coefficients ; and (3) determining the unknown coefficients by expressing the necessary quantities in Fourier series and applying the principle of harmonic balance Some examples are given to show the applicability of the technique presented

Symbolic matrix inversion with application to electronic circuits

Abstract: The problem of inverting matrices that contain as entries polynomials in several variables is considered. A method of inversion based on the multidimensional discrete Fourier transform of matrix sequences is developed. The method is particularly effective for a moderate number of symbolic variables. An example is given to illustrate its application to an electron amplifier. >

A Hankel transform approach to tomographic image reconstruction

Abstract: A relatively unexplored algorithm is developed for reconstructing a two-dimensional image from a finite set of its sampled projections. The algorithm, referred to as the Hankel-transform-reconstruction (HTR) algorithm, is polar-coordinate based. The algorithm expands the polar-form Fourier transform F(r, theta ) of an image into a Fourier series in theta ; calculates the appropriately ordered Hankel transform of the coefficients of this series, giving the coefficients for the Fourier series of the polar-form image f(p, phi ); resolves this series, giving a polar-form reconstruction; and interpolates this reconstruction to a rectilinear grid. The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm. >

The use of the FFT for the efficient solution of the problem of electromagnetic scattering by a body of revolution

Abstract: The authors consider the further enhancement of the computational efficiency in the electromagnetic scattering of large bodies of revolution (BOR) with a view to making it practical to solve large-body problems. The problem of the electromagnetic scattering of a perfect electrical conducting (PEC) BOR is considered, although the methods provided can be applied to multilayered dielectric bodies. In most methods used, the generation of the elements for the method-of-moments matrix consumes a major portion of the computational time. It is shown how this time can be significantly reduced by manipulating the expression for the matrix elements using the fast Fourier transform (FFT). A technique is also presented for extracting the singularity of the Green's function that appears within the integrands of the matrix diagonal. The extraction further enhances the use of the FFT and provides an exact analytic expression. Using this method, the computational time can be improved by at least an order of magnitude for large bodies, compared to existing algorithms. >

Detecting change points by Fourier analysis

Abstract: The cumulative sum (CUSUM) is a basic diagnostic tool in the analysis of change-point data. It is shown that Fourier analysis of the CUSUM can be a useful supplementary tool in such analyses. The technique is applied to three data sets that have appeared previously in the statistical literature.

Solar thermal storage systems using phase change materials

Abstract: Using Fourier series expansion of the involving temperatures and the forcing parameters i.e. the solar radiation and the ambient temperature, an iterative procedure has been developed to solve the heat transfer problem with moving boundaries. Calculations specific to a typical summer and winter day in Delhi have been presented for a numerical appreciation of the developed analysis. Experiments have been performed to validate the developed theoretical analysis. A good agreement is seen between theoretical and experimental results with in the domain of the applicability of theory.

On Fourier coefficients of Siegel modular forms.

Abstract: One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$ .

Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion

Abstract: Previously it was shown that one can solve the phase-retrieval problem from two intensities observed at the Fourier-transform plane of an object in one dimension by using the Fourier-series expansion. In this paper, an improved method using the logarithmic Hilbert transform and the Fourier series expansion is proposed. It is proved from the distribution of zeros in the complex plane that the Fourier-transform phase of Hermitian object functions cannot be retrieved by using the previous method but can be retrieved by using the method in this paper. The results reconstructed by the present method are also shown in computer simulations.

Chebyshev domain truncation is inferior to Fourier domain truncation for solving problems on an infinite interval

Abstract: “Domain truncation” is the simple strategy of solving problems onye [-∞, ∞] by using a large but finite computational interval, [− L, L] Sinceu(y) is not a periodic function, spectral methods have usually employed a basis of Chebyshev polynomials,T n(y/L). In this note, we show that becauseu(±L) must be very, very small if domain truncation is to succeed, it is always more efficient to apply a Fourier expansion instead. Roughly speaking, it requires about 100 Chebyshev polynomials to achieve the same accuracy as 64 Fourier terms. The Fourier expansion of a rapidly decaying but nonperiodic function on a large interval is also a dramatic illustration of the care that is necessary in applying asymptotic coefficient analysis. The behavior of the Fourier coefficients in the limitn→∞ for fixed intervalL isnever relevant or significant in this application.