Fourier Sums Research Articles

3D Image Reconstructions and the Nyquist–Shannon Theorem

Fracture surfaces are occasionally modelled by Fourier’s two-dimensional series that can be converted into digital 3D reliefs mapping the morphology of solid surfaces. Such digital replicas may suffer from various artefacts when processed inconveniently. Spatial aliasing is one of those artefacts that may devalue Fourier’s replicas. According to the Nyquist–Shannon sampling theorem the spatial aliasing occurs when Fourier’s frequencies exceed the Nyquist critical frequency. In the present paper it is shown that the Nyquist frequency is not the only critical limit determining aliasing artefacts but there are some other frequencies that intensify aliasing phenomena and form an infinite set of points at which numerical results abruptly and dramatically change their values. This unusual type of spatial aliasing is explored and some consequences for 3D computer reconstructions are presented. Fourier’s replicas of scanned surfaces correctly reproduce all morphological features only if the number N of harmonic terms in the Fourier sum does not exceed a critical value derived from the Nyquist frequency of the scanned original. Incorporating harmonic terms of higher frequencies in the Fourier sum, an abrupt increase of aliasing effects occurs whenever the frequencies in the Fourier sum reach even multiples of the Nyquist frequency. Illustration N = 50 represents a correct reproduction whereas illustrations N = 228 and N = 440 show critical abrupt increases of aliasing artifacts.

Order Estimates for the Best Approximations and Approximations by Fourier Sums in the Classes of Convolutions of Periodic Functions of Low Smoothness in the Uniform Metric

We obtain the exact-order estimates for the best uniform approximations and uniform approximations by Fourier sums in the classes of convolutions of periodic functions from the unit balls of the spaces L p , 1 ≤ p < ∞, with generating kernel Ψ β for which the absolute values of its Fourier coefficients ψ(k) are such that ∑ = 1 ∞ ψ p ′(k)k p ′ − 2 < ∞, $$ \frac{1}{p}+\frac{1}{p^{\prime }}=1 $$ , and the product ψ(n)n 1/p cannot tend to zero faster than power functions.

Approximation properties of Fejér- and de la Valleé-Poussin-type means for partial sums of a special series in the system

This paper is concerned with series of the form where is an even -periodic function with finite values and , Series of this type appear as a particular case of more general special series in ultraspherical Jacobi polynomials, which were first introduced and studied by the author. Partial sums of the form are shown to have a number of important properties, which give them an advantage over trigonometric Fourier sums of the form . Approximation properties of Fejér- and de la Valleé-Poussin-type means for the partial sums are studied. Bibliography: 7 titles.

Mismatched Decoding Metrics for Fiber Channels With Nonlinear Phase Noise

Aiming at reducing the complexity of the soft demapper for fiber channels dominated by nonlinear phase noise, we introduce the mismatched metrics $q^{\rm wg}$ and find optimal nonlinear correction factors for $q^{\rm wg} $ . The proposed mismatched metrics use simple arithmetic functions to replace the special functions and Fourier summations in the soft demapper. The numerical results show that the general mutual information (GMI) of $q^{\rm wg} $ can achieve the capacity of bit-interleaved coded modulation by using the good approximations to the probability density function of nonlinear fiber channels. Finally, we present a corrected metric $q^{\rm cr}$ to further improve the performance of $q^{\rm wg} $ in the middle nonlinear region. The numerical results show that the performance of $q^{\rm cr}$ outperforms $q^{\rm wg}$ and Gaussian channel metrics $q^{\rm gau}$ with pre- and post-nonlinear compensation.

New methods in time-resolved Laue pump-probe crystallography at synchrotron sources.

Newly developed methods for time-resolved studies using the polychromatic and in particular the pink-Laue technique, suitable for medium and small-size unit cells typical in chemical crystallography, are reviewed. The order of the sections follows that of a typical study, starting with a description of the pink-Laue technique, followed by the strategy of data collection for analysis with the RATIO method. Novel procedures are described for spot integration, orientation matrix determination for relatively sparse diffraction patterns, scaling of multi-crystal data sets, use of Fourier maps for initial assessment and analysis of results, and least-squares refinement of photo-induced structural and thermal changes. In the calculation of Fourier maps a ground-state structure model, typically based on monochromatic results, is employed as reference, and the laser-ON structure factors for the Fourier summations are obtained by multiplying the reference ground-state structure factors by the square root of the experimental ON/OFF ratios. A schematic of the procedure followed is included in the conclusion section.

Strong Summability of Fourier Transforms at Lebesgue Points and Wiener Amalgam Spaces

We characterize the set of functions for which strong summability holds at each Lebesgue point. More exactly, iffis in the Wiener amalgam spaceW(L1,lq)(R)andfis almost everywhere locally bounded, orf∈W(Lp,lq)(R) (1&lt;p&lt;∞,1≤q&lt;∞), then strongθ-summability holds at each Lebesgue point off. The analogous results are given for Fourier series, too.

Approximations by Fourier Sums on the Sets L ψ L P(∙)

We study some problems of imbedding of the sets of ψ-integrals of the functions f $$ \epsilon $$ L p(∙) and determine the orders of approximations of functions from these sets by Fourier’s sums.

On the norms of the integral means of spherical Fourier sums

The paper deals with the spherical Fourier sums \(S_r (f,x) = \sum\nolimits_{\left\| k \right\| \leqslant r} {\hat f} (k)e^{ik \cdot x}\) of a periodic function f in m variables and the strong integral means of these sums ((∫0R |Sr(f, x)|pdr)/R1/p for p ≥ 1. We establish the exact growth order as R → +∞ of the corresponding operators, i.e., the growth order of the quantities sup|f|≤1((∫0R|Sr(f, 0)|pdr)/R)1/p. The upper and lower bounds differ by their coefficients, which depend only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given.

Fourier Series and Twisted C*-Crossed Products

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C\(^*\)-dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products.

A variation norm Carleson theorem for vector-valued Walsh–Fourier series

We prove a variation norm Carleson theorem for Walsh–Fourier series of functions with values in certain UMD Banach spaces, sharpening a recent result of Hytönen and Lacey. They proved the pointwise convergence of Walsh–Fourier series of X -valued functions under the qualitative hypothesis that X has some finite tile type q&lt;\infty , which holds in particular if X is intermediate between another UMD space and a Hilbert space. Here we relate the precise value of the tile type index to the quantitative rate of convergence: tile type q of X is ‘almost equivalent’ to the L^p -boundedness of the r -variation of the Walsh–Fourier sums of any function f\in L^p([0,1);X) for all r&gt;q and large enough p .

High resolution spherical and ellipsoidal harmonic expansions by Fast Fourier Transform

High resolution transformations between regular geophysical data and harmonic model coefficients can be most efficiently computed by Fast Fourier Transform (FFT). However, a prerequisite is that the data grids are given in the appropriate geometrical domain. For example, if the data are situated on the ellipsoid at equi-angular reduced latitudes, spherical harmonic analysis can be employed and the coefficients subsequently converted by Jekeli’s transformation. This results in the spherical harmonic spectrum in the domain of geocentric latitudes. However, the data are most likely given at geodetic (ellipsoidal) latitudes which means that the FFT base needs to be shifted by latitude dependent phase lags in order to obtain the correct spherical harmonic spectrum. This requires appropriate sample rate conversion about the shifted latitudes by means of Fourier summation and cannot be treated efficiently by an FFT algorithm. In this article another solution is discussed instead. Since the variable heights between the spherical and ellipsoidal surfaces can be accurately approximated by a series of Tschebyshev polynomials, they can be convolved into the spherical basis. It will be shown how this new type of transformation to and from the ellipsoid in combination with Jekeli’s conversion of the spectra between the two surfaces allows eventually the sample rate conversion to shifted latitudes. This avoids the inexpedient Fourier summation mentioned previously. In this paper three applications for FFT in the domain of spherical and ellipsoidal surfaces, and using geocentric, reduced and geodetic latitudes are discussed. The Earth gravitational model EGM2008 of 5 arcminutes resolution has been used to demonstrate numerical results and computational advantages.

On strong means of spherical Fourier sums

On strong means of spherical Fourier sums

Growth estimates for arbitrary sequences of multiple rectangular Fourier sums

Growth estimates are obtained on a set of full measure for arbitrary sequences of rectangular partial sums of multiple trigonometric Fourier series.

Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions

We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes C ψ β, p of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces L p , 1 ≤ p < ∞, with generating fixed kernels Ψβ ⊂ L p′, $ \frac{1}{p}+\frac{1}{{p^{\prime}}}=1 $ , whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the L p -metric, 1 < p ≤ ∞, for the classes L ψ β,1 of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψβ ⊂ L p with functions from the unit ball in the space L 1. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.

Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam.

Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

The approximation characteristics of trigonometric sums Un,pψ of special type on the class Cβ,∞ψ of (ψ,β)-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class Cβ,∞ψ in the uniform metric. The range of parameters in which the sums Un,pψ give the order of the best uniform approximation on the classes Cβ,∞ψ is indicated.

On the convergence of multiple Haar series

We prove that the rectangular and spherical partial sums of the multiple Fourier–Haar series of an individual summable function may behave differently at almost every point, although it is known that they behave in the same way from the point of view of almost-everywhere convergence in the scale of integral classes: is the best class in both cases. We also find optimal additional conditions under which the spherical convergence of a multiple Fourier–Haar series (general Haar series, lacunary series) follows from its convergence by rectangles, and prove that these conditions are indeed optimal.

Numerical Buoyancy-Wave Model for Wave Stress and Drag Simulations in the Atmosphere

AbstractOrographic drag formation is investigated using a numerical wave model (NWM), based on the pressure-coordinate dynamics of non-hydrostatic HIRLAM. The surface drag, wave stress (vertical flux of horizontal momentum), and wave drag are split to the longitudinal and transverse components and presented as Fourier sums of their spectral amplitudes weighted with the power spectrum of relative orographic height. The NWM is accomplished, enabling a spectral investigation of the buoyancy wave stress, and drag generation by orography and is then applied to a cold front, characterised by low static stability of the upper troposphere, large vertical and directional wind variations, and intensive trapped wave generation downstream of obstacles. Resonances are discovered in the stress and drag spectra in the form of high narrow peaks. The stress conservation problem is revisited. Longitudinal stress conserves in unidirectional flow, 2D orography conditions, but becomes convergent for rotating wind or 3D orography. Even in the convergent case the vertical momentum flux from the troposphere to stratosphere remains substantial. The transverse stress never conserves. Disappearing at the surface and on the top, it realises the main momentum exchange between lower an upper parts of the troposphere. Existence of stationary stratospheric quasi-turbulence (SQT) is established above wind minimum in the stratosphere.

On the Lebesgue Inequality on Classes of $ \bar{\psi} $ -Differentiable Functions

We consider the deviations of Fourier sums in the spaces $ {C^{\bar{\psi}}} $ . The estimates of these deviations are expressed via the best approximations of the $ \bar{\psi} $ -derivatives of functions in the Stepanets sense. The sequences $ \bar{\psi} $ = (ψ1, ψ2) are quasiconvex.

Applications of Bivariate Fourier Series for Solving the Poisson Equation in Limited-Area Modeling of the Atmosphere: Higher Accuracy with a Boundary Buffer Strip Discarded and an Improved Order-Raising Procedure

Abstract Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (x, y) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width D, and analyze only the Fourier sums in the smaller domain (x, y) ∈ [D, 1 − D] ⊗ [D, 1 − D]. It is shown that the error in a one-dimensional Fourier series with N terms or in a two-dimensional series with N2 terms, is smaller by a factor of N on a boundary-buffer-discarded domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.