Terms Of Fourier Coefficients Research Articles

Approximation in periodic Gevrey spaces

The well-known Gevrey classes consist of C∞-functions on Rd whose derivatives satisfy certain growth conditions. For periodic functions these conditions can be expressed in terms of Fourier coefficients. Motivated by this observation, we introduce Gevrey spaces on the d-dimensional torus and study approximation numbers of their embeddings into L2. Our special emphasis is on the dependence on the dimension of the underlying domain, which is an important aspect in the numerical treatment of high-dimensional problems. In particular, we determine the exact rate of decay of the approximation numbers, together with optimal asymptotic constants, and establish preasymptotic estimates. Finally we translate our findings into the language of tractability notions from information-based complexity.

Functions with general monotone Fourier coefficients

This paper is a study of trigonometric series with general monotone coefficients in the class with . Sharp estimates are proved for the Fourier coefficients of integrable and continuous functions. Also obtained are optimal results in terms of coefficients for various types of convergence of Fourier series. For two-sided estimates are obtained for the -moduli of smoothness of sums of series with -coefficients, as well as for the (quasi-)norms of such sums in Lebesgue, Lorentz, Besov, and Sobolev spaces in terms of Fourier coefficients. Bibliography: 99 titles.

Weighted holomorphic mixed norm spaces in the unit disc defined in terms of Fourier coefficients

ABSTRACT Following the ideas of recent research in Karapetyants and Samko [Mixed norm variable exponent Bergman space on the unit disc. Complex Var Elliptic Equ. 2016;61:1090–1106], we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present a general approach for the weighted case. We study the boundedness of the weighted Bergman projection and discuss boundedness and compactness of Toeplitz operators with radial symbols. We also provide a characterization of functions in these spaces.

Band gap optimization of one-dimension elastic waveguides using spatial Fourier plane wave expansion coefficients

Periodic elastic waveguides, such as rods, beams, and shafts, exhibit frequency bands where wave reflections at impedance discontinuities cause strong wave attenuation by Bragg scattering. Such frequency bands are known as stop bands or band gaps. This work presents a shape optimization technique for one-dimensional periodic structures. The proposed approach, which aims to maximize the width of the first band gap, uses as tuning parameters the spatial Fourier coefficients that describe the shape of the cell cross-section variation along its length. Since the optimization problem is formulated in terms of Fourier coefficients, it can be directly applied to the Plane Wave Expansion (PWE) method, commonly used to obtain the dispersion diagrams, which indicate the presence of band gaps. The proposed technique is used to optimize the shape of a straight bar with both solid and hollow circular cross-sections. First, the optimization is performed using the elementary rod, the Euler-Bernoulli and Timoshenko beam, and the shaft theoretical models in an independent way. Then, the optimization is conducted to obtain a complete band gap in the dispersion diagrams, which includes the three wave types, i.e., longitudinal, bending, and torsional. All numerical results provided feasible shapes that generate wide stop bands in the dispersion diagrams. The proposed technique can be extended to two- and three-dimensional periodic frame structures, and can also be adapted for different classes of cost functions.

A modal boundary element method for axisymmetric acoustic problems in an arbitrary uniform mean flow

This paper presents a new numerical analysis approach based on an improved Modal Boundary Element Method (MBEM) formulation for axisymmetric acoustic radiation and propagation problems in a uniform mean flow of arbitrary direction. It is based on the homogeneous Modal Convected Helmholtz Equation (MCHE) and its convected Green’s kernel using a Fourier transform method. In order to simplify the flow terms, a general modal boundary integral solution is formulated explicitly according to two new operators such as the particular and convected kernels. Through the use of modified operators, the improved MBEM approach with flow takes a convective form of the general MBEM approach and has a similar form of the nonflow MBEM formulation. The reference and reduced Helmholtz Integral Equations (HIEs) are implicitly taken into account a new nonreflecting Sommerfeld condition to solve far field axisymmetric regions in a uniform mean flow. For isolating the singular integrations, the modal convected Green’s kernel and its modified normal derivative are performed partly analytically in terms of Laplace coefficients and partly numerically in terms of Fourier coefficients. These coefficients are computed by recursion schemes and Gauss-Legendre quadrature standard formulae. Specifically, standard forms of the free term and its convected angle resulting from the singular integrals can be expressed only in terms of real angles in meridian plane. To demonstrate the application of the improved MBEM formulation, three exterior acoustic case studies are considered. These verification cases are based on new analytic formulations for axisymmetric acoustic sources, such as axisymmetric monopole, axial and radial dipole sources in the presence of an arbitrary uniform mean flow. Directivity plots obtained using the proposed technique are compared with the analytical results.

Comprehensive Analysis and Optimized Control of Torque Ripple and Power Factor in a Three-Phase Mutually Coupled Switched Reluctance Motor With Sinusoidal Current Excitation

This article presents a comprehensive analysis of torque ripple and power factor for three-phase mutually coupled switched reluctance motors (MCSRMs) with sinusoidal current excitation. MCSRMs controlled by sinusoidal currents have the advantage of using the conventional voltage source inverter and the conventional vector control. However, MCSRMs are characterized by high torque ripple and low power factor. The torque harmonics due to the sinusoidal current excitation are investigated, and the effect of the current excitation angle on the motor saturation level and the power factor is analyzed. These analyses are then used in the development of an optimized control method to reduce torque ripple and to increase power factor and average torque. In the proposed control method, the power factor and phase voltage are calculated by the knowledge of the phase flux linkage and phase current. The phase flux linkage is represented in terms of Fourier coefficients, where these coefficients are functions of direct- and quadrature-axis currents. Hence, they are estimated from 2-D lookup tables (LUTs), which are independent of rotor position. Similarly, the Fourier coefficients of the torque harmonics are also estimated from the 2-D LUTs. The independence of the LUTs from rotor position reduces the size of the LUTs significantly. The proposed control method is validated by experiments on a 2-kW 12/8 MCSRM.

Sufficient Conditions for a Multidimensional System of Periodic Wavelets to be a Frame

We study multidimensional periodic wavelet systems with matrix dilations. We obtain conditions sufficient for such a system to be Bessel. The conditions are given in terms of Fourier coefficients. We propose a method for constructing a wavelet Riesz basis that starts with a suitable sequence of trigonometric polynomials.

Model-Based Spatial Harmonics Vector Compensation Method for Three-Phase Mutually Coupled Switched Reluctance Machine With Sinusoidal Current Excitation

This paper presents a spatial harmonics compensation method for mutually coupled switched reluctance machines (MCSRMs) with sinusoidal current excitation. The regulation by linear controllers to achieve sinusoidal currents in MCSRMs is challenging due to the substantial presence of spatial harmonics. The standard vector control with using Proportional-Integral controllers cannot effectively suppress the spatial harmonics of the current waveform due to the bandwidth limitations. In the proposed method, the essential voltage harmonics are added to the fundamental voltage component to create sinusoidal currents. The voltage harmonics are calculated from the flux linkage harmonics where voltage and flux linkage harmonics are represented as vectors in terms of Fourier coefficients. The Fourier coefficients of the flux linkage are function of direct-and quadrature-axis currents. Hence, they are in the form of two-dimensional look-up tables (LUTs). The used LUTs in the proposed method are independent of rotor position and they are obtained from the finite element analysis (FEA) model. The proposed spatial harmonics compensation method is validated using FEA and experiments for a 3-phase 12/8 MCSRM at generating and motoring modes of operation.

Comet 67P/Churyumov-Gerasimenko rotation changes derived from sublimation-induced torques

Context.The change in rotation period and the orientation of the rotation axis of comet 67P/Churyumov-Gerasimenko (67P) can be deduced with high precision from images taken by the scientific imaging instruments on board the Rosetta mission. Non-gravitational forces are a natural explanation for these data.Aims.We describe observed changes in orientation of the rotation axis and the rotation period of 67P. We explain them based on a sublimation model with a best fit for the surface active fraction (model P). Torque effects of periodically changing gas emissions on the surface are considered.Methods.We solved the equation of state for the angular momentum in the inertial and the body-fixed frames and provide an analytic theory of the rotation changes in terms of Fourier coefficients, which are generally applicable to periodically forced rigid-body dynamics.Results.The torque-induced changes in rotation state constrain the physical properties of the surface, the sublimation rate, and the local active fraction of the surface.Conclusions.We determine a distribution of the local surface active fraction in agreement with the rotation properties, period, and orientation of 67P. The torque movement confirms that the sublimation increases faster than the insolation toward perihelion. The derived relatively uniform activity pattern is discussed in terms of related surface features.

Fejér Sums and Fourier Coefficients of Periodic Measures

The Fejer sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejer sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejer sums at any two points whose difference is incommensurable with π.

On mixed norm Bergman–Orlicz–Morrey spaces

Abstract Following the ideas of our previous research, in this paper we continue the study of new Bergman-type spaces on the unit disc with mixed norm in terms of Fourier coefficients. Here we deal with the case where the sequence of norms of Fourier coefficients in the Orlicz–Morrey space in radial variable belongs to l q {l^{q}} . We study the boundedness of the Bergman projection and provide a description of functions in these spaces via the behavior of their Taylor coefficients.

An analytical plane-strain solution for surface uplift due to pressurized reservoirs

In this paper, we present an analytical plane strain solution for surface uplift above pressurized reservoirs. The solution is based on a Fourier representation of the reservoir pressure. The plane strain model is developed in two stages: First, an exact solution is derived for the displacement field for the reservoir alone subjected to a periodic overpressure distribution of one wavelength. This one-layer model forms the basis for the analytical plane strain solution for a two-layer model — a pressurized reservoir with an overburden. We give an example where numerically computed uplift is quite accurately estimated by a simple 1D estimate, except for in the near well area. The plane-strain solution is well suited to study conditions for when the simple 1D approximation of the uplift is accurate. A condition for the accuracy of the simple 1D approximation is first derived for just the reservoir expanded by a periodic overpressure distribution of one wavelength, which corresponds to one term in a Fourier series. The 1D estimate is accurate for wavelengths larger than 2π times the reservoir thickness. Then, a condition is derived for when the 1D estimate is accurate for the two-layer model. We show that the wavelength of the overpressure distribution must be larger than 2π times the maximum of the reservoir thickness and the overburden thickness for the 1D approximation to be accurate. We demonstrate how uplift is computed from a Fourier decomposition of the reservoir overpressure. The resulting uplift is analysed in terms of Fourier coefficients, using the knowledge of how a single wavelength behaves. The analytical results for the displacement field and the uplift are tested by comparison with finite element simulations, and the match is excellent.

On holomorphic projection for symplectic groups

We construct certain Casimir operators and study the spectral properties of their resolvents on L2(Γ\\Sp2(R)). We define non-holomorphic multi-variable Poincaré series of exponential type for symplectic groups and continue them analytically in case of genus two for the small weight four using the above resolvents. We apply our results to describe the holomorphic projection to the weight four holomorphic discrete series in terms of Fourier coefficients by using Sturm's operator. This paper is a study of a prototype for symplectic groups and special orthogonal groups.

Exploring limit behaviour of non-quadratic terms via H-measures. Application to small amplitude homogenisation

A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers of functions by means of H-measures. It is applied to the small amplitude homogenisation problem for a stationary diffusion equation, in which coefficients are assumed to be analytic perturbations of a constant, enabling formulæ for higher order correction terms in a general, non-periodic setting. Explicit expressions in terms of Fourier coefficients are obtained under periodicity assumption. The method allows of its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems.

On the absolute convergence of Fourier series and generalisation of Lipschitz spaces, defined by differences of fractional order

We obtain the necessary and sufficient conditions in terms of Fourier coefficients of $2\pi$-periodic functions $f$ with absolutely convergent Fourier series, for $f$ to belong to the generalized Lipschitz classes $H^{\omega, \alpha}_{\mathbb{C}}$, and to have the fractional derivative of order $\alpha$ ($0 < \alpha < 1$).

Parameterized data-driven fuzzy model based optimal control of a semi-batch reactor

A parameterized data-driven fuzzy (PDDF) model structure is proposed for semi-batch processes, and its application for optimal control is illustrated. The orthonormally parameterized input trajectories, initial states and process parameters are the inputs to the model, which predicts the output trajectories in terms of Fourier coefficients. Fuzzy rules are formulated based on the signs of a linear data-driven model, while the defuzzification step incorporates a linear regression model to shift the domain from input to output domain. The fuzzy model is employed to formulate an optimal control problem for single rate as well as multi-rate systems. Simulation study on a multivariable semi-batch reactor system reveals that the proposed PDDF modeling approach is capable of capturing the nonlinear and time-varying behavior inherent in the semi-batch system fairly accurately, and the results of operating trajectory optimization using the proposed model are found to be comparable to the results obtained using the exact first principles model, and are also found to be comparable to or better than parameterized data-driven artificial neural network model based optimization results.

Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system must be assumed. In the case of transverse isotropic media with a horizontal axis of symmetry, the P-wave reflectivity linearized approximation may be rewritten in terms of azimuthal FCs with the magnitude and phase of the different FCs corresponding to traditional AVAz attributes. Linear slip theory is used to show that the FCs can be interpreted similarly for the cases of a single set of parallel vertical fractures in isotropic media and in transverse isotropic media with a vertical axis of symmetry (VTI). The magnitude of the FCs depends on the fracture weakness parameters and the background media. For the case of vertical fractures in a VTI background, the AVOAz inverse problem is underdetermined, so extra information must be incorporated to determine how the weights are modified due to this background anisotropy. We evaluated this on a 3D data set from northwest Louisiana for which the main target was the Haynesville shale.

Recovery of the wave profile for irrotational periodic water waves from pressure measurements

Using a conformal hodograph transform and the governing equations for periodic traveling irrotational water waves, we obtain a relation between the bottom pressure and the free surface, expressed in terms of Fourier coefficients. This permits us to perform a formal asymptotic analysis that yields an approximate solution for the free surface provided the pressure at the flat bed is known.

The Weighted L 1-integrability of functions and the parseval equality with respect to multiplicative systems

In this paper we prove necessary and sufficient conditions for the weighted L1-integrability of functions defined on [0, 1) in terms of Fourier coefficients with respect to a multiplicative system of bounded type. These results are counterparts of trigonometric ones obtained by M. and S. Izumi and M. M. Robertson.

Nonparametric tests for stochastic ordering

We present two new tests for stochastic ordering in a standard two-sample scheme. We approach the problem via its reparametrization in terms of Fourier coefficients in some corresponding system of functions and combining the resulting empirical Fourier coefficients. The empirical Fourier coefficients can be seen to be the asymptotically optimal linear rank statistics for the local sequences of nonparametric alternatives related to the introduced system of functions. Therefore, our first construction of the test is via multiple testing. The second test is based on sum of squares of censored empirical Fourier coefficients with the number of summands determined via a new model selection rule. The selection rule is fully automatic. Extensive simulations show that the new solutions improve upon existing tests based on adjusted variants of classical Kolmogorov–Smirnov, Anderson–Darling and L 1-distance-based statistics, among others. We show that both tests control the error of the first kind for any fixed sample sizes and are capable of detecting essentially any alternative as the sample sizes are growing to infinity. We also discuss several aspects of our constructions, including possible efficiency calculations and asymptotic comparisons.