Series Expansion Of Functions Research Articles

Analog of the classical borel theorem for entire harmonic functions in ℝn and generalized orders

The article describes research on the growth of functions that are harmonic in the whole space ℝ n , n≥3, and thus they are called entire harmonic. A relation has been established between the maximum terms of entire functions of finite order in the plane, which are given by power series whose coefficients are somewhat connected. Also, the maximum modulus of a harmonic function in the space ℝ n is evaluated through the maximum modulus of some entire function in the plane, the coefficients which are expressed in terms of the coefficients of the expansion of the harmonic function in a series by Laplace spherical functions. These results made it possible to obtain an analog of the classical Borel theorem for entire harmonic functions of finite order in ℝ n . Besides, the study has revealed the most general characteristics of the growth of entire harmonic functions in ℝ n in terms of the uniform norm of Laplace spherical functions in the expansion of harmonic functions in series. Slow growth of the harmonic functions in the space has also been studied. The obtained results are analogous to the classical results that are known for entire functions of one complex variable. The research findings are important because harmonic functions occupy a special place not only in many mathematical studies but also in the application of mathematical analysis to physics and mechanics, where these functions often describe various stationary processes.

An Efficient Approximation of Spatial Correlation Based on Gauss–Hermite Quadrature

An efficient spatial correlation approximation for clustered multiple-input multiple-output channels based on a hierarchical angle structure is presented. To avoid complicated numerical integrations, different approximations have been proposed in the literature. One approximation is expressed in a high-order series expansion of Bessel functions, while the other suffers low accuracy for large angle spread. An approximation based on Gauss–Hermite quadrature (GHQ) is proposed, whose values are shown to match well with theoretical results when the antenna spacing is within a critical range. By exploring the properties of the abscissas and the weights of the GHQ, the proposed approximation can be effectively computed with only a few expansion terms, and its validity can be maintained over a large range of angle spread. The approximation errors of the proposed and existing methods are analyzed. Simulation results verify the accuracy of the analysis and the efficiency of the proposed approximation.

Exact spatial density of ideal Bose atoms in a two-dimensional isotropic harmonic trap

We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional (2D), and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series expansion of q-digamma functions in mathematics. We employ the analytical solution to calculate the spatial density of ideal Bose atoms in a 2D isotropic harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits a 2D dark spot in the center and a bright ring outside. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a 2D sharp peak with an extremely small radius. The third main finding is that the central density is a monotonically increasing function of the number of atoms N, but is a monotonically decreasing function of temperature T. The fourth main finding is that the number density of atoms of condensate exhibits a 2D Gaussian distribution in the center while the number density of atoms of vapor exhibits a 2D bright annulus with zero density at the origin.

Nonperturbative Series Expansion of Green's Functions: The Anatomy of Resonant Inelastic X-Ray Scattering in the Doped Hubbard Model.

We present a nonperturbative, divergence-free series expansion of Green's functions using effective operators. The method is especially suited for computing correlators of complex operators as a series of correlation functions of simpler forms. We apply the method to study low-energy excitations in resonant inelastic x-ray scattering (RIXS) in doped one- and two-dimensional single-band Hubbard models. The RIXS operator is expanded into polynomials of spin, density, and current operators weighted by fundamental x-ray spectral functions. These operators couple to different polarization channels resulting in simple selection rules. The incident photon energy dependent coefficients help to pinpoint main RIXS contributions from different degrees of freedom. We show in particular that, with parameters pertaining to cuprate superconductors, local spin excitation dominates the RIXS spectral weight over a wide doping range in the cross-polarization channel.

Analytical model of cracking due to rebar corrosion expansion in concrete considering the structure internal force

Based on the assumptions of uniform corrosion and linear elastic expansion, an analytical model of cracking due to rebar corrosion expansion in concrete was established, which is able to consider the structure internal force. And then, by means of the complex variable function theory and series expansion technology established by Muskhelishvili, the corresponding stress component functions of concrete around the reinforcement were obtained. Also, a comparative analysis was conducted between the numerical simulation model and present model in this paper. The results show that the calculation results of both methods were consistent with each other, and the numerical deviation was less than 10%, proving that the analytical model established in this paper is reliable.

Nonlinear PTO Effect on Performance of Vertical Axisymmetric Wave Energy Converter Using Semi-Analytical Method

Abstract The wave energy, as a clean and non-pollution renewable energy sources, has become a hot research topic at home and abroad and is likely to become a new industry in the future. In this article, to effectively extract and maximize the energy from ocean waves, a vertical axisymmetric wave energy converter (WEC) was presented according to investigating of the advantages and disadvantages of the current WEC. The linear and quadratic equations in frequency-domain for the reactive controlled single-point converter property under regular waves condition are proposed for an efficient power take-off (PTO). A method of damping coefficients, theoretical added mass and exciting force are calculated with the analytical method which is in use of the series expansion of eigen functions. The loads of optimal reactive and resistive, the amplitudes of corresponding oscillation, and the width ratios of energy capture are determined approximately and discussed in numerical results.

Solutions of Boundary-Value Problems for the Helmholtz Equation in Simply Connected Domains of the Complex Plane

The bases in the spaces of functions analytic in simply connected domains are constructed with the help of conformal mappings of these domains onto a circle. The obtained basis functions are biorthogonal to the Faber polynomials. By using the expansions of analytic functions in series in systems of basis functions, we determine the solutions of boundary-value problems for the Helmholtz equation whose boundary values coincide with the boundary values of these functions.

On-the-fly Doppler broadening method based on optimal double-exponential formula

As temperature changes constantly in nuclear reactor operation, on-the-fly Doppler broadening methods are commonly adopted for generating nuclear cross sections at various temperatures in neutron transport simulation. Among the existing methods, the widely used SIGMA1 approach is inefficient because it involves error function and Taylor series expansion. In this paper, we present a new on-the-fly Doppler broadening with optimal double-exponential formula based on SuperMC to improve efficiency with given accuracy. In this method, double-exponential formula in 1/16 steps is used for broadening cross section at low energy, with both accuracy and efficiency. Meanwhile, the Gauss–Hermite quadrature of different orders is used for broadening cross section at resonance energy. The method can generate neutron cross section rapidly and precisely at the desired temperature. Typical nuclide cross sections and benchmarking tests are presented in detail.

On a consistent finite-strain plate theory of growth

In this paper, a consistent finite-strain plate theory for growth-induced large deformations is developed. The three-dimensional (3D) governing system of the plate model is formulated through the variational approach, which is composed of the mechanical equilibrium equation and the constraint equation of incompressibility. Then, series expansions of the unknown functions in terms of the thickness variable are adopted. By using the 3D equilibrium equations and the surface boundary conditions, recursion relations for the expansion coefficients are successfully established. As a result, a 2D vector plate equation with three unknowns is obtained and the associated edge boundary conditions are proposed. It can be verified that the plate equation ensures the required asymptotic order for all the terms in the variations of the total energy functional. The weak formulation of the plate equation has also been derived for future numerical calculations. As applications of the plate theory, two examples regarding the growth-induced deformations and instabilities in thin hyperelastic plates are studied. Some analytical results are obtained in these examples, which can be used to describe the large deformations and reveal the bifurcation properties of the thin plates. Furthermore, the results obtained from the current plate theory are compared with those obtained from the classical Föppl-von Kármán plate theory, from which the efficiencies and advantages of the current plate theory can be demonstrated.

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.

ОБОБЩЕНИЕ УНИВЕРСАЛЬНОГО РЯДА ПО МНОГОЧЛЕНАМ ЧЕБЫШњВА

Chebyshev polynomials are widely used in theoretical and practical studies. Recently, they have become more signicant, particularly in quantum chemistry. In research [1] their important properties are described to provide faster convergence of expansions of functions in series of Chebyshev polynomials, compared with their expansion into a power series or in a series of other special polynomials or functions([1], p. 6). In this paper, a result associated with an approximation theory is presented. To some extent, the analogues of this result were obtained from other studies, such as in [2] [4], respectively for the power series, as well as the series in Hermite and Faber polynomials. With regard to the denition of the signicance of the series in Chebyshev polynomials listed above, the result of this research is of particular signicance in contrast to these analogues. More precisely, we can assume that the practical solution to the particular problems, can be solved much faster with the use of Chebyshev polynomials rather than the usage of such amounts related to power series [2] and the series in Hermite polynomials [3]. In addition, it is considered the rst synthesis of the universal series for polynomials with a density of one. The concept of a universal series of functions is associated with the notion of approximation of functions by partial sums of the corresponding rows. In [2] [19] the universal property of certain functional series are reviewed. In [2] [4], [18] a generalization of this property is considered. This paper generalizes the universality series properties in Chebyshev polynomials.

Chromatic derivatives and expansions with weights

Chromatic derivatives and series expansions of bandlimited functions have recently been introduced in signal processing and they have been shown to be useful in practical applications. We extend the notion of chromatic derivative using varying weights. When the kernel function of the integral operator is positive, this extension ensures chromatic expansions around every points. Besides old examples, the modified method is demonstrated via some new ones as Walsh-Fourier transform, and Poisson-wavelet transform. Moreover the chromatic expansion of a function in some $L^p$-space is investigated.

A General Structure of Linear-Phase FIR Filters With Derivative Constraints

In this paper, a general structure of linear-phase finite impulse response filters, whose frequency responses satisfy given derivative constraints imposed upon an arbitrary frequency, is proposed. It is comprised of a linear combination of parallelly connected subfilters, called the cardinal filters , with weighted coefficients being the successive derivatives of the desired frequency response at the constrained frequency. An advantage of such a cardinal filters design is that only the weighted coefficients are relevant to the desired frequency response but not the cardinal filters; hence, a dynamic adjustment of the filter system becomes feasible. The key to derive the coefficients of cardinal filters is the determination of the power series expansion of certain trigonometric-related functions. By showing the elaborately chosen trigonometric-related functions satisfy specific differential equations, recursive formulas for the coefficients of cardinal filters are subsequently established, which make efficient their computations. At last, a simple enhancement of the cardinal filters design by incorporating the mean square error minimization is presented through examples.

Single periodic Riemann problems for null-solutions to polynomially Cauchy–Riemann equations

In this paper, we focus on single periodic Riemann problems for a class of meta-analytic functions, i.e. null-solutions to polynomially Cauchy–Riemann equation. We first establish decomposition theorems for single periodic meta-analytic functions. Then, we give a series expansion of single periodic meta-analytic functions, and derive generalised Liouville theorems for them. Next, we introduce a definition of order for single periodic meta-analytic functions at infinity, and characterise their growth at infinity. Finally, applying the decomposition theorem for single periodic meta-analytic functions, we get explicit expressions of solutions and condition of solvability to Riemann problems for single periodic meta-analytic functions with a finite order at infinity.

СРЕДНЕКВАДРАТИЧЕСКОЕ ПРИБЛИЖЕНИЕ ФУНКЦИЙ РЯДАМИ ФУРЬЕ–БЕССЕЛЯ И ЗНАЧЕНИЯ ПОПЕРЕЧНИКОВ НЕКОТОРЫХ ФУНКЦИОНАЛЬНЫХ КЛАССОВ

It is known that many of the problems of mathematical physics, reduced to a differential equation with partial derivatives written in cylindrical and spherical coordinates, by using method of separation of variables, in particular, leads to the Bessel differential equation and Bessel functions. In practice, especially in problems of electrodynamics, celestial mechanics and modern applied mathematics most commonly used Fourier series in orthogonal systems of special functions. Given this, it is required to determine the conditions of expansion of functions in series into these special functions, forming in a given interval a complete orthogonal system. The work is devoted to obtaining accurate estimates of convergence rate of Fourier series by Bessel system of functions for some classes of functions in a Hilbert space L 2 := L 2 ([0 , 1] , x dx ) of square summable functions f : [0 , 1] → R with the weight x . The exact inequalities of Jackson–Stechkin type on the sets of L2 ( r ) 2 ( D ) , linking En − 1 ( f ) 2 — the best approximation of function f by partial sums of order n − 1 of the Fourier–Bessel series with the averaged positive weight of generalized modulus of continuity of m order Ω m (

Contractive maps on operator ideals and norm inequalities III

Let (I,⦀.⦀) be a norm ideal of operators equipped with a unitarily invariant norm ⦀.⦀. We discuss some generalized Lyapunov type norm inequalities for operators, which are motivated by the work of Bhatia and Drissi [8], Hiai and Kosaki [16] and Jocić [17]. We exploit integral representations and series expansions of certain functions to prove that certain ratios of linear operators acting on operators in I are contractive. This leads to several new and old norm inequalities for operators which were earlier in the matrix settings.

Unsteady Flow of Fluids With Arbitrarily Time-Dependent Rheological Behavior

This paper presents an analytical solution of the momentum equation for the unsteady motion of fluids in circular pipes, in which the kinematic viscosity is allowed to change arbitrarily in time. Velocity and flow rate are expressed as a series expansion of Bessel and Kelvin functions of the radial variable, whereas the dependence on time is expressed as Fourierlike series. The analytical solution for the velocity is compared with the direct numerical solution of the momentum equation in a particular case, verifying that the difference between analytical and numerical values of axial velocity is less than 1%, except near the discontinuity of the applied pressure gradient, where the typical behavior due to the Gibbs phenomenon is to be noted.

Rubber Composition–Properties Relationships during Tire Numerical Simulation and Design Optimization

ABSTRACT Numerical simulation based on finite element analysis is now widely used during the design optimization of tires, thereby drastically reducing the time investment in the design process and improving tire performance because it is obtained from the optimized solution. Rubber material models that are used in numerical calculations of stress–strain distributions are nonlinear and may include several parameters. The relations of these parameters with rubber formulations are usually unknown, so the designer has no information on whether the optimal set of parameters is reachable by the rubber technological possibilities. The aim of this work was to develop such relations. The most common approach to derive the equation of the state of rubber is based on the expansion of the strain energy in a series of invariants of the strain tensor. Here, we show that this approach has several drawbacks, one of which is problems that arise when trying to build on its basis the quantitative relations between the rubber composition and its properties. An alternative is to use a series expansion in orthogonal functions, thereby ensuring the linear independence of the coefficients of elasticity in evaluation of the experimental data and the possibility of constructing continuous maps of “the composition to the property.” In the case of orthogonal Legendre polynomials, the technique for constructing such maps is considered, and a set of empirical functions is proposed to adequately describe the dependence of the parameters of nonlinear elastic properties of general-purpose rubbers on the content of the main ingredients. The calculated sets of parameters were used in numerical tire simulations including static loading, footprint analysis, braking/acceleration, and cornering and also in design optimization procedures.

Holomorphic series expansion of functions of Carleman type on the interval [−1, 1

We characterize the functions of some Carleman classes on the unit interval [?1,1] as sums of holomorphic functions in specific neighborhoods of [?1,1]. As an application of our main theorem, we perform an alternative construction of the Dyn?kin?s pseudoanalytic extension for these Carleman classes on [?1,1].

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

Let $\mathcal{H}_\infty$ be the set of all ordinary Dirichlet series $D=\sum_n a_n n^{-s}$ representing bounded holomorphic functions on the right half plane. A multiplicative sequence $(b_n)$ of complex numbers is said to be an $\ell_1$-multiplier for $\mathcal{H}_\infty$ whenever $\sum_n |a_n b_n| < \infty$ for every $D \in \mathcal{H}_\infty$. We study the problem of describing such sequences $(b_n)$ in terms of the asymptotic decay of the subsequence $(b_{p_j})$, where $p_j$ denotes the $j$th prime number. Given a multiplicative sequence $b=(b_n)$ we prove (among other results): $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$ provided $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} < 1$, and conversely, if $b$ is an $\ell_1$-multiplier for $\mathcal{H}_\infty$, then $|b_{p_j}| < 1$ for all $j$ and $\overline{\lim}_n \frac{1}{\log n} \sum_{j=1}^n b_{p_j}^{*2} \leq 1$ (here $b^*$ stands for the decreasing rearrangement of $b$). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences $z$ in the infinite dimensional polydisk $\mathbb{D}^\infty$ (the open unit ball of $\ell_\infty$) for which every bounded and holomorphic function $f$ on $\mathbb{D}^\infty$ has an absolutely convergent monomial series expansion $\sum_{\alpha} \frac{\partial_\alpha f(0)}{\alpha!} z^\alpha$. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus $\mathbb{T}^\infty$.