A Novel Pulse Design Based on Hermite Functions for UWB Communications

This paper presents the design of a class of pulses that are based on Hermite functions for ultra-wideband communication systems. The presented class of pulses can not only meet the power spectral emission constraints of federal communications commission, but also have a short duration for multiple accesses. This paper gives closed form expressions of auto-and cross-correlation functions of the proposed pulses, which can be used to evaluate the performance of the correlator receiver. Furthermore, the paper investigates, under various channel conditions, the spectrum characteristic and the bit error rate of the pulses＇ wave forms. The investigation conditions include additive white Gaussian noise channels, multipleaccess interference channels, and fading multipath channels. Our results indicate that our systematic algorithm is flexible for designing ultra-wideband pulses that conform to spectral emission constraints and offer good bit error rate performance.

ON THE POINTWISE ESTIMATIONS OF APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY HERMITE INTERPOLATION

The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.

Two-Variable Hermite Function as Quantum Entanglement of Harmonic Oscillator＇s Wave Functions

We reveal that the two-variable Hermite function hm,n, which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator＇s wave functions. The Schmidt decomposition of hm,n is derived. It also turns out that hm,n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S（γ）Hm,n （μa1^＋, μa2^＋│00〉, which is the minimum uncertainty state for sum squeezing, in （ η│representation is calculated.

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.

GENERALIZED EXPANSIONS IN HILBERT SPACE

In this paper the authors give a definite meaning to any formal trigonometrical series and generalize it to all abstract Hilbert space. Then in the case L-2(-infinity + infinity) they discussed extensively the generalized expansion problem by Hermite functions, and applied to a non-strictly nonlinear hyperbolic system.

SOLUTIONS OF A SYSTEM OF FORCED BURGERS EQUATION IN TERMS OF GENERALIZED LAGUERRE POLYNOMIALS

In this article, we obtain explicit solutions of a linear PDE subject to a class of ra-dial square integrable functions with a monotonically increasing weight function｜x｜n-1eβ｜x｜2/2,β ≥ 0, x ∈ Rn. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n＆gt;1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.

ITERATIVE ALGORITHM FOR AXIALLY ACCELERATING STRINGS WITH INTEGRAL CONSTITUTIVE LAW

A numerical method is proposed to simulate the transverse vibrations of a viscoelastic moving string constituted by an integral law. In the numerical computation, the Galerkin method based on the Hermite functions is applied to discretize the state variables, and the Runge- Kutta method is applied to solve the resulting differential-integral equation system. A linear iterative process is designed to compute the integral terms at each time step, which makes the numerical method more efficient and accurate. As examples, nonlinear parametric vibrations of an axially moving viscoelastic string are analyzed.

New useful special function in quantum optics theory

By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min（m,n）∑l=0^min（m,n）n!m!（-1）^l/l!（n-1）!（m-l）!/Hn-l（x）y^m-l≡ n,m（x,y）.By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators（IWOP） we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.

Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains

In this paper,we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains.We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian.As a result,the“stiffness”matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N^(2))arithmetic operations.Moreover,the singular factor in a typical kernel function can be fully absorbed by the basis.With the aid of Fourier analysis,we can prove the convergence of the scheme.We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions.We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.

A Class Hermite Pseudospectral Approximate with ω(x) ≡ 1 and Application to Reaction-diffusion Equation

In this paper, we continue the discussion of [12] to establish the Hermite pseudospectral method with weight ω（x） = 1. As an application, we consider the pseudospectral approximation of the reaction-diffusion equation on the whole line, we prove the existence of the approximate attractor and give the error estimate for the approximate solution.

The best quadrature formula based on Hermite information for the class KW^(2)[a,b]

The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known.The best interpolation formula used to get the quadrature formula above is also found.Moreover,we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.

Some Recent Advances on SpectralMethods for Unbounded Domains

We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi,Laguerre and Hermite functions.A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out,both theoretically and computationally.A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.

A Sylvester-Based IMEXMethod via Differentiation Matrices for Solving Nonlinear Parabolic Equations

In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains,taking Hermite functions,sinc functions,and rational Chebyshev polynomials as basis functions.The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme,being of particular interest the treatment of three-dimensional Sylvester equations that we make.The resulting method is easy to understand and express,and can be implemented in a transparent way by means of a few lines of code.We test numerically the three choices of basis functions,showing the convenience of this new approach,especially when rational Chebyshev polynomials are considered.

Analysis of a moving collocation method for one-dimensional partial differential equations

A moving collocation method has been shown to be very effcient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4. In this paper, the relations between the method and the traditional collocation and finite volume methods are investigated. It is shown that the moving collocation method inherits desirable properties of both methods： the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method. Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems. Numerical results are given to demonstrate the convergence order of the method.

Homogeneous wavelets and framelets with the refinable structure

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper, we provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.