Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on Rⁿ${\mathbb{R}}^n$ with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.     

On a boundary value problem for delay differential equations of population dynamics and Chebyshev approximation

In this paper, a boundary value problem for delay differential equations of population dynamics is considered. We obtain approximate solutions by using Chebyshev polynomial series and Newton–Raphson's procedure and give the error estimation. The method of the error estimation has been obtained in an existence theorem proved by a part of the authors. We carry out some numerical experiments by a computer language MATLAB.     

On the problem of uniqueness of the trigonometric moment constants

Summary Given a real-valued function <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mu(x,y)$ of bounded variation in the sense of Hardy and Krause on the square $[0, 2\pi]\times [0, 2\pi]$, the sequence <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \mu_{m,n}:=\int^{2\pi}_0 \int^{2\pi}_0 e^{i(mx+ny)} \, d_x \, d_y \mu(x,y), \quad (m,n)\in \bZ^2, $$ may be called the sequence of trigonometric moment constants with respect to $\mu(x,y)$. We discuss the uniqueness of the expression of the sequence $\{\mu_{m,n}\}$ in terms of the function $\mu(x,y)$.     

求线性边值问题数值解的Chebyshevτ—方法

提出了一种求解二维线性边值问题的新的τ＿方法·对该问题进行了理论分析和数值求解·结果表明了本文方法的优点和有效性     

三对角行列式与Chebyshev多项式

给出了三对角行列式的几种算法,利用三对角行列式证明了两类Chebyshev多项式的几种显式.     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

The Legendre Galerkin Chebyshev collocation least squares for the elliptic problem

The Legendre Galerkin Chebyshev collocation least squares method is presented for a second‐order elliptic problem with variable coefficients. By introducing a flux variable, the original problem is rewritten as an equivalent first‐order system. The present method is based on the Legendre Galerkin method, but Chebyshev–Gauss–Lobatto collocation is used to deal with the variable coefficient and the right hand side terms. The coercivity and continuity of the method are proved and an error estimate in the ‐norm is derived. Some numerical examples are given to validate the efficiency and accuracy of the scheme. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1689–1703, 2016     

Simultaneous Polynomial Approximation in the Weighted Chebyshev Norm

Simultaneous approximation means that a given sufficiently smooth function g:[-1, 1] and its derivatives up to order q are approximated by a single polynomial p and its derivatives. This paper deals with new error estimates (in a weighted norm with explicit constants) and corresponding algorithms in the most interesting cases q = 1 and q = 2. The described method is based on the close relationship between algebraic and trigonometric polynomial approximation.     

Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is $y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m \in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$ where $\xi$ is the first root of $y(r)$. We recast this as a nonlinear eigenproblem, with three boundary conditions and $\xi$ as the eigenvalue allowing imposition of the extra boundary condition, by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x) y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. We find that a Newton-Kantorovich iteration always converges from an $m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x), \xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to discretize $x$. The Lane-Emden equation has branch point singularities at the endpoint $x=1$ whenever $m$ is not an integer; we show that the Chebyshev coefficients are $a_{n} \sim constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a Chebyshev truncation of $N=100$ always gives at least ten decimal places of accuracy — much more accuracy when $m$ is an integer. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table.     

Complete Hyper-elliptic Integrals of the First Kind and the Chebyshev Property

This paper is devoted to study the following complete hyper-elliptic integral of the first kind $$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$ where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$.     

一些包含Chebyshev多项式和Stirling数的恒等式

利用初等方法研究Chebyshev多项式的性质,建立了广义第二类Chebyshev多项式的一个显明公式,并得到了一些包含第一类Chebyshev多项式,第一类Stirling数和Lucas数的恒等式.     

Justification and numerical realization of the uniform method for finding point estimates of interval elicited scaling constants

The form of the utility function over multi-dimensional consequences depends on the point estimates of the scaling constants. Fuzzy rational decision makers elicit those in the form of uncertainty intervals. The paper proposes an analytical justification and a numerical realization of the uniform method that finds point estimates of interval scaling constants. The main assumption of the technique is that constants are uniformly distributed in their uncertainty intervals. The density of the constants’ sum is constructed using preliminarily chosen knots. A new numerical procedure to calculate the I type error p _value of a two-tail test for singularity of the constants’ sum is proposed. All numerical procedures are embodied into program functions. The application of the method is demonstrated in examples. The connection between precision and time for analysis is investigated. Comparison of the analytical uniform method and an earlier proposed simulation realization is also conducted.     

The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement

A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduced to a set of algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use.     

Finite-codimensional Chebyshev subspaces in the complex spaceC(Q)

We consider finite-condimensional Chebyshev subspaces in the complex spaceC(Q), whereQ is a compact Hausdorff space, and prove analogs of some theorems established earlier for the real case by Garkavi and Brown (in particular, we characterize such subspaces). It is shown that if the real spaceC(Q) contains finite-codimensional Chebyshev subspaces, then the same is true of the complex spaceC(Q) (with the sameQ). Translated fromMatermaticheskie Zametki, Vol. 62, No. 2, pp. 178–191, August, 1997. Translated by V. E. Nazaikinskii     

Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid

Let _N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf : _N+2m → ℝ by algebraic polynomials on the grid Ω _N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω _N+m and Ω _N , respectively, we construct a linear operatorY _{n+2m, N} =Y _{n+2m, N} (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω _N ):

The Bernstein Constant and Polynomial Interpolation at the Chebyshev Nodes

In this paper, we establish new asymptotic relations for the errors of approximation in L_p[−1,1], 0<p∞, of x^λ, λ>0, by the Lagrange interpolation polynomials at the Chebyshev nodes of the first and second kind. As a corollary, we show that the Bernstein constant

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is finite for λ>0 and .     

Lipschitz continuity and Gateaux differentiability of the best approximation operator in vector-valued Chebyshev approximation

When G is a finite-dimensional Haar subspace of C(X,R^k), the vector-valued functions (including complex-valued functions when k is 2) from a finite set X to Euclidean k-dimensional space, it is well-known that at any function f in C(X,R^k) the best approximation operator satisfies the strong unicity condition of order 2 and a Lipschitz (Hőlder) condition of order . This note shows that in fact the best approximation operator satisfies the usual Lipschitz condition of order 1 and has a Gateaux derivative on a dense set of functions in C(X,R^k).     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

Two- and three-field formulations for wave transmission between media with opposite sign dielectric constants

We consider a simplified scalar model problem related to Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic constants. We build two variational formulations equivalent to the model problem. We show that, under some suitable conditions, both formulations are well-posed since they fit into the coercive plus compact framework. Advantages over previous studies is the validity of the formulations in the general case of Lipschitz interface between the two media and L^∞$L^{\infty }$ dielectric and magnetic constants. An interesting feature of these formulations is that they allow a simple finite element numerical implementation.     

第二类变型Bessel函数的Chebyshev逼近

第二类变型Bessel函数Kn(z)在自变量趋于无穷时就是指数变小的，使用多项式逼近的方法求解往往误差很大．采用指数变换和J．P．Boyd的有理Chebyshev多项式计算第二类变型Bessel函数，得到了令人满意的在较大范围内有效的解．