On the orthogonality of the Chebyshev–Frolov lattice and applications

We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a power of two, i.e. \(d=2^m, m \in \mathbb {N}\), the resulting lattice is an admissible lattice in the sense of Skriganov. We prove that these lattices are orthogonal and possess a lattice representation matrix with orthogonal columns and entries not larger than 2 (in modulus). In particular, we clarify the existence of orthogonal admissible lattices in higher dimensions. The orthogonality property allows for an efficient enumeration of these lattices in axis parallel boxes. Hence they serve for a practical implementation of the Frolov cubature formulas, which recently drew attention due to their optimal convergence rates in a broad range of Besov–Lizorkin–Triebel spaces. As an application, we efficiently enumerate the Frolov cubature nodes in the d-cube \([-1/2,1/2]^d\) up to dimension \(d=16\).     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

Convergence of the variants of the Chebyshev–Halley iteration family under the Hölder condition of the first derivative

The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order p$p$, a convergence criterion of order 1+p$1+p$ for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros

In this paper we investigate the local convergence of Chebyshev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an arbitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero.     

Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, an efficient technique based on the Chebyshev spectral collocation method for computing the eigenvalues of fourth-order Sturm–Liouville boundary value problems is proposed. The excellent performance of this scheme is illustrated through four examples. Numerical results and comparison with other methods are presented.     

On the restricted Chebyshev–Boubaker polynomials

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev–Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.     

On the Chebyshev Function ψ(x)

Suppose that the Riemann hypothesis is valid, and let be the Chebyshev function. In this paper, we obtain the following bound for all >1 and positive integers .     

The Iyengar Type Inequalities with Exact Estimations and the Chebyshev Central Algorithms of Integrals

In this paper, both low order and high order extensions of the Iyengar type inequality are obtained. Such extensions are the best possible in the same sense as that of the Iyengar inequality. hzrthermore, the Chebyshev central algorithms of integrals for some function classes and some related problems are also considered and investigated.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

The Convergance Properties of Quasi Hemite－Fejer Interpolation Polynomial on the Disturbance Chebyshev Knot

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The Limits of a Chebyshev Type Theory of Restricted Range Approximation

Define two extended real-valued functions l(x)and u(x)on X≡[0,1]subjectto the following restrictions: (i)-∞≤l(x)相似文献   

Numerical solutions of random integral equation III: random Chebyshev polynomials and Fredholm equations of the second kind ∗

We consider, in this article, a numerical study of a certain class of Singular Cauchy integral equations with random nonhomogeneous term. The method is based on an approximation of the solution by random Chebyshev polynomials. Numerical results, based on simulation, of random forcing term are given and they are used (i) to determine the distribution of the random coefficients of the Chebyshev polynomial and (ii) to compare the mean of the random solution with the solution of the mean equation (which of course is deterministic)     

Chebyshev centers, ε-Chebyshev centers and the Hausdorff metric

In this paper, we affirmatively answer an open question raised by P.Szeptycki and Vlech in (9) and give a new characterization of p-uniformly convex Banach space. The Lipschitz stability of the set of -Chebyshev centers G(A) under the perturbations of A and G is also proved.     

加权Chebyshev逼近

本文考虑了由R. Smarzewski研究的加权Chebyshev逼近问题。首先对Smarzewski。提出的关于K-最大类加权逼近中的一个有意义而又相当困难的问题给出了肯定的回答。其次,解决了A-最大类加权逼近中类似的问题。     

The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.     

Equivalence of $n$-point Gauss–Chebyshev rule and $4n$-point midpoint rule in computing the period of a Lotka–Volterra system

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of restricted on , , respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period , is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at , , , , and , respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the -point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the -point midpoint rule to the integral (4.7). Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.