Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind. Postdoctoral researcher FWO-Flanders.     

关于勒让德多项式与契贝谢夫多项式间的关系

主要研究勒让德多项式与契贝谢夫多项式之间的关系的性质,利用生成函数和函数级数展开的方法,得出了勒让德多项式与契贝谢夫多项式之间的一个重要关系,这对勒让德多项式与契贝谢夫多项式的研究有一定的推动作用．     

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.     

Multivariate inequalities based on Sobolev representations

Here we derive very general multivariate tight integral inequalities of Chebyshev–Grüss, Ostrowski types and of comparison of integral means. These are based on well-known Sobolev integral representation of a function. Our inequalities engage ordinary and weak partial derivatives of the involved functions. We also give their applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions.     

Numerical solution of time-varying delay systems by Chebyshev wavelets

The solution of time-varying delay systems is obtained by using Chebyshev wavelets. The properties of the Chebyshev wavelets consisting of wavelets and Chebyshev polynomials are presented. The method is based upon expanding various time functions in the system as their truncated Chebyshev wavelets. The operational matrix of delay is introduced. The operational matrices of integration and delay are utilized to reduce the solution of time-varying delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Numerical solution of the abel integral equation

A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.     

第二类Chebyshev多项式系数的绝对值和的卷积

本文主要的目的是利用广义Fibonacci多项式的生成函数及其偏导数来研究第二类Cheby-shev多项式卷积的计算,并给出—个有趣的计算公式.     

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.     

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.     

Estimates of the trace of the inverse of a symmetric matrix using the modified Chebyshev algorithm

In this paper we study how to compute an estimate of the trace of the inverse of a symmetric matrix by using Gauss quadrature and the modified Chebyshev algorithm. As auxiliary polynomials we use the shifted Chebyshev polynomials. Since this can be too costly in computer storage for large matrices we also propose to compute the modified moments with a stochastic approach due to Hutchinson (Commun Stat Simul 18:1059–1076, 1989). In memory of Gene H. Golub.     

Generalized trigonometry and Chebyshev functions in finite fields

Trigonometry in finite fields was introduced by de Souza et al. and further developed by Lima and Panario and others, giving functions with many properties similar to trigonometric functions over the reals. Those explorations used a degree-2 extension of a base field. While this corresponds most closely to trigonometry over the reals, in finite fields we can have extensions of other degrees. In this paper we generalize the definitions of trigonometric functions and their related Chebyshev polynomials to arbitrary degrees and explore their properties. Many familiar results carry over into the generalized setting.     

Symmetrized Chebyshev polynomials

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves alternate) result that their coefficients are non-negative. As a corollary we find that and are positive definite functions. We further show that a Central Limit Theorem holds for the coefficients of our polynomials.

     

Regular two-point boundary value problems for the Schrödinger operator on a path

In this work we study the different type of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In any case, the Green function is given in terms of second kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Schödinger operator on a path.     

Dedekind和及第一类Chebyshev多项式

本文利用分析方法、Dedekind和及第一类Chebyshev多项式的算术性质,研究了一类关于Dedekind和及第一类Chebyshev多项式混合均值的渐近估计问题,并得到了一个较强的渐近公式.     

Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid

We construct an expansion of a discrete function in the form of a mixed series of Chebyshev polynomials. We obtain estimates of the approximation error of the function and its derivatives.     

Diagrammatic categorification of the Chebyshev polynomials of the second kind

We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.     

Horse paths, restricted 132-avoiding permutations, continued fractions, and Chebyshev polynomials

Several authors have examined connections among 132-avoiding permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we find analogues for some of these results for permutations π avoiding 132 and 1□23 (there is no occurrence π_i<π_j<π_j+1 such that 1?i?j-2) and provide a combinatorial interpretation for such permutations in terms of lattice paths. Using tools developed to prove these analogues, we give enumerations and generating functions for permutations which avoid both 132 and 1□23, and certain additional patterns. We also give generating functions for permutations avoiding 132 and 1□23 and containing certain additional patterns exactly once. In all cases we express these generating functions in terms of Chebyshev polynomials of the second kind.     

A new extension of Hermite matrix polynomials and its applications

In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced.     

Chebyshev spaces with polynomial blossoms

The use of extended Chebyshev spaces in geometric design is motivated by the interesting shape parameters they provide. Unfortunately the algorithms such spaces lead to are generally complicated because the blossoms themselves are complicated. In order to make up for this inconvenience, we here investigate particular extended Chebyshev spaces, containing the constants and power functions whose exponents are consecutive positive integers. We show that these spaces lead to simple algorithms due to the fact that the blossoms are polynomial functions. Furthermore, we also describe an elegant dimension elevation algorithm which makes it possible to return to polynomial spaces and therefore to use all the classical algorithms for polynomials. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hardy和Littlewood的一个三角和

本文对Hardy和Littlewood考虑的一个有限三角和做了进一步地研究.通过充分运用Chebyshev多项式和M?bius函数的性质,建立了该有限三角和的一个有趣的恒等式,并得到了一个精确的渐近公式.