Igor Rivin
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.
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Some new sufficient conditions for the weighted Chebyshev's inequality for real numbers to hold are provided.
We prove the Levin-Stečkin inequality using Chebyshev's inequality and symmetrization. Symmetry and slightly modified Chebyshev's inequality are also the key to an elementary proof of Clausing's inequality .
The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.
It is shown that some q-analogues of the Fibonacci and Lucas polynomials lead to q-analogues of the Chebyshev polynomials which retain most of their elementary properties.
We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind.
We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq \sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers $x_1,\ldots, x_n$.
We show that the only orthogonal polynomials with a generating function of the form $F(x z - αz^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(x w(z) - R(z))$.
We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the paper with some results concerning integrals of the generalized Chebyshev-II and Bernstein polynomials.
The classical form of Grüss' inequality, first published by G. Grüss in 1935, gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. In the subsequent years, many variants of this inequality appeared in the literature. The aim of this paper is to introduce a different approach, presenting a new Chebyshev-Grüss-type inequality and applying it to different well-known linear, not necessarily positive, operators. Some conjectures are presented as well. We also compare the new inequalities with some older results. This new approach…
The purpose of this note is to extend in a simple and unified way some results on orthogonal polynomials with respect to the weight function $$\frac{|T_m(x)|^p}{\sqrt{1-x^2}}\;,\quad-1<x<1\;,$$ where $T_m$ is the Chebyshev polynomial of the first kind of degree $m$ and $p>-1$.
We make a number of comments on Chebyshev polynomials for general compact subsets of the complex plane. We focus on two aspects: asymptotics of the zeros and explicit Totik--Widom upper bounds on their norms.
We construct a simple closed-form representation of degree-ordered system of bivariate Chebyshev-I orthogonal polynomials $\mathscr{T}_{n,r}(u,v,w)$ on simplicial domains. We show that these polynomials $\mathscr{T}_{n,r}(u,v,w),$ $r=0,1,\dots,n;$ $n\geq 0$ form an orthogonal system with respect to the Chebyshev-I weight function.
We prove Szegő-Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau-Widom and DCT conditions.
Properties of two Chebyshev-like polynomial sequences
研究通过修改第二类切比雪夫多项式生成函数得到两个多项式序列，获得其判别式和结式恒等式及部分不可约性结果，还指出与两类切比雪夫多项式的关联。
The generalized complex numbers can be realized in terms of $2\times2$ or higher-order matrices and can be exploited to get different ways of looking at the trigonometric functions. Since Chebyshev polynomials are linked to the power of matrices and to trigonometric functions, we take the quite natural step to discuss them in the context of the theory of generalized complex numbers. We also briefly discuss the two-variable Chebyshev polynomials and their link with the third-order Hermite 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. We further show that a Central Limit Theorem holds for our polynomials.
Orthogonal polynomials have very useful properties in the solution of mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize the generalized Chebyshev-type polynomials of the first kind $\mathscr{T}_{n}^{(M,N)}(x),$ then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials $B_{k}^{n}(x).$ We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.
The classical form of Grüss' inequality was first published by G. Grüss in 1935 and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. After that many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. First and second moments of the operators are used to explain the situation.
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
We prove mean convergence of the Fourier series in Akhiezer-Chebyshev polynomials in $L^p$, $p>1$, using a weighted inequality for the Hilbert transform in an arc of the unit circle.
Bounds for weighted Chebyshev and residual polynomials on subsets of $\mathbb{R}$
本文研究实轴子集上加权 Chebyshev 多项式与残差多项式的上下界，并将其用于在 Parreau–Widom 集框架中建立 Szegő 型定理。
We determine which sets saturate the Szeg}o and Schiefermayr lower bounds on the norms of Chebyshev Polynomials. We also discuss sets that saturate the Totik--Widom upper bound.
We give a few examples of Chebyshev polynomials that appeared in mathematical problems from the 16th and 17th century. The main example is the famous equation of Adrianus Romanus (Adriaan van Roomen) containing a polynomial of degree $45$.
The article is concerned with polynomials $g(x)$ whose graphs are "partially packed" between two horizontal tangent lines. We assume that most of the local maximum points of $g(x)$ are on the first horizontal line, and most of the local minimum points on the second horizontal line, except several "exceptional" maximum or minimum points, that locate above or under two lines, respectively. In addition, the degree of $g(x)$ is exactly the number of all extremum points $+1$. Then we call $g(x)$ a multipartite Chebyshev polynomial associated with the two lines. Under a certain condition, we show th…
Chebyshev polynomials on a Jordan arc
该研究描述平面解析若尔当弧上切比雪夫多项式的渐近行为，结合多种多项式与不等式证明，解决相关猜想并得到其Szegő-Widom渐近结果。
We study residual polynomials, $R_{x_0,n}^{(\mathfrak{e})}$, $\mathfrak{e}\subset\mathbb{R}$, $x_0\in\mathbb{R}\setminus\mathfrak{e}$, which are the degree at most $n$ polynomials with $R(x_0)=1$ that minimize the $\sup$ norm on $\mathfrak{e}$. New are upper bounds on their norms (that are optimal in some cases) and Szegő--Widom asymptotics under fairly general circumstances. We also discuss several illuminating examples and some results in the complex case.
We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, $T_n(x) \pm 1$ and $U_n(x) \pm 1$. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$ using minimal polynomials of $\cos(\frac{2π}{d})$.
We show how polynomial mappings of degree k from a union of disjoint intervals onto [-1,1] generate a countable number of special cases of a certain generalization of the Chebyshev Polynomials. We also derive a new expression for these generalized Chebyshev Polynomials for any number of disjoint intervals from which the coefficients of x^n can be found explicitly in terms of the end points and the recurrence coefficients. We find that this representation is useful for specializing to the polynomial mapping cases for small k where we will have algebraic expressions for the recurrence coefficien…
Chebyshev polynomials related to Jacobi weights
本文研究Jacobi权对应的Chebyshev多项式，确定相关Widom因子的单调性，并揭示单位圆与区间[-1,1]上加权Chebyshev多项式的关系，补充了Bernstein等人的经典工作。
Chebyshev Expansion of the Zernike Polynomials
本文提出了一种将泽尼克多项式展开为切比雪夫多项式的方法，并给出了相关系数的闭式表达和递推关系。
We sum multivariate generating functions composed of products of Chebyshev polynomials of the first and the second kind. That is, we find closed forms of expressions of the type $\sum_{j\geq0}ρ^{j}\prod_{m=1}^{k}T_{j+t_{m}}% (x_{m})\prod_{m=k+1}^{n+k}U_{j+t_{m}}(x_{m}),$ for different integers $t_{m},$ $m=1,...,n+k.$ We also find a Kibble-Slepian formula of $n$ variables with Hermite polynomials replaced by Chebyshev polynomials of the first or the second kind. In all the considered cases, the obtained closed forms are rational functions with positive denominators. We show how to apply the obt…
The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kampè de Fèriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol type numbers and polynomials. Using Euler's formula, relations among special functions, Hermite-type polynomials, the Chebyshev polynomials and the Dickson polynomials are given. Using generating functions and their functional equations, various formulas and identities are given. With help of computational formula for new families of special polynomials, some o…
Chebyshev polynomials corresponding to a vanishing weight
本文研究单位圆上对应消失权重$(z-1)^s$的加权切比雪夫多项式，将Lachance等结论推广至非整数$s$，并联系双纽线等类别，关键推广了Erdős–Lax不等式。
Askey and Gasper (1976) proved a trigonometric inequality which improves another trigonometric inequality found by M. S. Robertson (1945). Here these inequalities are reformulated in terms of the Chebyshev polynomial of the first kind $T_n$ and then put into a one-parametric family of inequalities. The extreme value of the parameter is found for which these inequalities hold true. As a step towards the proof of this result we establish the following complement to the finite increment theorem specialized to $T_n^{\prime}$: $$ T_n^{\prime}(1)-T_n^{\prime}(x)\geq (1-x)\,T_n^{\prime\prime}(x)\,,\q…
In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn (z, N) as the degree grows to infinity. Global asymptotic formulas are obtained as n \rightarrow \infty, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.
Change of basis for generalized Chebyshev Koornwinder's type polynomials of first kind
本文提供了第一类广义切比雪夫Koornwinder型多项式以固定次数n的伯恩斯坦基表示的显式形式，并研究了其与伯恩斯坦多项式之间的基变换。
For $0<α<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^α(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of $P_{m}(z)$ outside the interval $(-1,1)$ is bounded by a constant independent of $m$.
Exploring the zeros of real self-reciprocal polynomials by Chebyshev polynomials
本文研究了与一阶切比雪夫准正交多项式相关、最多有两个零点在单位圆外的实自互多项式，探讨其零点分布、简单性和单调性。
Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where $\|p\|_\infty := \sup_{|x| \le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.
Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deter…
Consistent rational approximations of power series, trigonometric series and series of Chebyshev polynomials
本文研究三角级数与切比雪夫多项式级数的一致有理逼近，建立三角 Hermite-Padé 多项式的存在唯一性准则并给出显式形式，同时构造例子说明 Hermite-Jacobi 逼近与 Hermite-Padé 逼近可以不同。
Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x), \end{equation*} with initial value $P_{0}^{g,h}(x)=1$ encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov--Okounkov polynomials. In this paper we prove that for $g(n)=n$ and fixed $h$ we obtain orthogonal polynomial seq…
In this paper, we give a sharp lower bound for the minimum deviation of the Chebyshev polynomial on a compact subset of the real line in terms of the corresponding logarithmic capacity. Especially if the set is the union of several real intervals, together with a lower bound for the logarithmic capacity derived recently by A.Yu.\,Solynin, one has a lower bound for the minimum deviation in terms of elementary functions of the endpoints of the intervals. In addition, analogous results for compact subsets of the unit circle are given.
We consider semi-infinite Jacobi matrices corresponding to a point interaction for the discrete Schrödinger operator. Our goal is to find explicit expressions for the spectral measure, the resolvent and other spectral characteristics of such Jacobi matrices. It turns out that their spectral analysis leads to a new class of orthogonal polynomials generalizing the classical Chebyshev polynomials.
We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of po…
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