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 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 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.
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 consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets $\frak{e} \subset \mathbb{R}$ (that is, the monic polynomials minimizing the sup-norm, $\Vert T_n \Vert_{\frak{e}}$, on $\frak{e}$). We resolve a $45+$ year old conjecture of Widom that for finite gap subsets of $\mathbb{R}$, his conjectured asymptotics (which we call Szegő-Widom asymptotics) holds. We also prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leq Q C({\frak{e}})^n$ (where $C(\frak{e})$ is the logarithmic capacity of $\frak{e}$) for a class of $\frak{e}$'s with an infinite number of compone…
There is a vast theory of Chebyshev and residual polynomials and their asymptotic behavior. The former ones maximize the leading coefficient and the latter ones maximize the point evaluation with respect to an $L^\infty$ norm. We study Chebyshev and residual extremal problems for rational functions with real poles with respect to subsets of $\overline{\mathbb{R}}$. We prove root asymptotics under fairly general assumptions on the sequence of poles. Moreover, we prove Szegő--Widom asymptotics for sets which are regular for the Dirichlet problem and obey the Parreau--Widom and DCT conditions.
Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called $\textrm{Szeg\H o}$-Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalizat…
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set $A$. We find that for Riesz kernels $1/|x-y|^s$ with $s>p-2$ the optimum unconstrained configurations concentrate close to the set $A$ and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained p…
For the Chebyshev-Stirling numbers, a special case of the Jacobi-Stirling numbers, asymptotic formulae are derived in terms of a local central limit theorem. The underlying probabilistic approach also applies to the classical Stirling numbers of the second kind. Thereby a supplement of the asymptotic analysis for these numbers is established.
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.
This paper contains a general theory for asymptotic expansions of type (*) f(x)=a_1 phi_1(x)+...+a_n phi_n(x)+o(phi_n(x)), x tends to x_0, n>=3, where the asymptotic scale phi_1(x)>>phi_2(x)>>...>>phi_n(x), x tends to x_0, is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x_0. "Factorizational theory" refers to proofs being based on various types of factorizations of a differential operator associated to (phi_1,...,phi_n), hence we preliminarly collect various results concerning the concept of Chebyshev asymptotic scale, associated disconjugate operators and…
$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{ρ(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where $ρ(x)$ is a positive $m$ times continuously differentiable function on $[-1,1]$, $m\geq3$.
The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points $x=0,1,..., N-1$, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for $t_{n}(aN,N+1)$ in the double scaling limit, namely, $N\rightarrow\infty$ and $n/N\rightarrow b$, where $b\in(0,1)$ and $a\in(-\infty,\infty)$. One expansion involves the confluent hypergeometric function and holds uniformly for $a\in[0,1/2]$, and the other involves the Gamma function and holds uniforml…
We survey results on Chebyshev polynomials centered around the work of H. Widom. In particular, we discuss asymptotics of the polynomials and their norms and general upper and lower bounds for the norms. Several open problems are also presented.
We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation…
The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded $1$ on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik-Widom bounds for the extremal value, thereby providing a complete asym…
We raise a conjecture that asymptotics for Chebyshev polynomials in a complex domain can be given in terms of the reproducing kernels of a suitable Hilbert space of analytic functions in this domain. It is based on two classical results due to Garabedian and Widom. To support this conjecture we study asymptotics for Ahlfors extremal polynomials in the complement to a system of intervals on $\mathbb R$, arcs on $\mathbb T$, and its continuous counterpart.
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…
In 1969 Harold Widom published his seminal paper, which gave a complete description of orthogonal and Chebyshev polynomials on a system of smooth Jordan curves. When there were Jordan arcs present the theory of orthogonal polynomials turned out to be just the same, but for Chebyshev polynomials Widom's approach proved only an upper estimate, which he conjectured to be the correct asymptotic behavior. In this note we make some clarifications which will show that the situation is more complicated.
Ratio asymptotics for matrix orthogonal polynomials with recurrence coefficients $A_n$ and $B_n$ having limits $A$ and $B$ respectively (the matrix Nevai class) were obtained by Durán. In the present paper we obtain an alternative description of the limiting ratio. We generalize it to recurrence coefficients which are asymptotically periodic with higher periodicity, and/or which are slowly varying in function of a parameter. Under such assumptions, we also find the limiting zero distribution of the matrix orthogonal polynomials, generalizing results by Durán-López-Saff and Dette-Reuther to the…
In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introdu…
In this paper, we discuss asymptotic relations for the approximation of $\left\vert x\right\vert ^α,α>0$ in $L_{\infty}\left[ -1,1\right] $ by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula. As a consequence, we further deduce an asymptotic relation for the Approximation error when $α\rightarrow\infty$. Finally, we present connections of our results together with some recent work of Ganzburg [5] and Lubinsky [10], by presenting nume…
In this note we compute the logarithmic energy of points in the unit interval $[-1,1]$ chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expresion for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.
We study the zero distribution of non-orthogonal polynomials attached to $g(n)=s(n)=n^2$: \begin{equation*} Q_n^g(x)= x \sum_{k=1}^n g(k) \, Q_{n-k}^g(x), \quad Q_0^g(x):=1. \end{equation*} It is known that the case $g=id$ involves Chebyshev polynomials of the second kind. The zeros of $Q_n^s(x)$ are real, simple, and are located in $(-6\sqrt{3},0]$. Let $N_n(a,b)$ be the number of zeros between $-6 \sqrt{3} \leq a < b \leq 0$. Then we determine a density function $v(x)$, such that \begin{equation*} \lim_{n \rightarrow \infty} \frac{N_n(a,b)}{n} = \int_a^b v(x) \,\, \mathrm{d}x. \end{equation*…
