Related papers: Asymptotics of Chebyshev Polynomials, IV. Comments…

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…

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…

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 prove Szeg\H{o}-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 derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound…

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…

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…

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set $K$ in the complex plane are established. These estimates are exact (up to a constant factor) in the case where $K$ consists of a finite number of…

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of…

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…

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all…

The estimates of the uniform norm of the Chebyshev polynomial associated with a compact set $K$ consisting of a finite number of continua in the complex plane are established. These estimates are exact (up to a constant factor) in the case…

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…

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence…

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…

We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's…

We employ the generalized Remez algorithm, initially suggested by P. T. P. Tang, to perform an experimental study of Chebyshev polynomials in the complex plane. Our focus lies particularly on the examination of their norms and zeros. What…

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets.…

In this note, we provide a wide range of upper bounds for the moduli of the zeros of a complex polynomial. The obtained bounds complete a series of previous papers on the location of zeros of polynomials.

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given…