We consider the problem of maximizing the sum of squares of the leading coefficients of polynomials $P_{i_1}(x),\ldots ,P_{i_m}(x)$ (where $P_j(x)$ is a polynomial of degree $j$) under the restriction that the sup-norm of $\sum_{j=1}^m P_{i_j}^2(x)$ is bounded on the interval $[-b,b]$ ($b>0$). A complete solution of the problem is presented using duality theory of convex analysis and the theory of canonical moments. It turns out, that contrary to many other extremal problems the structure of the solution will depend heavily on the size of the interval $[-b,b]$.
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…
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…
Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the obtained minimax, equioscillation, and characterization results extend those of Bojanov, Fenton, Hardin, Kendall, Saff and Ambrus, Ball, Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev.
Let $Γ$ be a $C^{2+}$ Jordan arc and let $Γ_0$ be the open arc which consists of interior points of $Γ$. We find concrete upper and lower bounds for the limit of Widom factors for $L_2(μ)$ extremal polynomials on $Γ$ which was given in [18]. In addition, we show that the upper bound for the limit supremum of Widom factors for the weighted Chebyshev polynomials which was obtained in [18] can be improved once two normal derivatives of the Green function do not agree at one point $z\in Γ_0$. We also show that if $Γ_0$ is not analytic then we have improved upper bounds.
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.
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 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…
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…
Chebyshev systems and Sturm oscillation theory for discrete polynomials
Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）；Tula State University（图拉国立大学）；ICREA（加泰罗尼亚高级研究院）；Centre de Recerca Matemàtica（数学研究中心）；Universitat Autónoma de Barcelona（巴塞罗那自治大学）
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.
First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients $a_0,...,a_l,$ $b_0,...,b_l \in \mathbb R$ find that one with least maximum norm on $[0,2 π].$ We show that the minimal polynomial is on $[0,2 π]$ asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the $τ_j = a_j + i b_j.$ As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linea…
A comprehensive study of periodic trajectories of billiards within ellipsoids in $d$-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line. By leveraging deep, but yet not widely known results of the Krein-Levin-Nudelman theory of generalized Chebyshev polynomials, fundamental properties of billiard dynamics are proven for any $d$, viz., that the sequences of winding numbers are monotonic. By employing the potential theory we pr…
We summarize researches - in great deal jointly with my host Y. Sarantopoulos and his PhD. students V. Anagnostopoulos and A. Pappas - started by a Marie Curie fellowship in 2001 and is still continuing. The project was to study multivariate polynomial inequalities. In the course of work we realized the role of the ``generalized Minkowski functional'', to which we devoted a throughout survey. Resulting from this, infinite dimensional extensions of Chebyshev's extremal problems were tackled successfully. Investigating Bernstein-Markov constants for homogeneous polynomials of real normed spaces …
