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Béla Nagy

Publications and source records attributed to Béla Nagy.

13 recordsLinked to original sources
arXiv2023

On the weighted trigonometric Bojanov-Chebyshev extremal problem

We investigate the weighted Bojanov-Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing $\|T\|_{w,C({\mathbb T})}$, where $w$ is a sufficiently nonvanishing, upper bounded, nonnegative weight function, the norm is the corresponding weighted maximum norm on the torus ${\mathbb T}$, and $T$ is a trigonometric polynomial with prescribed multiplicities $ν_1,\ldots,ν_n$ of root factors $|\sin(π(t-z_j))|^{ν_j}$. If the $ν_j$ are natural numbers and their sum is even, then $T$ is indeed a trigonometric polynomial and the case when all the $ν_j$ are 1 covers the Chebyshev extremal problem. Our result will be more general, allowing, in particular, so-called generalized trigonometric polynomials. To reach our goal, we invoke Fenton's sum of translates method. However, altering from the earlier described cases without weight or on the interval, here we find different situations, and can state less about the solutions.

math.CA↗
arXiv2023

Minimal degree rational open up mappings and related questions

We establish the existence and uniqueness of rational conformal maps of minimal degree $n+1$ for opening up $n$ arcs. In earlier results, the degree was exponential in $n$. We also discuss two related problems. (a) We establish existence of rational functions of minimal degree with prescribed critical values, and show that the number of (suitably normalized) rational functions is given in terms of the Hurwitz numbers. (b) We consider the problem of finding rational functions of minimal degree with prescribed critical points, where we establish existence of solutions by considering certain polynomial equations, and where the number of normalized solutions is bounded from above by a Catalan number. We illustrate our results with two examples.

math.CV↗
arXiv2023

Intertwining of maxima of sum of translates functions with nonsingular kernels

In previous papers we investigated so-called sum of translates functions $F({\mathbf{x}},t):=J(t)+\sum_{j=1}^n ν_j K(t-x_j)$, where $J:[0,1]\to \underline{\mathbb{R}}:={\mathbb{R}}\cup\{-\infty\}$ is a "sufficiently nondegenerate" and upper-bounded "field function", and $K:[-1,1]\to \underline{\mathbb{R}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, and also satisfying the singularity condition $K(0)=\lim_{t\to 0} K(t)=-\infty$. For node systems ${\mathbf{x}}:=(x_1,\ldots,x_n)$ with $x_0:=0\le x_1\le\dots\le x_n\le 1=:x_{n+1}$, we analyzed the behavior of the local maxima vector ${\mathbf{m}}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j({\mathbf{x}}):=\sup_{x_j\le t\le x_{j+1}} F({\mathbf{x}},t)$. Among other results we proved a strong intertwining property: if the kernels are also decreasing on $(-1,0)$ and increasing on $(0,1)$, and the field function is upper semicontinuous, then for any two different node systems there are $i,j$ such that $m_i({\mathbf{x}}) m_j({\mathbf{y}})$. Here we partially succeed to extend this even to nonsingular kernels.

math.CA↗
arXiv2023

On the weighted Bojanov-Chebyshev Problem and the sum of translates method of Fenton

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.

math.CA↗
arXiv2023

A homeomorphism theorem for sums of translates

For a fixed positive integer $n$ consider continuous functions $ K_1,\dots$, $ K_n:[-1,1]\to \mathbb{R}\cup\{-\infty\}$ that are concave and real valued on $[-1,0)$ and on $(0,1]$, and satisfy $K_j(0)=-\infty$. Moreover, let $J:[0,1]\to \mathbb{R}\cup\{-\infty\}$ be upper bounded and such that $[0,1]\setminus J^{-1}(\{-\infty\})$ has at least $n+1$ elements, but it is arbitrary otherwise. For $x_0:=0<x_1<\dots< x_n \le x_{n+1}:=1$, so called nodes, and for $t\in [0,1]$ consider the sum of translates function $F(x_1,\ldots,x_n,t):=J(t)+\sum_{j=1}^n K_j(t-x_j)$, and the vector of interval maximum values $m_j:=m_j(x_1,\ldots,x_n):=\max_{t\in [x_j,x_{j+1}]}F(x_1,\ldots,x_n,t)$ ($j=0,1,\ldots,n$). We describe the structure of the arising interval maxima as the nodes run over the $n$-dimensional simplex. Applications presented here range from abstract moving node Hermite-Fejér interpolation for generalized algebraic and trigonometric polynomials via Bojanov's problem to more abstract results of interpolation theoretic flavour.

math.CA↗
arXiv2021

Bernstein- and Markov-type inequalities

This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets.

math.CV↗
arXiv2020

Minimal energy point systems on the unit circle and the real line

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.

math.CA↗
arXiv2017

A potential theoretic minimax problem on the torus

We investigate an extension of an equilibrium-type result, conjectured by Ambrus, Ball and Erdélyi, and proved recently by Hardin, Kendall and Saff. These results were formulated on the torus, hence we also work on the torus, but one of the main motivations for our extension comes from an analogous setup on the unit interval, investigated earlier by Fenton. Basically, the problem is a minimax one, i.e. to minimize the maximum of a function $F$, defined as the sum of arbitrary translates of certain fixed "kernel functions", minimization understood with respect to the translates. If these kernels are assumed to be concave, having certain singularities or cusps at zero, then translates by $y_j$ will have singularities at $y_j$ (while in between these nodes the sum function still behaves realtively regularly). So one can consider the maxima $m_i$ on each subintervals between the nodes $y_j$, and look for the minimization of $\max F = \max_i m_i$. Here also a dual question of maximization of $\min_i m_i$ arises. This type of minimax problems were treated under some additional assumptions on the kernels. Also the problem is normalized so that $y_0=0$. In particular, Hardin, Kendall and Saff assumed that we have one single kernel $K$ on the torus or circle, and $F=\sum_{j=0}^n K(\cdot-y_j)= K + \sum_{j=1}^n K(\cdot-y_j)$. Fenton considered situations on the interval with two fixed kernels $J$ and $K$, also satisfying additional assumptions, and $F= J + \sum_{j=1}^n K(\cdot-y_j)$. Here we consider the situation (on the circle) when \emph{all the kernel functions can be different}, and $F=\sum_{j=0}^n K_j(\cdot- y_j) = K_0 + \sum_{j=1}^n K_j(\cdot-y_j)$. Also an emphasis is put on relaxing all other technical assumptions and give alternative, rather minimal variants of the set of conditions on the kernel.

math.CA↗
arXiv2016

Bernstein- and Markov-type inequalities for rational functions

Asymptotically sharp Bernstein- and Markov-type inequalities are established for rational functions on $C^2$ smooth Jordan curves and arcs. The results are formulated in terms of the normal derivatives of certain Green's functions with poles at the poles of the rational functions in question. As a special case (when all the poles are at infinity) the corresponding results for polynomials are recaptured.

math.CV↗
arXiv2016

Bernstein type inequalities for rational functions on analytic curves and arcs

Borwein and Erdélyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erdélyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.

math.CA↗
arXiv2015

Polynomial and rational inequalities on Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein-Erdélyi inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.

math.CV↗
arXiv2014

Some notes on $L^{p}$ Bernstein inequality when $0<p<1$

Recently, Nagy-Toókos and Totik-Varga proved an asymptotically sharp $L^{p}$ Bernstein type inequality on union of finitely many intervals. We extend this inequality to the case when the power $p$ is between $0$ and $1$; such sharp Bernstein type inequality was proved first by Arestov.

math.CA↗