A conjecture of Gy. Petruska

Summary We give a negative answer to a conjecture of Gy. Petruska.     

The norm of minimal polynomials on several intervals

Using works of Franz Peherstorfer, we examine how close the $n$th Chebyshev number for a set $E$ of finitely many intervals can get to the theoretical lower limit $2\mathrm{cap}{\left(E\right)}^n$.     

Weighted polynomial inequalities

For the weights exp (?|x|^λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP _n can decrease on [?1,1] ifP _n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL ^p spaces.     

Uniform approximation by positive operators on infinite intervals

В работе устанавливае тся оценка (*) |L _n (???| ≦Kω _? (?;α _n) для положительных оп ераторов, определенн ых на конечном или бесконе чном интервале (a,b), гдеL _n(1,χ)≡1,L _n((t?χ)²;χ)≦K? ²(χ)α _n ² (x∈(a,b)) ;и \(\omega _\varphi (f;\delta ) = \mathop {\sup }\limits_{0 \leqq h \leqq \delta ,x \pm h\varphi (x) \in (a,b)} \left| {f(x - h\varphi (x)) - 2f(x) + f(x + h\varphi (x))} \right|\) модуль гладкости?, св язанный с ? (функция? удовлетворяет некот орым условиям регуля рности). С помощью (*) для некотор ых {L _n } получена характеристика тех ф ункций?, для которыхL _n (?)??=o(1) равном ерно на (a, b). Наконец, рассматриваются слу чай насыщения и случай так называем ой неоптимальной апп роксимации. Результаты применяю тся к операторам Саса —Миракяна, Баскакова, Мейер-Кëни га и Целлера, гамма и бета операторам, а также к н екоторым операторам типа свер тки.     

Sharpening of Hilbert’s lemniscate theorem

In this paper, we extend Hilbert’s lemniscate theorem to touching systems of curves. The result allows finding sharp constants in Bernstein type inequalities. Supported by OTKA TS44782. Supported by NSF grant DMS-0097484 and by OTKA T/034323, TS44782.     

Summability in the classes Hω

Mathematical Notes -     

Extensions of Szegö's theory of orthogonal polynomials,II

Let {? _n (dμ)} be a system of orthonormal polynomials on the unit circle with respect to a measuredμ. Szegö's theory is concerned with the asymptotic behavior of? _n (dμ) when logμ'∈L ¹. In what follows we will discuss the asymptotic behavior of the ratio φ_n(dμ ₁)/φ_n(dμ ₂) off the unit circle in casedμ ₁ anddμ ₂ are close in a sense (e.g.,dμ ₂=g dμ ₁ whereg≥0 is such thatQ(e ^it )g(t) andQ(e ^it )/g(t) are bounded for a suitable polynomialQ) and μ ₁ ^′ >0 almost everywhere or (a somewhat weaker requirement) lim _n→∞Φ _n (dμ ₁,0)=0, for the monic polynomials Φ _n . The consequences for orthogonal polynomials on the real line are also discussed.