### A Markov Type Inequality for Higher Derivates of Polynomials.

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Let ${r}_{n}*\in {\mathcal{R}}_{nn}$ be the best rational approximant to $f\left(x\right)={x}^{\alpha}$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of ${r}_{n}*$ lie on the negative axis ${\mathbb{R}}_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function ${e}_{n}=f-{r}_{n}*$ on [0,1], and survey related convergence results.

In this paper we calculate the constants of strong uniqueness of minimal norm-one projections on subspaces of codimension k in the space ${l}_{\infty}^{\left(n\right)}$. This generalizes a main result of W. Odyniec and M. P. Prophet [J. Approx. Theory 145 (2007), 111-121]. We applied in our proof Kolmogorov’s type theorem (see A. Wójcik [Approximation and Function Spaces (Gdańsk, 1979), PWN, Warszawa / North-Holland, Amsterdam, 1981, 854-866]) for strongly unique best approximation.

We give an overview of the behavior of the classical Hilbert Transform H seen as an operator on Lp(R) and on weak-Lp(R), then we consider other operators related to H. In particular, we discuss various versions of Discrete Hilbert Transform and Fourier multipliers periodized in frequency, giving some partial results and stating some conjectures about their sharp bounds Lp(R) → Lp(R), for 1 < p < ∞.

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set $E\subset {\u2102}^{N}$. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function ${\Phi}_{E}$. Moreover, we show that one of these extremal-like functions is equal to ${\Phi}_{E}$ if E is a nonpluripolar set with $li{m}_{n\to \infty}M\u2099{\left(E\right)}^{1/n}=1$ where $M\u2099\left(E\right):=sup{\left|\right|\left|gradP\right|\left|\right|}_{E}/{\left|\right|P\left|\right|}_{E}$, the supremum is taken over all polynomials P of N variables of total...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots ${t}_{j}$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let ${V}_{n,\nu}$ be the space of piecewise linear continuous functions on the torus with knots ${t}_{j}:0\le j\le N-1$. Finally, let ${P}_{n,\nu}$ be the orthogonal projection operator from L²([0,1)) onto ${V}_{n,\nu}$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|{P}_{n,\nu}:{L}^{\infty}\to {L}^{\infty}\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|{P}_{n,\nu}:{L}^{\infty}\to {L}^{\infty}\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical Franklin...