Related papers: A discrete weighted Markov--Bernstein inequality f…

Let ${\mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${\mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let…

Let $w_{\lambda}(t)=(1-t^2)^{\lambda-1/2}$, $\lambda>-1/2$, be the Gegenbauer weight function, and $\Vert\cdot\Vert$ denote the associated $L_2$-norm, i.e., $$ \Vert f\Vert:=\Big(\int_{-1}^{1}w_{\lambda}(t)\vert f(t)\vert^2\,dt\Big)^{1/2}.…

The classical A. Markov inequality establishes a relation between the maximum modulus or the $L^{\infty}\left([-1,1]\right)$ norm of a polynomial $Q_{n}$ and of its derivative: $\|Q'_{n}\|\leqslant M_{n} n^{2}\|Q_{n}\|$, where the constant…

Let $w_{\lambda}(t) := (1-t^2)^{\lambda-1/2}$, where $\lambda > -\frac{1}{2}$, be the Gegenbauer weight function, let $\|\cdot\|_{w_{\lambda}}$ be the associated $L_2$-norm, $$ \|f\|_{w_{\lambda}} = \left\{\int_{-1}^1 |f(x)|^2…

We prove two-sided estimates for the best (i.e., the smallest possible) constant $\,c_n(\alpha)\,$ in the Markov inequality $$ \|p_n'\|_{w_\alpha} \le c_n(\alpha) \|p_n\|_{w_\alpha}\,, \qquad p_n \in {\cal P}_n\,. $$ Here, ${\cal P}_n$…

Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}. $$…

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the…

We obtain the following dimension independent Bernstein-Markov inequality in Gauss space: for each $1\leq p<\infty$ there exists a constant $C_p>0$ such that for any $k\geq 1$ and all polynomials $P$ on $\mathbb{R}^{k}$ we have $$ \| \nabla…

We investigate the best constant $J(n,d)$ such that Jackson's inequality \[ \inf_{\mathrm{deg}(g) \leq d} \|f - g\|_{\infty} \leq J(n,d) \, s(f), \] holds for all functions $f$ on the hypercube $\{0,1\}^n$, where $s(f)$ denotes the…

We consider the well-known method of least squares on an equidistant grid with $N+1$ nodes on the interval $[-1,1]$ with the goal to approximate a function $f\in\mathcal{C}\left[-1,1\right]$ by a polynomial of degree $n$. We investigate the…

Let $\|\cdot\|$ denote the minimum distance to an integer. For $0<\gamma< 1$, $\theta>0$ and $(\alpha, \beta) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}$ we study when \begin{equation*} \|\alpha p^{\gamma}+\beta \|<p^{-\theta},…

Let $a\in (0, \infty)$, $\gamma(a)$ be the Generalized Euler-Mascheroni Constant, and let \begin{align*} &x_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n}{a},\\…

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

Let $w_\alpha(t) := t^{\alpha}\,e^{-t}$, where $\alpha > -1$, be the Laguerre weight function, and let $\|\cdot\|_{w_\alpha}$ be the associated $L_2$-norm, $$ \|f\|_{w_\alpha} = \left\{\int_{0}^{\infty} |f(x)|^2…

A sequence of real numbers $\{x_{n}\}_{n\in \mathbb{N}}$ is said to be $\alpha \beta$-statistically convergent of order $\gamma$ (where $0<\gamma\leq 1$) to a real number $x$ \cite{a} if for every $\delta>0,$ $$\underset{n\rightarrow…

Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,...,f_n:{-1,1}^n\to X$ satisfy \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n…

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset $P$ of width $2$. This constant is defined as $\delta(P) = \max_{(x, y)\in P^2}\min\{\mathbb{P}(x\prec y),…

Recently Talagrand [T] estimated the deviation of a function on $\{0,1\}^n$ from its median in terms of the Lipschitz constant of a convex extension of $f$ to $\ell ^n_2$; namely, he proved that $$P(|f-M_f| > c) \le 4 e^{-t^2/4\sigma ^2}$$…

We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$ \sum_{k=1}^{n}\Big(\frac{1}{k}\sum_{j=1}^{k}a_j\Big)^2\leq d_n\,\sum_{k=1}^{n}a_k^2, \qquad (a_1,\ldots,a_n) \in \mathbb{R}^n $$ and $$…

Let $p \in (1,\infty)$, $\alpha\in \mathbb{R}$, and $\Omega\subsetneq \mathbb{R}^N$ be a $C^{1,\gamma}$-domain with a compact boundary $\partial \Omega$, where $\gamma\in (0,1]$. Denote by $\delta_{\Omega}(x)$ the distance of a point $x\in…